13D Trigonometry - Weebly...Figure 3 shows a tetrahedron ABCE. The horizontal base is the isosceles...

1 3D Trigonometry

1988-1 Paper(1) Q.11

1

V

A

B

C

3 cm

3 cm

4 cm

Fig. 1

In Fig. 1, the edges V A, V B and V C of the tetrahedron V ABC are mutually perpendicular.

Given that V A = V C = 3 cm and V B = 4 cm, calculate

(a) the volume, in cm3, of the tetrahedron, (2)

(b) the length, in cm to 2 decimal places, of AC, (1)

(c) the angle, in degrees to the nearest tenth of a degree, between the planes BAC and V AC,

(5)

(d) the area, in cm2 to 2 decimal places, of triangle ABC, (4)

(e) the perpendicular distance, in cm to one decimal place, from V to the plane ABC. (3)

1 Compiled on 24/4/2018 by Steve Cheung

1988-6 Paper(1) Q.11

2 Given that x = sin θ◦ − 2 cos θ◦

and y =√

3 sin θ◦, where 0 6 θ < 180,

(a) show that x2 + y2 = 2(sin θ◦ − cos θ◦)2 + 2.

Deduce the minimum value of (x2 + y2) and the value of θ for which it occurs. (4)

The tetrahedron ABCD is such that AB, AC and AD are mutually perpendicular, AB = AC = 3

cm and AD = 4 cm. Calculate, to the nearest 0.1◦,

(b) the angle between DB and the plane ABC, (3)

(c) the angle between the planes ABC and DBC. (8)


1989-1 Paper(1) Q.12

3

A

S

Q

P

R

North

150 m

40◦

Fig. 3

In Fig. 3, the points A, Q, and S are in the same horizontal plane. Q is 150 m due north of A

with ∠QAS = 40◦ and the bearing of S from Q is 120◦.

(a) Calculate, to the nearest metre, the lengths of AS and QS. (7)

PQ and RS are vertical towers. Given that the angle of elevation of P from A is 25◦,

(b) calculate, to the nearest metre, the length of PQ. (3)

Given that the angle of elevation of R from A is 18◦,

(c) calculate, to the nearest metre, the length of RS. (2)

(d) Calculate, to the nearest degree, the angle of elevation of P from R. (3)


1989-6 Paper(1) Q.13

4

A B

C

E

F

D

Fig. 3

Figure 3 shows a tetrahedron ABCE. The horizontal base is the isosceles 4ABC with AB = AC

and BC = 14 cm. The point F lies on BC such that AF is perpendicular to BC. The vertex E

is vertically above the point D, which lies on AF such that AD = 10 cm and DF = 14.4 cm.

The edges BE and EC are each 25 cm.

(a) Calculate the length of EF . (2)

(b) Show that the height DE of the tetrahedron is 19.2 cm. (1)

(c) By using appropriate right-angled triangles, calculate, to 1 decimal place, the lengths of AB

and AE. (3)

(d) Calculate, to the nearest half degree, the size of ∠AEB. (3)

(e) Calculate, to the nearest degree, the angle that BE makes with the base ABC. (3)

(f) Calculate, to the nearest degree, the angle made between the faces ABC and EBC. (3)


1990-1 Paper(1) Q.9

5 A, B and C are 3 points on horizontal ground such that AB = 20 m, BC = 30 m and ∠ABC =

110◦.

(a) Find, in m to 3 significant figures, the distance AC. (4)

(b) Find, to the nearest degree, the size of ∠BAC. (4)

A vertical pole BP of height 10 m is erected at B. The point M is the foot of the perpendicular

from B to AC and a taut wire joins P to M . Find

(c) the length, in m to 3 significant figures, of BM , (3)

(d) the angle, in degrees to 1 decimal place, between PM and the ground. (4)

1990-6 Paper(1) Q.12

6 Solve, for 0 6 x < 360, giving your answers to 1 decimal place, where appropriate, the equations

(a) tan 2x◦ = −1, (3)

(b) 2 cos2 x◦ = sinx◦ cosx◦, (4)

(c) 3 sinx◦ + 3 = cos2 x◦. (3)

A cuboid has a square horizontal base of side 2 cm and a height of 3 cm. Calculate, in degrees to

3 significant figures, the angle that a diagonal of the cuboid makes with

(d) the base (2)

(e) a vertical face of the cuboid. (3)


1991-1 Paper(1) Q.10

7

A C

B

V12 cm

12 cm

5 cm

110◦

Fig. 1

Figure 1 shows the tetrahedron V ABC with face ABC on a horizontal plane and edge V A vertical.

AB = AC = 12 cm, V A = 5 cm and ∠BAC = 110◦. Calculate

(a) the length, in cm to 2 decimal places, of the perpendicular from A to BC, (2)

(b) the length, in cm to 2 decimal places, of BC, (3)

(c) the angle, in degrees to the nearest tenth of a degree, between the plane V BC and the plane

ABC, (3)

(d) the distance, in cm to 2 decimal places, from A to the plane V BC, (3)

(e) the area, in cm2 to 1 decimal place, of 4V BC. (3)


1991-6 Paper(1) Q.13

8

B

X

Y

T

125◦

Fig. 1

Figure 1 shows two cables joining T , the top of a vertical radio transmitter mast, to points X and

Y , which are in the same horizontal plane as B, the foot of the mast. The angle of elevation of T

from X is 35◦. The angle of elevation of T from Y is 25◦. The distance of X from B is 50 m and

∠XBY is 125◦.

Calculate, in m to one decimal place,

(a) the height of the mast, (2)

(b) the distance of Y from the foot of the mast B, (2)

(c) the distance between points X and Y . (3)

F is the foot of the perpendicular from B to XY .

(d) Calculate, in m to one decimal place, the length FB. (2)

(e) Calculate, in degrees to the nearest 0.1◦, the angle between planes BXY and TXY . (3)

(f) Calculate, in m to one decimal place, the perpendicular distance of B from the plane TXY .

(3)


1991-6 Paper(2) Q.5

9

B

A

C

D

X

5 cm

5 cm

4 cm

Fig. 1

Figure 1 shows the tetrahedron ABCD in which the edges BA, BC and BD are mutually per-

pendicular, AD = CD = 5 cm and BD = 4 cm. The mid-point of AC is X.

(a) Calculate, in cm to 3 significant figures, the length of BX. (2)

(b) Calculate, to the nearest tenth of a degree, the angle between planes ABC and ADC. (3)


1993-6 Paper(1) Q.14

10

C B

A

D

Fig. 3

Figure 3 shows a tetrahedron ABCD. The triangle ABC lies in a horizontal plane and D is

vertically above C. AB = 12 cm, AD = 15 cm and DB = 11 cm.

(a) Calculate, in degrees to 2 decimal places, the sizes of ∠DBA. (4)

(b) Hence calculate, in cm2 to one decimal place, the area of 4ABD. (3)

Given also that the angle between DA and the horizontal plane is 30◦, calculate

(c) the length, in cm, of CD, (2)

(d) the angle, in degrees to one decimal place, between the planes ABC and ABD. (6)


1994-1 Paper(1) Q.13

11 A symmetrical pyramid stands on its base, which is a regular pentagon of side 20 cm. The height

of the pyramid is 50 cm. Calculate, to 3 significant figures,

(a) the angle, in degrees between a sloping edge and the base, (5)

(b) the length, in cm, of a sloping edge, (3)

(c) the angle, in degrees, between a sloping face and the base, (4)

(d) the area, in cm2, of a sloping face. (3)

1995-1 Paper(1) Q.12

12 Three points A, B and C are on flat horizontal ground. A is 500 m due north of B and B is 700

m due north of C. A vertical pole DE has its base D on the ground and E is the top of the pole.

D is to the east of the line ABC and DE = 40 m, AD = 900 m and CD = 1100 m.

(a) Find, to the nearest 0.1◦, the bearing of D from C. (4)

(b) Find the length, in m to 3 significant figures, of BD. (3)

(c) Find, to the nearest 0.1◦, the angle of elevation of E from B. (3)

The base of a tree is at the point F on the horizontal ground where F is on a bearing of 115◦

from B and 100◦ from C. A woman walks from B to F at a constant speed of 2.5 m/s.

(d) Find, to the nearest minute, the time she takes to walk from B to F . (5)


1996-1 Paper(1) Q.14

13 In the tetrahedron ABCD, AB = 1 cm, ∠DAB = ∠DAC = 90◦, ∠ABD = ∠CDB = 50◦ and

∠ADC = 42◦.

Calculate

(a) the length of CB, in cm to 3 significant figures, (11)

(b) the angle between the planes DAB and DAC, in degrees to 1 decimal place. (4)

1996-6 Paper(1) Q.9

14

A

D

B

C

E

M

N

Fig. 1

Figure 1 shows a pyramid ABCDE with its rectangular base ABCD lying in a horizontal plane.

AB = 8 cm, BC = 14 cm, EA = EB = 10 cm and EC = ED = 12 cm. M is the mid-point of

AB and N is the mid-point of DC. Calculate

(a) the length, in cm to 2 decimal places, of (i) EM , (ii) EN , (3)

(b) the size of the angle, 0.1◦, between the plane EAB and the base of the pyramid, (4)

(c) the height, in cm to 2 decimal places, of E above the base of the pyramid, (3)

(d) the area, in cm2 to one decimal place, 4EBD. (5)


1997-6 Paper(1) Q.10

15

A

B

C

D

M

Fig. 3

Figure 3 shows a tetrahedron ABCD. The triangle ABC lies in a horizontal plane and D is

vertically above A. AB = 7 cm, AC = 9 cm, AD = 11 cm and cos∠BAC = 0.6. Calculate

(a) the length, in cm to 3 decimal places, of BC, (3)

(b) the area, in cm2, of 4ABC, (3)

(c) the length, in cm to 3 decimal places, of the perpendicular from A to BC, (2)

(d) the angle, to the nearest 0.1◦, between the planes ABC and BCD. (2)

The point M is the mid-point of BC.

(e) Calculate the length, in cm to 2 decimal places, of AM . (5)


2007-1 Paper(2) Q.11

16

B

C

E

D

F I

HG

A

12 cm

16 cm

4 cm

Figure 1

Figure 1 shows a paperweight which consists of a cuboid BCDEFGHI and a right pyramid

ABCDE. The height of the pyramid 8 cm, BF = 4 cm, BE = 12 cm and DE = 16 cm.

(a) Find, to 3 significant figures, the length of

(i) GE,

(ii) AB. (6)

Calculate, in degrees to the nearest 0.1◦, the size of the angle

(b) between GE and the plane FGHI, (3)

(c) between AB and the plane BCDE, (3)

(d) between the plane ABC and the plane BCHI. (5)


2007-6 Paper(1) Q.9

17

A

B

D

C

E H

GF

P

Q

Figure 1

Figure 1 shows a cuboid with a rectangular top ABCD.

AB = 5 cm, AB = 8 cm and AE = 4 cm.

The mid-point DH is P and the mid-point of CG is Q.

(a) Find, to 3 significant figures, the length of AG. (2)

Calculate, in degrees to one decimal place, the acute angle

(b) between AG and the plane EFGH, (3)

(c) between the plane ABQP and the plane EFQP , (3)

(d) between the plane BCH and the plane EFGH, (3)

(e) between AG and CE. (4)


2008-1 Paper(1) Q.10

18

C

D

B

A

V

18 cm

10 cm

10 cm

Figure 2

Figure 2 shows a right pyramid V ABCD. The base ABCD of the pyramid is a square of side 10

cm and V A = V B = V C = V D = 18 cm.

(a) Find, in cm to 3 significant figures, the height of the pyramid. (3)

(b) Find, to the nearest 0.1◦, the size of the angle between V A and the plane ABCD. (3)

(c) Find, to the nearest 0.1◦, the size of the angle between the plane V AB and the plane ABCD.

(3)

(d) Find, in cm to 3 significant figures, the length of the perpendicular from B to V A. (4)

(e) Find, in cm to the nearest 0.1◦, the size of the angle between the plane V AB and the plane

V AD. (4)


2008-6 Paper(2) Q.11

19

A

B

D

C

V

12 cm

Figure 1

Figure 1 shows a right pyramid with vertex V and square base ABCD, of side 12 cm. The size

of angle AV C is 90◦.

(a) Show that the height of the pyramid is 6√

2 cm. (4)

(b) Find, in cm, the length of V A. (3)

(c) Find, in cm, the exact length of the perpendicular from D to V A. Give your answer in the

form p√q, where p and q are integers and q is prime. (3)

Find, in degrees to 1 decimal place, the size of

(d) the angle between the plane V AB and the base ABCD, (3)

(e) the angle between the plane V AB and the plane V AD. (4)


2009-6 Paper(2) Q.9

20

A

B

D

C

V

X

Figure 1

Figure 1 shows a hollow right pyramid V ABCD with a square base of side 4 cm.

The point X is the mid-point of CD.

The sloping edge V A makes an angle of 60◦ with the base ABCD.

Find, in cm to 3 significant figures,

(a) the height of the pyramid, (4)

(b) the length of V A, (3)

(c) the length of V X. (3)

(d) Find, in degrees, to one decimal place, the angle between the plane V CD and the base

ABCD. (3)

A sphere is inside the pyramid and is touching all five plane faces of the pyramid.

(e) Find, in cm to 3 significant figures, the radius of the sphere. (3)


2010-1 Paper(1) Q.11

21

A

B

D

C

E H

GF

V

Figure 2

Figure 2 shows a solid V ABCDEFGH which consists of a cuboid ABCDEFGH and a right

pyramid V ABCD.

AB = 5 cm, BC = 12 cm, EC = 17 cm.

The height of the pyramid is 10 cm.

Calculate, in cm to 3 significant figures, the length of

(a) AE, (3)

(b) V A. (3)

Find, in degrees to the nearest 0.1◦, the size of the angle between

(c) EC and the plane ABCD, (3)

(d) the plane V AB and the plane ABGH, (4)

(e) the plane V AB and the plane V CD. (4)


2011-1 Paper(2) Q.5

22

10 cm

15 cm

Figure 2

Figure 2 shows a right pyramid with a square base of side 10 cm.

The length of each sloping edge is 15 cm.

Calculate, in degrees to the nearest 0.1◦, the size of the angle between a triangular face and the

base. (6)


2011-6 Paper(1) Q.10

23

AD

B C

N

V

8x

6x

Figure 2

Figure 2 shows the pyramid V ABCD. The base ABCD is a rectangle with CD = 6x cm and

AD = 8x cm. The diagonals of the base intersects at the point N . The edges V A, V B, V C and

V D are all of equal length. The angle between V A and the base ABCD is 60◦.

Find, in terms of x,

(a) the height, V N , of the pyramid, (4)

(b) the length of V A. (3)

Find, in degrees to the nearest 0.1◦,

(c) the size of the angle between the planes AV B and ABCD, (3)

(d) the size of the angle between the planes BVD and AV C. (3)

The volume of the pyramid is 1110 cm3.

(e) Find, to the nearest whole number, the value of x. (3)


2012-1 Paper(1) Q.9

24

A

B

C

D

Figure 1

Figure 1 shows a triangular pyramid ABCD.

∠BAC = ∠DAC = ∠BAD = 90◦

AD = 5 cm, AC = 8 cm and AB = 6 cm.

(a) Find, in degrees to the nearest 0.1◦, the size of ∠BDC. (6)

(b) Find, to 3 significant figures, the area of triangle BDC. (3)

(c) Find the area of triangle DAC. (1)

The point E lies on CD so that AE is perpendicular to CD.

(d) Find the exact length of AE. (2)

(e) Hence, or otherwise, find in degrees to the nearest 0.1◦, the size of the angle between the

planes DAC and BDC. (4)


2012-1 Paper(2) Q.2

25

A

B C

D

EF

X

V

Figure 1

Figure 1 shows a right pyramid with vertex V and base ABCDEF which is a regular hexagon.

The diagonal AD of the base is 10 cm and X is the mid-point of AD. The height V X of the

pyramid is 12 cm.

(a) Find the length of V A. (2)

(b) Find, in degrees to a 1 decimal place, the size of the angle between the plane V AB and the

base. (4)


2012-6 Paper(2) Q.11

26

A B

DC

E F

H G

P

Q

4 cm

4 cm

12 cm

10 cm

Figure 1

Figure 1 shows a truncated right pyramid. The base ABCD is a square with sides of length 10cm. The top EFGH is a square with sides of length 4 cm. The base is parallel to the top andAE = BF = CG = DH.The point P is on the line AC such that angle APE is a right-angle and EP = 12 cm.

(a) Find, in centimetres, the exact length of

(i) AC (ii) EG (iii) AP (6)

(b) Find, in centimetres to 3 significant figures, the length of AE. (2)

(c) Find, in degrees to 1 decimal place, the angle between the line AE and the plane ABCD.(2)

The point Q is one the line AB. Angle AQP is a right-angle.

(d) (i) Show that PQ = 3 cm.

(ii) Write down, in centimetres, the length of AQ. (2)

(e) Find, in degrees to 1 decimal place, the angle between the line A and the line AB. (2)

(f) Find, in degrees to 1 decimal place, the angle between the plane ABFE and the plane ABCD.(3)


2013-6 Paper(1) Q.9

27

A

C

E

D

B

F

P

Figure 3

Figure 3 shows a triangular prism ABCDEF .

ACDE is a rectangle. In triangle ABC, AC = 12 cm, ∠BAC = 60◦ and ∠BCA = 30◦

(a) Find the exact length of BC. (3)

The point P lies on the line AC and ∠BPC = 90◦

(b) Show that BP = 3√

3 cm. (2)

The angle between the plane AFC and the plane ACDE is 25◦

(c) Find, to 3 significant figures, the length of BF . (3)

(d) Find the size of the angle between the line BD and the plane ACDE, giving your answer in

degrees to 1 decimal place. (4)

(e) Find, to 3 significant figures, the volume of the prism ABCDEF . (2)


2014-1 Paper(2) Q.7

28

E

F

B

H

DC

IJ

A

G

5 cm

8 cm

10 cm

15 cm

P

Figure 3

Figure 3 shows a prism ABCDEFGHIJ which consists of a triangular prism ABEFGH on top

of a cuboid BCDEFHIJ .

AB = AE = 5 cm, EB = 8 cm, ED = 10 cm, CI = 15 cm

P is the mid-point of DC.

Calculate, in cm to 3 significant figures,

(a) the length of PG, (3)

(b) the length of AC. (2)

Find, in degrees to the nearest 0.1◦,

(c) the size of the angle between PG and the plane CDJI, (3)

(d) the size of the angle between the plane AGIC and the plane CDJI. (3)


2014-6 Paper(1) Q.10

29

E

B

D

C

F I

HG

A

3 cm

4 cm

5 cm

Figure 2

A paperweight ABCDEFGHI consists of a cuboid BCDEFGHI and a right pyramid ABCDE

as shown in Figure 1.

EF = 3 cm, FI = 4 cm, IH = 5 cm

The volume of the pyramid is equal to the volume of the cuboid.

(a) Show that the height of the pyramid is 9 cm. (2)

Find, in cm to 3 significant figures, the length of

(b) AE, (3)

(c) EH. (2)

Find, in degrees to the nearest 0.1◦, the size of

(d) the angle between AE and the plane EBCD, (3)

(e) the obtuse angle between the plane ABE and the plane BEIH. (5)


2015-1 Paper(2) Q.9

30

A

B

C

D

13 cm

10 cm

E

Figure 2

Figure 2 shows a triangular pyramid ABCD.

AB = BC = CA = 10 cm and DA = DB = DC = 13 cm.

The point E is the midpoint of AC.

(a) Find the exact length of

(i) DE

(ii) BE (4)

(b) Find, in degrees to 1 decimal place, the size of the angle between the line BD and the line

DE. (3)

(c) Find, in degrees to 1 decimal place, the size of the angle between the line BD and the plane

ABC. (3)

(d) Find, in degrees to 1 decimal place, the size of the angle between the plane ADC and the

plane ABC. (2)

(e) Find, to 3 significant figures, the volume of the pyramid ABCD. (3)


2015-6 Paper(2) Q.7

31

A

D

B

C

F G

HE

V

8 cm

6 cm

Figure 2

Figure 2 shows a solid V ABCDEFGH which is formed by joining a cuboid ABCDEFGH to a

right pyramid V ABCD. The height of the cuboid and the height of the pyramid are both h cm

and FG = 8 cm and GH = 6 cm. The total volume of the solid is 256 cm3.

(a) Show that h = 4 (2)

(b) Find, in cm to 3 significant figures, the length of V F . (3)

Find, to the nearest 0.1◦,

(c) the angle between V A and the plane ABCD, (3)

(d) the acute angle between the plane V AB and the plane ABHE. (4)


2016-1 Paper(2) Q.12

32

A

D

B

C

H

G

E

F

16 cm

20 cmM

Figure 3

Figure 3 shows a right prism ABCDEFH. The cross section ABCD of the prism is a trapezium

with AB = DC. The point M lies on AD and BM is perpendicular to AD.

AB = 8 cm CD = 8 cm BC = 8 cm AD = 16 cm DE = 20 cm

Given that BM = p√q cm where q is a prime number,

(a) find the value of p and the value of q. (3)

(b) Find the size of angle BAM in degrees. (2)

Find, in degrees to the nearest 0.1◦

(c) the size of the angle between EB and the plane ADEH, (4)

(d) the size of the angle between the plane BCEH and the plane ADEH. (3)


2016-6 Paper(1) Q.3

33 A right pyramid ABCDE has a square base ABCD of side 10 cm.

The height of the pyramid is 8 cm.

(a) Find, to 3 significant figures, the length of AE. (3)

(b) Find, in degrees to the nearest degree, the size of the angle between the plane ABE and the

base ABCD. (3)


2017-1 Paper(2) Q.10

34

A

E

C

D

BJ

H

F

G

I

8 cm

6 cm

10 cmM

Figure 1

Figure 1 shows a right prism ABCDEFGHIJ . The base, DEFG, is horizontal and is a rectangle

with DG = EF = 10 cm. The midpoint of ED is M .

The planes ABCDE and JIHGF are vertical.

AE = CD = GH = FJ = 8 cm

AB = BC = HI = IJ = 6 cm

Angle BAC = 30◦

(a) Show that the length of MD is 3√

3 cm. (2)

(b) Show that the length of BM , the height of the prism, is 11 cm. (2)

(c) Find, in cm to 3 significant figures, the length BG. (3)

Find, in degrees to 1 decimal place

(d) the size of the angle between the planes BCHI and CHFE, (3)

(e) the size of the angle between the planes ABIJ and BEFI. (5)


2017-6 Paper(2) Q.10

35

A D

BC

F

G H

E

N

3 cm

8 cm

Figure 2

Figure 2 shows a cuboid ABCDEFGH with EF = 8 cm and EH = 3 cm.

The angle between the diagonal AH of the cuboid and the plane ABCD is 45◦.

The midpoint of CH is N .

Find, in cm to 3 significant figures,

(a) the length of CH, (4)

(b) the length of AH, (3)

(c) the length of FN . (3)

Find, in degrees to 1 decimal place, the size of

(d) the angle between the plane BCEF and the plane FGHE, (3)

(e) angle FNG. (3)


2018-1 Paper(2) Q.11

36

A

D

B

CO

E

12 cm

10 cm

8 cm

h cm

Figure 6

A pyramid with a rectangular base ABCD and vertex E is shown in Figure 6.

The rectangular base is horizontal with AB = 12 cm and BC = 8 cm.

The diagonals of the base intersect at the point O.

The vertex of E of the pyramid is vertically above O.

The height of the pyramid is h cm and AE = BE = CE = DE = 10 cm.

(a) Show that h = 4√

3 (3)

(b) Find, in degrees to 1 decimal place, the size of angle OCE. (2)

The angle between OE and the plane CBE is θ◦

(c) Show that cos θ◦ =2√

7

7(3)

The point P is the midpoint of BC and the point Q is the midpoint of CE.

(d) Find, in degrees to 1 decimal place, the size of the angle between the plane OPQ and the

plane EPQ. (4)


2 Alpha Beta

1988-1 Paper(2) Q.3

1 The roots of the equation

2x2 − x+ 3 = 0

are α and β. Without solving the equation, find the value of α2β + αβ2. (4)

1989-1 Paper(2) Q.10

2 Given that the equation x2 − 5x+ 7 = 0 has roots α and β,

(a) find a quadratic equation, with integer coefficients, whose roots are1

αand

1

β. (6)

(b) Find constant p and q such that

x2 + 3x+ 8 = (x+ p)2 + q

Hence show that x2 + 3x+ 8 > 0 for all real values of x. (4)

(c) Use your answer to (a) to find the minimum value of the x2 + 3x+ 8. (2)

(d) Confirm your answer to (c) by using a calculus method. (2)

(e) Sketch the curve whose equation is y = x2 + 3x+ 8. (1)

1989-6 Paper(2) Q.1

3 Given that roots of the equation

3x2 − 2x− 4 = 0

are α and β, find and equation whose roots are α2 and β2. (4)


1990-1 Paper(2) Q.4

4 The roots of the equation y2 − 2y − 5 = 0 are α and β.

(a) State the value of α+ β. (1)

Given that both x1 and x2 satisfy the equation

(lnx)2 − 2 lnx− 5 = 0,

(b) show that lnx1 + lnx2 = 2, (1)

(c) deduce the value of x1x2. (3)

1990-1 Paper(2) Q.9

5 The equation 2x2 + 6x+ 3 = 0 has roots α and β.

(a) Show that α2 + β2 = 6. (1)

(b) Form an equation whose roots are 2α2 and 2β2. (8)

For the equation x2 + 2(p+ 1)x+ p− 1 = 0,

(c) write down, in terms of p, the condition that the roots of the equation are real. (3)

(d) Show that this condition is satisfied by all real values of p. (1)

Given that the roots of this equation are γ and −γ,

(e) determine the value of p. (3)


1991-1 Paper(2) Q.12

6 f(x) = x2 + 3x+ 5 = (x+A)2 +B.

(a) Find the values of the constants A and B and hence deduce the minimum value of f(x). (4)

g(x) = x2 + kx+ 2k − 3, where k is a constant.

(b) Find the range of values of k for which the equation g(x) = 0 has no real solutions. (4)

(c) Given that the equation g(x) = 0 has roots α and β, form a quadratic equation whose roots

are (2α+ β) and (α+ 2β). (7)

1991-6 Paper(2) Q.9

7 The equation x2 + px+ (p+ 1) = 0 has roots α and β.

(a) Form a quadratic equation with roots (α+ 2) and (β + 2). (4)

(b) Form another quadratic equation with roots

(1

α+ β

)and

(1

β+ α

). (7)

Given that α− β = 1,

(c) find the possible values of p. (4)

1993-6 Paper(2) Q.1

8 Given that the roots of the equation 5x2 − 18x+ p = 0 are α and 5α, find the value of p. (3)


1994-1 Paper(2) Q.3

9 Given that α and β are the roots of the equation

2x2 + 5x− 1 = 0,

find a quadratic equation with roots α+ 2 and β + 2. (5)

1994-1 Paper(2) Q.12

10 f(x) = 4x2 + 12x− 9 = A(x+B)2 + C,

where A, B, and C are constants.

(a) Find the values of A, B, and C and hence deduce the minimum value of f(x). (4)

Given that

g(x) =1

f(x),

(b) using your answers to (a), or otherwise, find the coordinates of the stationary point of the

curve with equation y = g(x) and state whether it is a maximum or minimum. (3)

(c) Find the range of values of k for which the equation

f(x) = k

has no real roots. (2)

Given that α and β are the roots of the equation f(x) = 0,

(d) find a quadratic equation with roots(α+

2

β

)and

(β +

2

α

).

(6)


1995-1 Paper(2) Q.9

11 f(x) = 2x2 − 8x+ 3.

(a) Express f(x) in the form a(x− 2)2 − b, where a and b are constants to be found. (2)

(b) Hence state the minimum value of f(x) and the value of x which gives this minimum value.

(2)

The roots of the equation f(x) = 0 are α and β.

(c) Find the value of α2 + β2. (3)

Given that g(x) = x2 − px + q, where p and q are constants, and that the roots of the equation

g(x) = 0 are 3α+ β and 3β + α,

(d) calculate the values of p and q. (4)

(e) For your values of p and q express g(x) in the form (x+ r)2− s, where r and s are constants

to be found. (2)

(f) Hence write down the maximum value of1

g(x)and the value of x which gives this maximum

value. (2)

1995-6 Paper(2) Q.1

12 f(x) = 4x2 + kx− 2k,

where k is a negative constant. Given that one root of the equation f(x) = 0 is double the other

root, find the value of k. (4)

1996-1 Paper(2) Q.1

13 f(x) = 3x2 + kx+ 100, where k is a positive constant.

Given that one root of the equation f(x) = 0 is three times the other root, find the value of k. (4)


1996-1 Paper(2) Q.9

14 f(x) = x2 + 3x− 2.

The line, l, with equation y = 2x + k, where k is a constant, is a tangent to the curve, C, with

equation y = f(x).

(a) Find the value of k. (3)

(b) Using your value of k, find the coordinates of the point, P , where l meets C. (3)

Given that α and β are the roots of the equation f(x) = 0, find a quadratic equation, with integer

coefficients, with roots,

(c) α2 and β2, (4)

(d) (α− β)2 and (α+ β)2. (5)

1996-6 Paper(2) Q.9

15 f(x) = 3x2 + 6x− 7.


(a) form a quadratic equation, with integer coefficients, whose roots are2

αand

2

β, (6)

(b) form a quadratic equation, with integer coefficients, whose roots are α(α+ 1) and β(β + 1).

(5)

Given that f(x) can be expressed in the form A(x+B)2 − C, where A, B, and C are constants,

(c) find the values of A, B and C. (2)

(d) Hence write down the minimum value of f(x) and the value of x for which it occurs. (2)


1997-6 Paper(2) Q.12

16 2x2 − 6x+ 13 = A(x+B)2 + C

where A, B and C are constants.

(a) Determine the values of A, B and C. (4)

f(x) =1

2x2 − 6x+ 13.

Using your answers to (a), or otherwise,

(b) find, giving your answer as an exact fraction, the maximum value of f(x), (1)

(c) state the value of x at which the maximum value occurs. (1)

Given that α and β are the roots of the equation 2x2−5x+p = 0, where p is a non-zero constant,

(d) obtain a quadratic equation with roots

(1 +

1

α

)and

(1 +

1

β

). (6)

Given that the equation found in (d) has two equal roots,

(e) find the value of p. (3)

2007-1 Paper(2) Q.9

17 f(x) = 2x2 + px+ 3, where p is a constant.

The equation f(x) = 0 has roots α and β. Without solving the equation,

(a) form a quadratic equation, with integer coefficients, which has roots α2β2 and1

α2β2, (3)

(b) form, in terms of p, a quadratic equation which has roots α2 and β2. (5)

Given that 3 is a root of the equation found in part (b), find

(c) the value of the other root of the equation, (2)

(d) the possible values of p. (3)


2007-6 Paper(2) Q.9

18 (a) Show that

(i) (α− β)(α2 − αβ + β2) = α3 + β3,

(ii) (α− β)(α2 + αβ + β2) = α3 − β3. (3)

The equation 2x2 + 7x + 4 = 0 has roots α and β, where α > β. Without solving the equation,

calculate the value of

(b) (α− β)2, (3)

(c) α3 + β3. (2)

Hence

(d) find the value of α3 − β3, giving your answer in the form k√m, where k is rational and m is

a prime number. (2)

(e) form a quadratic equation, with integer coefficients, which has rootsα2

βand

β2

α. (4)

2008-1 Paper(1) Q.6

19 f(x) = 3x2 − 6x+ p.

The equation f(x) = 0 has roots α and β. Without solving the equation f(x) = 0.

(a) form a quadratic equation, with integer coefficients, which has roots (α+ β) and1

α+ β. (4)

(b) form a quadratic equation which has rootsα+ β

αand

α+ β

β. (4)

Given that 3 is a root of the equation found in part (b), find

(c) the value of p, (2)

(d) the other root of the equation. (2)


2008-6 Paper(2) Q.7

20 f(x) = x2 + kx− 5, k ∈ R.


(a) Find, in terms of k where appropriate, the value of

(i) α2 + β2, (ii) α2β2. (4)

Given that 5(α2 + β2) = 7α2β2,

(b) find the possible values of k. (2)

Using the positive value of k found in part (b), and without solving the equation f(x) = 0,

(c) form a quadratic equation, with integer coefficients, which has roots1

α2and

1

β2. (5)

2009-6 Paper(2) Q.8

21 f(x) = 3− 5x− 7x2

(a) Show that f(x) can be written in the form A−B(x+C)2, stating the values of A, B and C.

(4)

(b) Write down the maximum value of f(x) and the value of x for which this maximum occurs.

(2)

The equation f(x) = 0 has roots α and β.

Without solving the equation find, as exact fractions,

(c) α2 + β2, (3)

(d)α

β+β

α. (3)

(e) Form a quadratic equation, with integer coefficients, which has rootsα

βand

β

α. (2)


2010-6 Paper(2) Q.8

22 (a) Show that

(α− β)(α2 + αβ + β2) = α3 − β3

(α+ β)(α2 − αβ + β2) = α3 + β3

(2)

f(x) = x2 − 2x− 5 The roots of the equation f(x) = 0 are α and β, where α > β.

Without solve the equation, calculate the value of

(b) α2 + β2, (3)

(c) (α− β)2. (2)

Hence

(d) calculate the value of α3 + β3, (2)

(e) calculate the exact value of α3 − β3, giving your answer in the form k√

6 (2)

(f) form an equation with roots (α− β)2 and (α+ β)2. (4)


2011-1 Paper(2) Q.10

23 f(x) = 3x2 − 6x− 2


Without solving the equation, form an equation with integer coefficients,

(a) with roots αβ and1

αβ, (6)

(b) with roots 2α+ β and α+ 2β. (5)

(c) Express f(x) in the form f(x) = A(x+B)2 +C, stating the values of the constants A, B and

C. (3)

(d) Hence write down

(i) the minimum value of f(x),

(ii) the value of x for which this minimum occurs. (2)

2011-6 Paper(2) Q.10

24 The roots of the equation x2+6x+2 = 0 are α and β, where α > β. Without solving the equation

(a) find

(i) the value of α2 + β2

(ii) the value of α4 + β4 (5)

(b) Show that α− β = 2√

7 (3)

(c) Factorise completely α4 − β4 (2)

(d) Hence find the exact value of α4 − β4 (2)

Given that β3 = A+B√

7 where A and B are positive constants

(e) find the value of A and the value of B. (2)


2012-1 Paper(1) Q.8

25 The equation x2 +mx+ 15 = 0 has roots α and β and the equation x2 + hk + k = 0 has rootsα

β

andβ

α

(a) Write down the value of k (1)

(b) Find an expression of h in terms of m (6)

Given that β = 2α+ 1

(c) find the two possible values of α (3)

(d) Hence find the two possible values of m (3)

2012-6 Paper(1) Q.4

26 The equation 2x2 − 7x+ 4− 0 has roots α and β

Without solving this equation, form a quadratic equation with integer coefficients which has roots

α+1

βand β +

1

α(8)

2013-1 Paper(1) Q.10

27 f(x) = 2x2 − 5x+ 1

The equation f(x) = 0 has roots α and β. Without solving the equation

(a) find the value of α2 + β2 (3)

(b) show that α4 + β4 =433

16(2)

(c) form a quadratic equation with integer coefficients which has roots(α2 +

1

α2

)and

(β2 +

1

β2

)(7)


2013-6 Paper(1) Q.6

28 The equation x2 + px+ 1 = 0 has roots α and β

(a) Find, in terms of p, an expression for

(i) α+ β

(ii) α2 + β2

(iii) α3 + β3 (6)

(b) Find a quadratic equation, with coefficients expressed in terms of p, which has roots α3 and

β3 (2)

2014-1 Paper(1) Q.10

29 f(x) = x2 + (k − 3)x+ 4

The roots of the equation f(x) = 0 are α and β

(a) Find, in terms of k, the value of α2 + β2 (3)

Given that

4(α2 + β2) = 7α2β2

(b) without solving the equation f(x) = 0, form a quadratic equation, with integer coefficients,

which has roots1

α2and

1

β2(5)

(c) find the possible values of k. (5)


2014-6 Paper(1) Q.8

30 f(x) = 3x2 + px− 7


(a) Without solving the equation

(i) write down the value of α2β2

(ii) find, in terms of p, α2 + β2 (4)

Given that 3α− β = 8

(b) find the possible values of p. (5)

Given also that p is negative,

(c) form an equation with roots1

α2and

1

β2(3)

2015-1 Paper(2) Q.6

31 The equation 2x2 + px− 3 = 0, where p is a constant, has roots α and β.

(a) Find the value of

(i) αβ

(ii)

(α+

1

β

)(β +

1

α

)(4)

(b) Find, in terms of p,

(i) α+ β

(ii)

(α+

1

β

)+

(β +

1

α

)(4)

Given that

(α+

1

β

)+

(β +

1

α

)= 2

(α+

1

β

)(β +

1

α

)(c) find the value of p. (1)

(d) Using the value of p found in part (c), find a quadratic equation, with integer coefficients,

which has roots

(α+

1

β

)and

(β +

1

α

). (2)


2015-6 Paper(1) Q.5

32 (a) Show that (α+ β)(α2 − αβ + β2) = α3 + β3 (1)

The roots of the equation 2x2 + 6x− 7 = 0 are α and β where α > β

Without solving the equation,

(b) find the value of α3 + β3 (4)

(c) show that α− β =√

23 (2)

(d) Hence find the exact value of α3 − β3 (2)

2016-1 Paper(2) Q.5

33 Given that α+ β = 5 and α2 + β2 = 19

(a) show that αβ = 3 (2)

(b) Hence form a quadratic equation, with integer coefficients, which has roots α and β (2)

(c) Form a quadratic equation, with integer coefficients, which has rootsα

βand

β

α(5)


2016-6 Paper(1) Q.9

34 f(x) = 3x2 − 5x− 4


(a) Without solving the equation f(x) = 0, form an equation, with integer coefficients, which has

(i) rootsα

βand

β

α

(ii) roots 2α+ β and α+ 2β (11)

(b) Express f(x) in the form A(x+B)2 +C, stating the values of the constants A, B and C. (3)

(c) Hence, or otherwise, show that the equation f(x) = −8 has no real roots. (2)

2017-1 Paper(1) Q.9

35 The equation 3x2 − 4x+ 6 = 0 has roots α and β.

(a) Without solving the equation, write down

(i) the value of α+ β

(ii) the value of αβ (2)

(b) Without solving the equation, show that α3 + β3 = −152

27(3)

(c) Form a quadratic equation, with integer coefficients, that has rootsα

β2and

β

α2(5)


2017-6 Paper(2) Q.8

36 f(x) = x2 + px+ 7 p ∈ RThe roots of the equation f(x) = 0 are α and β

(a) Find, in terms of p where necessary,

(i) α2 + β2 (ii) α2β2 (4)

Given that 7(α2 + β2) = 5α2β2

(b) find the possible values of p (2)

Using the positive value of p found in part (b) and without solving the equation f(x) = 0

(c) form a quadratic equation with roots2p

α2and

2p

β2(5)

2018-1 Paper(1) Q.9

37 It is given that α and β are such that α+ β = −5

2and αβ = −5

(a) Form a quadratic equation with integer coefficients that has roots α and β (2)

Without solving the equation found in part (a)

(b) find the value of

(i) α2 + β2

(ii) α3 + β3 (5)

(c) Hence form a quadratic equation with integer coefficients that has roots(α− 1

α2

)and

(β − 1

β2

)(6)


3 Antiderivative

1989-6 Paper(1) Q.14

1 The curve with equation y = f(x) is such thatdy

dx= 3x2 − 10x+ 3. The curve passes through the

point with coordinates (−1, 0).

(a) Find f(x) and show that the curve passes through the point with coordinates (0, 9). (4)

(b) Show that f(x) can be written in the form

f(x) = (x+ p)(x− q)2

where p and q are positive numbers. (3)

(c) Deduce the solutions f(x) = 0. (1)

(d) Determine the coordinates of any turning points on the curve. (4)

(e) Hence sketch the graph of y = f(x), showing

(i) the coordinates of the points where the curve meets the coordinate axes,

(ii) the coordinates of the turning points of the curve. (3)


1989-6 Paper(2) Q.10

2 (a) Differentiate e2x sin 3x with respect to x and, hence, find the equation of the tangent at the

origin to the curve with equation y = e2x sin 3x. (5)

(b) By using the identity tanx =sinx

cosxshow that

d

dx(tanx) = 1 + tan2 x.

(3)

(c) Hence show that∫

tan2 x = tanx− x+ k, where k is an arbitrary constant. (2)

(d) The finite region bounded by the x-axis, the line x = π4 and the curve with equation y = tanx

is rotated through 360◦ about the x-axis. Find the volume generated, leaving your answer in

terms of π. (5)


1990-1 Paper(2) Q.12

3 In 4ABC, AB = BC = y cm, AC = p cm and ∠ABC = 2x.

(a) Express p in terms of y and sinx. (2)

(b) Using the cosine rule in 4ABC, show that

cos 2x = 1− 2 sin2 x.

(3)

(c) Show that

∫2 sin2 x dx = x− 1

2sin 2x+ k, where k is a constant. (2)

O x

y

P

Q

R

π

Fig. 2

Figure 2 shows the graph of y = 2 sin2 x in the interval 0 6 x 6 π. The curve touches the x-axis

at O and Q, and the side PR of the rectangle OPRQ touches the curve at its maximum point.

(d) State the coordinates of the point P . (1)

(e) Show that the area enclosed by the curve and the x-axis, in the given interval, is equal to

one half of the area of the rectangle OPRQ. (3)

(f) Calculate the coordinates of the points of intersection, in the interval 0 6 x 6 π, of y =

2 sin2 x and y = cos 2x. (5)


1990-6 Paper(1) Q.2

4 (a) Differentiate 4 + 3√x with respect to x. (2)

(b) Find

∫ (5 +

1

x2

)dx. (2)

1991-1 Paper(1) Q.4

5 Given that

dy

dx= (x− 3)2.

and that y = 11 at x = 6,

(a) find an expression for y in terms of x, (4)

(b) find the value of y at x = 3. (1)

1991-6 Paper(2) Q.4

6 Given that y =1 + sinx

cosx, where 0 6 x <

π

2, show that

dy

dx=

1

1− sinx. (5)


1991-6 Paper(2) Q.11

7 (a) Using the identities,

sin (A+B) = sinA cosB + cosA sinB

and cos (A+B) = cosA cosB − sinA sinB

show that sin 2A = 2 sinA cosA

and cos 2A = 2 cos2A− 1

(3)

(b) Solve, for 0 6 x 6 2π, giving your answers in radians, the equations

(i) sin 2x sinx = cosx,

(ii) sin

(x+

π

3

)+ sin

(x− π

3

)= 1. (7)

(c) Find

∫ √(1 + cos 2x) dx. (2)

(d) In 4ABC, AB = 3 cm, AC = 5 cm, ∠ABC = 2θ◦ and ∠ABC = θ◦. Calculate, to 3

significant figures, the values of θ. (3)

1993-6 Paper(2) Q.3

8 (a) Evaluate

∫(sinx+ cos 3x) dx. (2)

(b) Hence calculate, to 3 significant figures.

∫ π3

π6

(sinx+ cos 3x) dx. (2)

1993-6 Paper(2) Q.5

9 The volume, V cm3, of a sphere of radius r cm, is giving by the formula V =4

3πr3.

An x% error, where x is small, is made in measuring the radius of the sphere.

Find an estimate for the percentage error in the value of V . (5)


1994-1 Paper(1) Q.8

10 Given that y = (3x+ 1x)2

(a) finddy

dx, (2)

(b) evaluate

∫ 3

2y dx. (5)

1994-1 Paper(1) Q.14

11 The curve with equation y = f(x) is such that

dy

dx= 6x(x+ 1).

The passes through the point with coordinates (0,−1).

(a) Find f(x). (4)

f(x) can be written in the form f(x) = (px− q)(x+ r)2.

(b) Find the values of p, q, and r. (2)

(c) Find the coordinates of the points where the curve meets the coordinate axes. (2)

(d) Determine the coordinates of the turning points of the curve. (3)

(e) Hence sketch the graph of y = f(x) showing the points where the curve meets the coordinate

axes and the coordinates of the turning points. (2)

(f) Calculate the finite area bounded by the curve and the x-axis. (2)


1995-6 Paper(2) Q.12

12 f(x) = 2x2 + 3x+ 4.

(a) Express f(x) in the form A(x+B)2 + C. (4)

Hence

(b) write down the minimum value of f(x), (1)

(c) the value of x for which f(x) takes this minimum value. (1)

(d) Confirm your answers to (b) and (c) by using a calculus method. (3)

(e) Sketch the curve with equation y = f(x) showing clearly the coordinates of the minimum

point and the coordinates of any points where the curve crosses the coordinate axes. (2)

(f) Find

∫f(x) dx. (2)

(g) Hence calculate the area of the finite region bounded by the curve with equation y = f(x),

the lines x = 1, x = 2 an the x-axis. (2)

1995-6 Paper(2) Q.13

13 (a) Using the formula cos (A+B) = cosA cosB − sinA sinB, show that cos 6x = 1 − 2 sin2 3x.

(3)

(b) Hence show that

∫4 sin2 3x dx =

1

3(6x− sin 6x) + c, where c is a constant. (3)

(c) Evalute

∫ π4

04 sin2 3x dx, giving your answer in terms of π. (2)

(d) Find, in terms of π, the coordinates of the points at which the normal to the curve with

equation y = x+ sin 2x at the point (π, π) cuts the coordinate axes. (7)


1996-1 Paper(1) Q.6

14 (a) Differentiate 2 +3√x

with respect to x. (3)

(b) Find

∫(2− x)3 dx. (3)

1996-6 Paper(2) Q.2

15 Find

(a)

∫6 sin 3x dx, (1)

(b)

∫(e2x + 1)2 dx (3)


1997-6 Paper(2) Q.10

16 Using the identity cos (A+B) = cosA cosB − sinA sinB,

(a) solve, in radians to 3 decimal places, for 0 6 x < 2π, the equation

(4)

cos (x+ π3 ) = 3 sinx,

(b) show that cos 2x = 1− 2 sin2 x, (3)

(c) solve, in radians to 3 decimal places, for 0 6 x < 2π, the equation

cos 2x = 2 sinx.

(4)

(d) Find∫

sin2 xdx. (2)

(e) Use your answer to (d) to evaluate ∫0

π3 sin2 x dx,

giving your answer in terms of π. (2)

2013-6 Paper(2) Q.3

17 (a) (i) Find

∫ (1 + 3x− 2

x2

)dx

(ii) Hence show that

∫ 2

1

(1 + 3x− 2

x2

)dx = 4

1

2(4)

(b) (i) Find

∫3 sin 2x dx


∫ π6

03 sin 2x dx =

3

4(4)


2014-6 Paper(2) Q.10

18 Using the identities cos (A+B) = cosA cosB − sinA sinB


(a) (i) show that cos 2A = 1− 2 sin2A

(ii) write down an expression for sin 2A in terms of sinA and cosA (4)

(b) Hence show that sin 3A = 3 sinA− 4 sin3A (4)

(c) Solve, for 0 6 x 6 π, the equation 16 sin3 x− 12 sinx+ 1 = 0

Give your answers correct to 3 significant figures. (4)

(d) Find

∫(24 sin3 θ + 6 cos θ) dθ (2)

(e) Hence evaluate

∫ π3

0(24 sin3 θ + 6 cos θ) dθ, giving your answer in the form a+ b

√c, where a,

b and c are integers. (2)


2015-6 Paper(1) Q.8




(ii) express sin 2A in terms of sinA and cosA, simplifying your answer. (4)


(c) Solve, for −90◦ 6 A 6 90◦, the equation

8 sin3A− 6 sinA = 1

(4)

(d) (i) Find

∫sin3 θ dθ

(ii) Evaluate

∫ π4

0sin3 θ dθ, giving your answer in the form

a− b√

2

c, where a, b and c are

integers. (5)

2016-1 Paper(1) Q.1

20 f(x) = 3x3 + 2 sinx− 4

x2where x 6= 0

(a) Find f ′(x) (3)

(b) Find∫

f(x) dx (4)


2016-6 Paper(2) Q.9

21 sin (A+B) = sinA cosB + cosA sinB

cos (A+B) = cosA cosB − sinA sinBUsing the above identities

(a) show that cos 2θ = 2 cos2 θ − 1 (3)

(b) find a simplified expression for sin 2θ in terms of sin θ and cos θ (1)

(c) show that cos 3θ = 4 cos3 θ − 3 cos θ (4)

Hence, or otherwise,

(d) solve, for 0 6 θ 6 π giving your answer in terms of π, the equation

6 cos θ − 8 cos3 θ + 1 = 0

(4)

(e) find

(i)

∫(8 cos3 θ + 4 sin θ) dθ

(ii) the exact value of

∫ π3

0(8 cos3 θ + 4 sin θ) dθ (4)


2017-6 Paper(1) Q.9

22 Using

cos (A+B) = cosA cosB − sinA sinB

(a) show that cos2 θ =1

2(cos 2θ + 1) (2)

f(θ) = 8 cos4 θ + 4 cos2 θ − 5

(b) show that f(θ) = cos 4θ + 6 cos 2θ (4)

Hence

(c) solve, for 0◦ 6 x < 180◦, the equation

8 cos4 x+ 4 cos2 x− 6 cos 2x = 4.5

(4)

(d) find

(i)∫

f(θ) dθ

(ii) the exact value of∫ π

30 f(θ) dθ (5)


4 Arc and Sector

2011-6 Paper(2) Q.6

1

O

A

B

8 cm

6 cm

Figure 1

Figure 1 shows a circle, centre O, with radius 8 cm. The arc AB has length 6 cm.

(a) Find, in radians, the size of angle AOB. (2)

(b) Find the area of the sector AOB. (2)

(c) Find, to 3 significant figures, the area of the shaded segment. (3)


2012-1 Paper(2) Q.4

2

O B

A

1.2 rad

Figure 2

Figure 2 shows an arc AB of a circle with centre O. The arc subtends an angle of 1.2 radians at

O and the area of the sector AOB is 15 cm2.

Find

(a) the radius of the circle, (2)

(b) the length of the arc AB, (2)

(c) the area of the shaded segment, giving your answer to 3 significant figures. (3)


2012-6 Paper(1) Q.6

3

O

P

Q

r cmθ rad

Figure 1

The points P and Q lie on the circumference of a circle with center O and radius r cm. Angle

POQ = θ radians. The segment shaded in Figure 1 has area A cm2.

(a) Show that A =1

2r2(θ − sin θ) (3)

When angle POQ is increased to (θ + δθ) radians, where δθ is small, the area of the shaded

segment is increased to (A+ δA) cm2, where δA is small.

(b) Show that δA =1

2r2(1− cos θ)δθ (3)

For a circle of radius 4 cm, the area of the shaded segment is increased by 0.05 cm2 when angle

POQ increases by 0.02 radians.

(c) Find, to 1 decimal place, an estimate for θ (4)


2013-1 Paper(2) Q.1

4

O

A

B

8 cm

2 cm

2θ

Figure 1

Figure 1 shows the sector, AOB of a circle with centre O and radius 8 cm. A circle of radius 2

cm touches the lines OA and OB and the arc AB. Angle AOB is 2θ radians, 0 < θ <π

4.

(a) Find, to 4 significant figures, the values of θ (3)

(b) Find, to 3 significant figures, the area of the region shaded in Figure 1. (3)

2013-6 Paper(1) Q.1

5 A circle has centre O and radius 12 cm. The sector AOB of the circle has area 126 cm2. Find the

length of the arc AB. (4)


2014-1 Paper(2) Q.4

6

O

A

B10 cm

θ rad

Figure 1

Figure 1 shows a sector of a circle of radius 10 cm and centre O. The area of triangle OAB is 20

cm2 and the size of the angle AOB is θ radians.

Find, to 3 significant figures,

(a) the value of θ, (2)

(b) the length of the arc AB, (2)

(c) the area of the shaded segment. (3)


2014-6 Paper(2) Q.1

7

O

A

B

6 cm

Figure 1

Figure 1 shows the sector OAB of a circle. The circle has centre O and radius 6 cm. The area of

the sector 12 cm2.

(a) Find, in radians, the size of angle AOB. (2)

(b) Find, cm, the length of the arc AB. (2)


2015-1 Paper(1) Q.4

8

O

A

B

5 cm

1.8 radians

Figure 2

Figure 2 shows the sector AOB of a circle of radius 5 cm. The centre of the circle is O and the

angle AOB is 1.8 radians.

(a) Find the length of the arc AB. (1)

(b) Find the area of the sector AOB. (2)


2015-6 Paper(2) Q.6

9

O

D

A

C

B

θ rad

Figure 1

Figure 1 shows a sector OAB of the circle with centre O and radius 10 cm.

The points C and D lie on OB and OA respectively and CD is an arc of the circle with centre O

and radius 6 cm. The size of angle AOB is θ radians. The shaded region is bounded by the arcs

AB and CD and the lines AD and BC.

The area of the shaded region is S cm2.

(a) Show that S = 32θ. (3)

The size of angle AOB is increasing at a constant rate of 0.2 rad/s.

(b) Find the rate of increase of S. (2)

When the area of the shaded region is 20 cm2

(c) calculate the perimeter of the shaded region. (5)


2016-1 Paper(2) Q.2

10 The sector OAB of a circle, centre O, has area 48 cm2.

The length of the arc AB is 8 cm and the size of angle AOB is θ radians.

Find

(i) the radius of sector OAB

(ii) the value of θ (5)

2017-1 Paper(1) Q.1

11

θ

r cm

Figure 1

Figure 1 shows a sector of a circle. The circle has radius r cm and the sector has angle θ radians.

The sector has an arc length of 18π cm and an area of 126π cm2.

Find

(i) the value of r,

(ii) the exact value of θ. (5)


2018-1 Paper(2) Q.1

12

O

A

B

12 cm

θ radians

Figure 1

Figure 1 shows the sector AOB of a circle with center O and radius 12 cm. The angle AOB is θ

radians and the area of the sector is 192 cm2

Calculate

(a) the value of θ, (2)

(b) the length, in cm, of the arc AB. (2)


5 Area Below Curve

1988-1 Paper(2) Q.14

1

x

y

y = sin 2xy = sin (2x+π

3)

O A

B

The figure shows the graphs of

y = sin 2x and y = sin (2x+π

3)

in the interval −π66 x 6

π

2.

(a) Determine the coordinates of the point A. (3)

(b) By using the basic addition formula

sin (P +Q) = sinP cosQ+ cosP sinQ,

show that the x-coordinate of the point B isπ

6. (7)

(c) Calculate the area of the shaded region. (5)


1988-6 Paper(1) Q.9

2

x

y

O

A

P

R

Fig. 1

Figure 1 shows the curve y = x2 − 1 and the line y = 14− 2x. The line cuts the y-axis at A and

the curve at P .

(a) Show that the coordinates of P are (3, 8). (2)

(b) Calculate the area of the shaded region R. (7)

The normal to the curve at P meets the y-axis at N .

(c) Calculate the area of 4APN . (6)


1989-6 Paper(1) Q.11

3

x

y

PO

Q

y = 4x3 + x2 − 2x+ 1

y = 8x2 + 1

A

B

C

Fig. 1

Figure 1 shows, for x > 0, the graphs of y = 8x2 + 1 and y = 4x3 + x2 − 2x + 1. The curvesintersect at the points P and Q.

(a) State the coordinates of P and show that the x-coordinate of Q is 2. (3)

Region A is bounded by the curve with equation y = 4x3 + x2 − 2x+ 1, the x-axis and the linesx = 0 and x = 2.

(b) Show that the area of region A is 1623 . (3)

Region B lies in the first quadrant and is bounded by the two curves.

(c) Calculate the area of region B. (4)

Region C is bounded by the curve with equation y = 8x2 + 1, the y-axis and the line through Qparallel to the x-axis.

(d) Calculate, in terms of π, the volume generated when region C is rotated through 360◦ aboutthe y-axis. (5)

1989-6 Paper(2) Q.6

4 Find the area of the finite region bounded by the x-axis, the lines x = −1 and x = 1 and the

curve with equation y = e4x, giving your answer to 3 significant figures. (5)


1990-1 Paper(2) Q.11

5

O A x

y

D

B

C

S

R

y = 2x(3− x)

Fig. 1

The curve with equation y = 2x(3− x) crosses the x-axis at O and A.

(a) State the coordinates of the point A. (1)

A straight line, which crosses the y-axis at the point B with coordinates (0, 5), meets the curveat the points C and D, as shown in Fig. 1. The coordinates of the point D are (k, k).

(b) Show that the value of k is 212 . (1)

(c) Find the equation of the straight line passing through B and D. (2)

(d) Show that the x-coordinate of the point C is 1. (2)

(e) Calculate the area of the shaded region R. (5)

(f) Calculate, in terms of π, the volume generated when the shaded region S, bounded by thecurve, the x-axis and the line x = 1, is rotated through 360◦ about the x-axis. (4)


1990-1 Paper(2) Q.12




cos 2x = 1− 2 sin2 x.

(3)

(c) Show that

∫2 sin2 x dx = x− 1


O x

y

P

Q

R

π

Fig. 2









1990-6 Paper(1) Q.9

7

O A C x

y

B

x+ y = 2

y2 = 3x− 2

Fig. 3

Figure 3 shows, for x > 0, y > 0, the curve with equation y2 = 3x− 2 and the line with equation

x + y = 2. The curve meets the x-axis at the point A. The line meets the curve at the point B,

and the x-axis at the point C.

(a) State the coordinates of A and C. (2)

(b) Calculate the coordinates of B. (3)

(c) Calculate the area of the region bounded by the curve, the x-axis, the y-axis and the line

through B parallel to the x-axis. (4)

(d) Calculate, in terms of π, the volume of the solid generated when the shaded region bounded

by the curve, the line x+ y = 2 and the x-axis is rotated through 360◦ about the x-axis. (6)


1991-1 Paper(1) Q.12

8

O A C

B

x

y

y = x2 − 4x+ 3

D

Fig. 2

Figure 2 shows part of the curve with equation

y = x2 − 4x+ 3,

cutting the x-axis at A.

(a) Calculate the coordinates of A. (2)

The curve has gradient 4 at the point B.


The normal at B cuts te x-axis at C and intersects the curve again at D.

(c) Find an equation of the line BC. (2)

(d) Find the coordinates of C and D. (4)

(e) Calculate the area of the shaded region. (5)


1991-6 Paper(2) Q.7

9 (a) Differentiate x3 cos 2x with respect to x. (3)

(b) Calculate the area of the region bounded by the curve with equation y = e12x, the y-axis, the

x-axis and the line x = ln 4. (3)

1991-6 Paper(2) Q.12

10 The line with equation y = x cuts the curve C with equation 8y = x2 at the origin O and at the

point P .

(a) Find the coordinates of P . (1)

(b) Calculate the area of the finite region bounded by C and the line. (4)

The region bounded by C and the line y = h is rotated through 180◦ about the y-axis.

(c) Show that the volume of the solid generated is 4πh2. (2)

A set of coordinate axes are graduated with each unit being 1 cm and the point P defined as

above. A bowl is made with an inner surface in the shape formed by rotating the arc OP of the

curve with equation 8y = x2 through 360◦ about the y-axis. The bowl is held with the vertical

through O being an axis of symmetry and water is poured in at a rate of 24 cm3/s.

(d) At the instant when the depth of water is 2 cm calculate, leaving your answer in terms of π,

the rate, in cm/s, at which the depth of water is increasing. (5)

(e) Calculate, to the nearest second, the time taken to fill the bowl. (3)


1992-1 Paper(1) Q.12

11 f(x) = x3 + 3x2 − 24x+ 28.

(a) Show that (x− 2) is a factor of f(x).

(b) Show further that f(x) = (x− 2)2(x+ 7).

(c) Find the coordinates of the maximum and minimum points on the curve with equation

y = f(x), distingusing between them.

(d) Sketch the graph of y = f(x) showing clearly the coordinates of

(i) the turning points,

(ii) the points in which the curve meets the axes.

(e) Calculate the area of the finite region bounded by the curve and the x-axis.


1993-6 Paper(1) Q.12

12

O x

y

A

D

B

C

Fig. 2

Figure 2 shows part of the curve with equation y = 7 + 6x − x2. The point A has coordinates

(0, 19) and AD is parallel to the tangent to the curve at the point on the curve at which x = 4.

AD cuts the curve at B and C.

(a) Show that an equation of the line AD is y + 2x = 19. (4)

(b) Find the coordinates of B and C. (4)



1993-6 Paper(2) Q.11

13 f(x) = 8e3x + 27e−3x.

(a) Find the rate of change of f(x) with respect to x when x = 12 , giving your answer to 2 decimal

places. (4)

(b) Find the area of the finite region bounded by the curve with equation y = f(x), the x-axis

and the lines x = 13 and x = −1

3 , giving your answer in terms of e. (5)

(c) By using the substitution z = e3x solve, for x > 0, the equation f(x) = 35, giving your answer

to 2 decimal places. (6)

1994-1 Paper(1) Q.14


dy

dx= 6x(x+ 1).


(a) Find f(x). (4)









1995-1 Paper(1) Q.9

15

O

A

B

x

y

Fig. 2

Figure 2 shows part of the curve with equation y = 6x− x2. The point A has coordinates (2, 8).

The normal to the curve at A intersects the curve again at B. Find

(a) an equation of the line AB, (4)

(b) the coordinates of B. (5)

The shaded region is bounded by the curve and the line AB.

(c) Show that the area of the shaded region is 22948 . (6)


1995-6 Paper(2) Q.12

16 f(x) = 2x2 + 3x+ 4.


Hence






(f) Find

∫f(x) dx. (2)




1996-1 Paper(1) Q.10

17

O x

y

P

Q

A

Fig. 2

Figure 2 shows the line with equation y = 2x− 2 and the curve with equation y = −x2 + 3x+ 4

which intersect at the points P and Q.

(a) Find the coordinates of P and Q. (4)

The shaded region A is bounded by the curve, the line and the x-axis.

(b) Calculate the area of A. (11)


1996-6 Paper(1) Q.10

18

O

A

x

y

P

Fig. 2

Figure 2 shows part of the curve C with equation y = 2x3 − 13x2 + 24x. The point P , on C, has

coordinates (1, 13) and the normal to the curve at P cuts the y-axis at A.

(a) Find an equation of the line AP . (6)

(b) Write down the coordinates of A. (2)

(c) Calculate, as an exact fraction, the finite area bounded by C, AP and the y-axis. (7)


1997-6 Paper(1) Q.9

19

O

A

B

x

y

Fig. 2

Figure 2 shows part of the curve, C, with equation y = 6x− 2x2. The curve C cuts the x-axis at

O and A. The normal, l, to C at A cuts C again at the point B.

(a) Write down the coordinates of A. (1)

(b) Obtain an equation for l. (5)

(c) Find, as an exact fraction, the x-coordinate of B. (4)

The shaded region is bounded by C, l and the y-axis.

(d) Show that the area of the shaded region is 934 . (5)


1997-6 Paper(2) Q.5

20

O A x

y

R

Fig. 1

Figure 1 shows part of the curve with equation y = k cos 3x, where x is in radians and k is a

constant. The curve cuts the x-axis at the point A.


(b) Find, as a multiple of k, the area of the shaded region R bounded by the x-axis, the y-axis

and the curve. (4)


2007-6 Paper(1) Q.10

21

O RP Q

S

T

x

yl

Figure 2

Figure 2 shows the curve with equation y = k+ 7x− x3, where k is a constant. The curve crosses

the x-axis at the points P , Q, and R. Given that R has coordinates (3, 0), find

(a) the value of k, (2)

(b) the coordinates of P and the coordinates of Q. (3)

The curve crosses the y-axis at the point S. The line l passes through P and S.

(c) Find an equation for l. (3)

The line l meets the curve again at the point T .

(d) Find the coordinates of T . (3)

(e) Calculate the area of the region shown in Figure 2. (7)


2013-1 Paper(1) Q.11

22 f(x) = x3 + px2 + qx+ 6 p, q ∈ ZGiven that f(x) = (x− 1)(x− 3)(x+ r)

(a) find the value of r. (1)


(b) find the value of p and the value of q. (3)

O

A

1 3

B

x

y

C

Figure 2

Figure 2 shows the curve C with equation y = f(x) which crosses the x-axis at the points with

coordinates (3, 0) and (1, 0) and at the point A. The point B on C has x-coordinate 2

(c) Find an equation of the tangent to C at B. (5)

(d) Show that the tangent at B passes through A. (2)

(e) Use calculus to find the area of the finite region bounded by C and the tangent at B. (5)


2013-6 Paper(1) Q.8

23

OA B

R

S

lC

x

y

Figure 2

Figure 2 shows the curve C with equation y = 15 + 2x− x2

The curve crosses the x-axis at the points A and B.

(a) Find the x-coordinate of A and the x-coordinate of B. (3)

(b) Use calculus to find the area of the finite region bounded by C and the x-axis. (4)

The line l with equation y = x+ 9 intersects C at the points R and S.

(c) Find the x-coordinate of R and the x-coordinate of S. (3)

(d) Use calculus to find the area of the region bounded by C, the line l and the x-axis, shown

shaded in Figure 2. (4)


2016-1 Paper(1) Q.9

24

OA B x

y

S

Figure 2

Figure 2 shows the curve S with equation y = 8− 2x− x2

The curve S crosses the x-axis at the points A and B.


(b) Use calculus to find the area of the finite region bounded by S and the x-axis. (4)

The curve T with equation y = x2 + x+ 6 intersects S.

(c) Find the x-coordinates of the points of intersection of S and T . (2)

(d) Use calculus to find the a area of the finite region bounded by S and T . (4)


6 Area between Curves

1990-1 Paper(1) Q.8

1

x

y

O

B

y = 2x

A

y = 6x− x2

Fig. 1

Figure 1 shows the curve with equation y = 6x− x2 and the straight line y = 2x. Find

(a) the coordinates of the point B, (2)

(b) the coordinates of the point A, (2)

(c) the area of the shaded region. (3)


2008-1 Paper(1) Q.8

2

O x

yP

Q

Figure 1

Figure 1 shows the curve with equation y = f(x) where f ′(x) = 3x2− 4x− 4. Given that the curve

passes through the point with coordinates (1, 0).

(a) find f(x). (3)

The curve has a maximum point at P and a minimum point Q.

(b) Find the exact value of the coordinates of

(i) P , (ii) Q. (3)

(c) Write down an equation for

(i) the tangent at P ,

(ii) the normal at Q. (2)

(d) Find the exact value of the finite area enclosed by the curve between the points P and Q,

the tangent at P and the normal Q. (7)


2010-6 Paper(1) Q.9

3

O x

y

P Q R

C

S

Figure 1

f(x) = x3 + ax2 + bx+ d, a, b, d ∈ Z

Figure 2 shows the curve C with equation y = f(x).

The curve C crosses the x-axis at the points P , Q, and R.

The x-coordinates of P , Q and R are −3,−1 and 2 respectively.

The point S on C has x-coordinate −2.

(a) Find the value of a, the value of b and the value of d. (4)

The line l is the tangent to C at S.

(b) Find an equation for l giving your answer in the form y = px+ q. (5)

(c) Hence show that l passes through R. (1)

(d) Find the area of the region enclosed by C and l. (5)


2012-6 Paper(2) Q.5

4 The curve R has equation y = x2 − 7x+ 10

The curve S has equation y = −x2 + 7x− 2

(a) Find the coordinates of each of the two points where the curves R and S intersect. (4)

(b) Find the area of the finite region bounded by the curve R and the curve S. (4)

2014-6 Paper(2) Q.9

5 f(x) = x3 + 5x2 + px− q p, q ∈ ZGiven that (x+ 2) and (x− 1) are factors of f(x),

(a) form a pair of simultaneous equations in p and q, (2)

(b) show that p = 2 and find the value of q, (3)

(c) factorise f(x) completely. (1)

(d) Sketch the curve with equation y = f(x) showing the coordinates of the points where the

curve crosses the x-axis. (2)

The curve with equation y = x3 + 2x2 + 4x meets the curve with equation y = f(x) at two points

A and B. The x-coordinate of A is −4

3and the x-coordinate of B is 2

(e) Use algebraic integration to find, to 3 significant figures, the area of the finite region bounded

by the two curves. (5)


2015-1 Paper(1) Q.7

6 The curve C has equation y = x2 + 3

The point A with coordinates (0, 3) and the point B with coordinates (4, 19) lie on C, as shown

below in Figure 3.

O

A

B

x

y

C

Figure 3

The finite area enclosed by the arc AB of curve C, the axes and the line with equation x = 4 is

rotated through 360◦ about the x-axis.

(a) Using algebraic integration, calculate, to 1 decimal place, the volume of the solid generated.

(6)

O

A

B

x

y

C

Figure 4

(b) Using algebraic integration, calculate the region between the chord AB and the arc AB of

C, shown shaded in Figure 4. (6)


2015-6 Paper(2) Q.8

7

O

C

2 4x

y

l

R

Figure 3

Figure 3 shows part of the curve C with equation y = x3 + ax2 + bx+ c

The curve passes through the origin O and the points with coordinates (2, 0) and (4, 0).

(a) Show that c = 0 (1)

(b) Find the value of a and the value of b. (3)

The point P with x-coordinate 3 lies on C. The line l passes through O and meets C at P .

(c) Show that l is the tangent to C at P . (4)

The finite region R, shown shaded in Figure 3, is bounded by C and by l.

(d) Use algebraic integration to find the area of R. (5)


2016-1 Paper(1) Q.9

8

OA B x

y

S

Figure 2

Figure 2 shows the curve S with equation y = 8− 2x− x2

The curve S crosses the x-axis at the points A and B.


(b) Use calculus to find the area of the finite region bounded by S and the x-axis. (4)

The curve T with equation y = x2 + x+ 6 intersects S.

(c) Find the x-coordinates of the points of intersection of S and T . (2)

(d) Use calculus to find the a area of the finite region bounded by S and T . (4)

2016-6 Paper(2) Q.6

9 (a) Use algebra to find the coordinates of the points of intersection of the curve with equation

y = x2 + 2x− 6 and the line with equation y = 5x+ 4 (5)

(b) Use algebra integration to find the exact area of the finite region bounded by the curve and

the line. (5)


2017-6 Paper(2) Q.9

10

OA B(2, 0) x

y

C

Figure 1

The curve C with equation y = x3−4x2−4x+16 crosses the x-axis at the point with coordinates

(2, 0) and at the points A and B, as shown in Figure 1. The coordinates of the points A and B

are (a, 0) and (b, 0) respectively.

(a) Find the value of a and the value of b. (4)

The point D lies on C and has x coordinate 0

The line l is the tangent to C at the point D.

(b) Find an equation of l. (5)

(c) Show that l passes through B. (1)

(d) Use algebraic integration to find the area of the finite region bounded by l and C. (5)


2018-1 Paper(2) Q.9

11

O (a, 0) (b, 0)(−2, 0) x

y

C

P

Figure 5

Figure 5 shows the curve C with equation y = x3 − 2x2 − 5x+ 6

The curve C crosses the x-axis at the points with coordinates (−2, 0), (a, 0) and (b, 0)

(a) (i) Show that a = 1

(ii) Find the value of b. (4)

The point P on C has x-coordinate 2 and the line l is the tangent to C at P .

(b) Show that l crosses the x-axis at the point with coordinates (−2, 0) (6)

(c) Use algebraic integration to find the exact area of the finite region bounded by C and l. (4)


7 Arithmetic Series

1988-6 Paper(1) Q.10

1 (a) In a geometric series, (x − 1), (x + 1) and (x + 9) are consecutive terms. Find the value of

the common ratio of the series. (5)

The sum of the first n terms of an arithmetic series is 6014. Given that the sum of the first term

and the nth term is 124.

(b) calculate the value of n, (3)

(c) show that the value of the 49th term is 62. (2)

Given also that the value of the 61st term is 77, find

(d) the value of the first term of the series, (2)

(e) the common difference, (1)

(f) the sum of the first 80 terms. (2)


1988-6 Paper(2) Q.9

2 (a) Write down the common ratio of the geometric series G,

e+ e12 + 1 + ... .

(2)

(b) Calculate, to 3 significant figures, the sum of the first six terms of the series. (3)

(c) Write down, in its simplest form, the common difference of the arithmetic series A,

log3 2 + log3 6 + log3 18 + ... .

(3)

(d) Show that the sum of the first ten terms of A is 10 log3 2 + 45 and evaluate this to 2 decimal

places. (4)

(e) One of these two series has a sum to infinity. Calculate, to 2 decimal places, this sum. (3)

1989-1 Paper(1) Q.4

3 The first three terms of an arithmetic series are 2, 412 and 7.

Find

(a) the 50th term of the series, (3)

(b) the sum of the first 50 terms. (2)


1989-1 Paper(1) Q.11

4 The first three terms of a series, S, are (m− 4), (m+ 2) and (3m+ 1).

Given, also, that S is an arithmetic series,

(a) find m. (2)

Using your value of m,

(b) write down the first four terms of the arithmetic series. (1)

Given, instead, that S is a geometric series,

(c) find the two possible values of m, (5)

(d) write down the first four terms of each of the two geometric series obtained with your values

of m, (2)

(e) state the value of the common ratio of each of the series. (2)

One of these two geometric series has a sum to infinity.

(f) Find the sum to infinity of that series. (3)

1989-6 Paper(1) Q.10

5 (a) The second term of an arithmetic series is −2. The sum of the first and seventh terms of

the series is equal to the sum of the first eight terms. Find the value of the first term of the

series. (7)

The numbers1

t,

1

t− 1and

1

t+ 2are the first three terms of a geometric series.

(b) Find the value of t. (4)

(c) Show that the common ratio of this series if −13 . (2)

(d) Deduce the sum to infinity of the series. (2)


1990-1 Paper(1) Q.13

6 The sixth term of an arithmetic series is 25 and the eighteenth term is −11.

(a) Find the first term and the common difference of the series. (5)

(b) Find the smallest value of n for which the sum of the first n terms of the series is negative.

(4)

(c) After n years £300 invested at 8% per annum compound interest amounts to £300(1.08)n.

A man invests £300 on January 1st each year for 10 consecutive years at this rate, leaving

his money to accumulate. Find, to the nearest £, the total sum due to him to January 1st

of the eleventh year. (6)

1990-6 Paper(1) Q.11

7 The first term of an arithmetic series is a and the common difference is d. The first, fourth and

sixth terms of this series are also the first three terms of a geometric series.

(a) Show that a+ 9d = 0. (4)

The sixth term of the arithmetic series is −6.

(b) Calculate the values of a and d. (4)

(c) Find the sum of the first 50 terms of the arithmetic series. (3)

(d) Calculate the value of the common ratio of the geometric series. (2)

(e) Find the sum to infinity of the geometric series. (2)


1991-1 Paper(1) Q.9

8 The rth term of an arithmetic series is giving by

2r − 5, r = 1, 2, 3, ... .

(a) Write down the first four terms of this series and find

Sn =n∑r=1

(2r − 5)

in terms of n. (7)

Given that Sn = 165 find the value of n.

(b) In a geometric series, (x + 1), (x + 3) and (x + 4) are the first, second and third terms

respectively. Calculate the value of x and hence write down the numerical values of the

common ratio and the first term of the series.

Calculate the numerical value of the sum to infinity of the series. (8)

1991-1 Paper(2) Q.10

9 The first three terms of an arithmetic series are lg x, lg 2(x+ 1) and lg 4(x+ 6) respectively.

(a) Find the value of x. (4)

(b) Find the value of the common difference of this series. (2)

The equation of a curve is y = e−x sinx.

(c) Show that the values of the x-coordinates of the turning points of the curve are the solutions

of tanx = 1. (3)

The points A, B and C on the curve have x-coordinatesπ

4,

5π

4and

9π

4respectively.

(d) Show that the y-coordinates of the points A, B and C respectively are consecutive terms of

a geometric series. (4)

(e) State the value of the common ratio of this series. (2)


1991-6 Paper(1) Q.10

10 The first three terms of an arithmetic series of positive terms have sum 24 and product 440. Find

(a) the values of each of the first three terms, (6)


The first three terms of a geometric series have sum 21 and product 216. Given that the terms

are increasing, find

(c) the value of each of the first three terms, (6)

(d) the sum of the first 15 terms. (2)

1992-1 Paper(1) Q.11

11 The sum of the first two terms of a geometric series of positive terms is 825 , and the sum to infinity

of the series if 834 . For this series find the value of

(a) the first term,

(b) the common ratio.

(c) An arithmetic series and a geometric series each have a second term equal to 6 and a third

term equal to q. The first term of the arithmetic series is (2p − 15); the first term of the

geometric series is p. Given that the first term of both series is positive, find the values of p

and q.


1994-1 Paper(1) Q.10

12 The seventh term of an arithmetic series is 712 . The sum of the third term and the fifth term of

the series of zero.

Find

(a) the value of the first term of the series, (4)

(b) the sum of the first sixteen terms of the series. (3)

The numbers t, (t− 7) and9

tare the first three terms of a geometric series in which all the terms

are positive.

Calculate

(c) the value of t, (4)

(d) the sum to infinity of the series. (4)


1995-1 Paper(1) Q.13

13 The sum of the first four terms of an arithmetic series is 20 and the fifth term is five terms the

second term. Given that the first term of the series is a and the common difference of the series

is d.

(a) form a pair of simultaneous equation in a and d, (2)

(b) solve these equations to find a and d, (3)

(c) find the sum of the first 40 terms of the series. (2)

Given that (p2 − 4), (5p− 2) and 27 are, respectively, the first three terms of a geometric series,

(d) find the two possible values of p. (4)

(e) For each of your values of p, write down the first three terms of the series. (2)

Given also that the series has a sum to infinity, find

(f) the common ratio of the series. (2)


1996-1 Paper(1) Q.9

14 Given that x, (x + y), and (2x + 2) are respectively the first three terms of an arithmetic series

and that 2x, x, and 2(x − y), x 6= 0, are respectively the first three terms of a geometric series,

show that

(a) the common ratio of the geometric series is 0.5. (2)

Find

(b) the value of x, (4)

(c) the value of y, (2)

(d) the sum to infinity of the geometric series, (2)

(e) the sum of the first 121 terms of the arithmetic series, (3)

(f) the value of the 87th term of the arithmetic series. (2)

1996-6 Paper(1) Q.4

15 The third term of an arithmetic series is 10 and the ninth term is 46. Find

(a) the common difference of the series, (2)

(b) the first term of the series, (1)

(c) the sum of the first 12 terms of the series. (2)


1997-6 Paper(1) Q.14

16 Sn =n∑r=1

(50− 4r).

(a) Write down the first three terms of the series. (1)

(b) Calculate the value of S20. (3)

(c) Find n such that Sn = 0. (3)

The sum of the first four terms of a geometric series of positive terms is 78.336 and the sum to

infinity of the series is 90. Calculate

(d) the common ratio of the series, (4)

(e) the first term of the series, (2)

(f) the difference, to 3 significant figures, between the sum to infinity of the series and the sum

of the first thirty terms. Give your answer in standard form. (2)

2007-1 Paper(1) Q.5

17 (a) Find, in terms of n,n∑r=1

(7r − 3). (3)

(b) Hence, or otherwise, evaluate30∑r=15

(7r − 3). (3)

Given thatn∑r=1

(7r − 3) = 1020,

(c) find the value of n. (3)


2007-1 Paper(2) Q.2

18 The sum of the first 10 terms of an arithmetic series is 295, and the sum of the first 8 terms of

the same series is 196. Find


(b) the first term of the series. (1)

2007-6 Paper(1) Q.8

19 The 15th term of an arithmetic series is 46. The sum of the first 20 terms is 650. Find



(c) the least number of terms for which the sum of the series is greater than 1000. (4)

Given that the series has 40 terms,

(d) find the sum of the last 10 terms of the series. (3)


2008-1 Paper(2) Q.5

20 The fourth term of an arithmetic series is four times the eighth term. The sum of the first four

terms is 164. Find

(a) the common difference of the series. (4)


The sum of the first n terms of the series if Sn.

(c) Find the greatest value of n for which Sn is positive. (4)

2008-6 Paper(2) Q.2

21 Find the sum of all the integers from 5 to 195 inclusive which are not multiples of 5. (5)

2008-6 Paper(2) Q.9

22 The third and fifth terms of an arithmetic series are given by log pq4 and log pq8 respectively,

q 6= 1.

The common difference of the series is b log q, where b is a constant. Find

(a) the value of b, (3)


The sum of the first n terms of the series can be written in the form s log pqr.

(c) Express r and s in terms of n. (4)

Given that the sum of the first 16 terms of the series is 10 times the sum of the first 4 terms of

the series.

(d) show that log p = 5 log q. (4)


2009-6 Paper(1) Q.7

23 In an arithmetic series, the sum of the sixth and seventh terms is equal to five times the sum of

the first and second terms. The fourth term of the series is 15.

(a) Find, for the series

(i) the first term,

(ii) the common difference. (5)

(b) Find the sum of the 10th to 25th terms inclusive of the series. (4)

2010-1 Paper(1) Q.7

24 The sum, Sn, of the first n terms of an arithmetic series is given by Sn = n4 (13 + 7n).

Find

(a) the first term of the series, (1)

(b) the rth term of the series, (3)

(c) the common difference of the series. (2)

The pth term of the series is a multiple of the first term.

(d) Given that p 6= 1, find the least value of p. (3)


2010-6 Paper(1) Q.5

25 The sum, Sn, of the first n terms of an arithmetic series is given by Sn = n(2n+ 1).

For this series,

(a) prove that the first term is 3, (2)

(b) find the common difference, (3)

(c) find the value of the 25th term. (2)

2011-1 Paper(1) Q.10

26 The 9th term of an arithmetic series is five times the second term.

(a) Show that the 19th term is five times the 4th term. (4)

Given also that the sum of the first 12 terms of the series is 300,

(b) show that the first term is 3. (3)

(c) find the common difference of the series. (1)

The sum of the first p terms of the series is less than 2000.

(d) Find the greatest value of p. (5)

2011-1 Paper(2) Q.3

27 (a) Show thatn∑r=1

r =n(n+ 1)

2(1)

(b) Hence or otherwise find the sum of all the integers from 1 to 250 inclusive which are not

multiples of 3. (4)


2011-6 Paper(1) Q.6

28 The third term of an arithmetic series is 70 and the sum of the first 10 terms of the series is 450

(a) calculate the common difference of the series. (4)

The sum of the first n terms of the series is Sn

Given that Sn > 350

(b) find the set of possible values of n. (6)

2011-6 Paper(2) Q.8

29 The sum of the first and third terms of a geometric series is 100

The sum of the second and third term is 60

(a) Find the two possible values of the common ratio of the series. (5)

Given that the series is convergent, find


(c) the least number of terms for which the sum is greater than 159.9 (4)


2012-1 Paper(2) Q.10

30 The sum of the first and third terms of a geometric series G is 104

The sum of the second and third terms of G is 24

Given that G is convergent and that the sum to infinity is S, find

(a) the common ratio of G (4)

(b) the value of S (4)

The sum of the first and third terms of another geometric series H is also 104 and the sum of the

second and third terms of H is 24

The sum of the first n terms of H is Sn

(c) Write down the common ratio of H (1)

(d) Find the least value of n for which Sn > S (6)

2012-6 Paper(1) Q.5

31 The first four terms of an arithmetic series, S, are

loga 2 + loga 4 + loga 8 + loga 16

(a) Write down an expression for the rth term of S. (1)

(b) Find an expression for the common difference of S. (2)

The sum of the first n terms of S is Sn

(c) Show that Sn =1

2n(n+ 1) loga 2 (2)

The first four terms of a second arithmetic series, T , are


The sum of the first n terms of T is Tn

(d) Find Tn − Sn and simplify your answer. (4)


2013-1 Paper(1) Q.9

32 The sum Sn of the first n terms of an arithmetic series is given by Sn = n(2n+ 3). The first term

of the series is a.

(a) Show that a = 5 (2)

(b) Find the common difference of the series. (3)

(c) Finn the 12th term of the series. (2)

Given that 1 + Sp+4 = 2Sp

(d) find the value of p. (4)

2013-6 Paper(1) Q.7

33 An arithmetic series has first term a and common difference d. The nth term of ther series is tn

and the sum of the first n terms of the series is Sn

(a) Write down an expression in terms of a and d for

(i) t58

(ii) S13 (2)

Given that t58 = S13

(b) show that d = −4

7a (2)

(c) show that t176 = S21 (4)

(d) find the value of r when tr = 5t9 (3)


2014-1 Paper(1) Q.4

34 The sum of the first n terms of an arithmetic series is 2n(n+ 3)

Find


(b) the common difference of the series, (3)

(c) the 25th term of the series. (2)

2014-6 Paper(1) Q.4

35 The 3rd term of an arithmetic series is 108 and the 12th term is 54

Find




(c) Show that Sn = 3n(41− n) (3)

Given that Sn = 1200

(d) find the two possible values of n. (4)


2015-1 Paper(2) Q.7

36 The first term of an arithmetic series is −14 and the common difference is 4

(a) Find the 15th term of the series. (2)

(b) Find the sum of the first 25 terms of the series. (3)

The sum of nine consecutive terms of the series is 1422

(c) Find the smallest of these nine terms. (5)

2015-6 Paper(1) Q.4

37 The sum Sn of the first n terms of an arithmetic series is given by Sn = 2n(10− n)

(a) Write down the first term of the series. (1)

(b) Find the common difference of the series. (2)

Given that Sn > −50

(c) (i) write down an inequality satisfied by n,

(ii) hence find the largest value of n for which Sn > −50 (4)

2016-1 Paper(1) Q.4

38 An arithmetic series has first term p and common difference p where p 6= 0

A geometric series also has first term p. The common ratio of this geometric series is r.

The sum of the first three terms of the arithmetic series is equal to the sum of the first three terms

of the geometric series.

Given that r > 0

show that r =−1 +

√21

2(5)


2016-1 Paper(2) Q.8

39 The nth term of an arithmetic series is tn where tn = 2n− 3


(a) Show that Sn = n(n− 2) (4)

(b) Find the value of n such that 5tn+2 = 3Sn−3 (5)

2016-6 Paper(1) Q.4

40 The nth term of an arithmetic series is tn and the sum of the first n terms of the series is Sn

Given that S2 =2

3t5 and that S4 = t10 + 3

(a) find

(i) the common difference of the series,

(ii) the first term of the series. (5)

Given also that Sp+2 − Sp = 110

(b) find the value of p. (3)


2017-1 Paper(2) Q.6

41 The sum of the first 21 terms of an arithmetic series is 987 and the 8th term of the series is 35

The first term of the series is a and the common difference is d.

(a) Find the value of

(i) a,

(ii) d. (5)

The sum, Sn, of the first n terms of the series is given by Sn =n∑r=1

(Ar +B), where A and B are

integers.

(b) Find the value of

(i) A,

(ii) B. (3)

(c) Find the least value of n such that Sn > 2000 (5)

2017-6 Paper(2) Q.11

42 (a) Show that log pq4 − log pq2 = log pq6 − log pq4 (3)

Given that log pq2 and log pq4 are the second and third terms of an arithmetic series, find


(c) the sum of the first n terms of the series.

Give your answers in the form n log pqs, expressing s in terms of n. (4)


8 Asymptote

1988-1 Paper(2) Q.6

1 State the equations of the two asymptotes of the curve

y = 2 +1

x, where x ∈ R and x 6= 0.

Sketch the curve showing clearly where it crosses a coordinate axis and how it approaches the

asymptotes. (6)

1988-6 Paper(2) Q.11

2 (a) Find the set of values of x for which

x2 − 2x > 0.

(3)

(b) Given that f(x) =x2 − 2x

x− 3, x 6= 3,

and using your answer to (a), or otherwise, find the set of values of x for which f(x) > 0. (4)

(c) Calculate, to one decimal place, the x-coordinate of each of the two turning points of the

graph of y = f(x). (5)

(d) Sketch the graph of y = f(x) showing clearly any asymptote which is parallel to a coordinate

axis. (3)


1989-6 Paper(2) Q.14

3 f(x) =2x2 + 6

x− 1, x ∈ R, x 6= 1.

(a) Find the set of values of x for which f(x) > 0. (2)

(b) Write down the equation of that asymptote of the curve with equation y = f(x) which is

parallel to the y-axis. (1)

(c) Find the coordinates of the maximum and minimum points of the curve with equation y =

f(x), distingusing between these points. (7)

(d) Hence sketch the curve with equation y = f(x), showing the results you have in (a), (b), and

(c). (5)

1990-1 Paper(2) Q.10

4 f(x) = (2x− 1)(2x− 3).

(a) Express f(x) in the form p(x+ q)2 + r, where p, q and r are constants. (4)

(b) Deduce the minimum value of f(x) and state the value of x for which it occurs. (2)

(c) Sketch the graph of y = f(x), showing the coordinates of the points of intersection with the

coordinate axes and the coordinates of the minimum point.

g(x) =1

(2x− 1)(2x− 3).

(2)

(d) State the equations of the asymptotes of the curve with equation y = g(x). (2)

(e) Show that1

2

(1

2x− 3− 1

2x− 1

)= g(x). (1)

(f) Using the result in part (e), or otherwise, find the gradient of the tangent to the curve with

equation y = g(x) at the point with coordinates (0, 13). (4)


1991-1 Paper(2) Q.8

5 (a) Sketch the graph of y =2x+ 5

x, x 6= 0. (2)

(b) State the equations of the asymptotes for this curve. (2)

(c) On the same set of axes sketch the graph of y = x− 2. (1)

(d) Calculate the values of the x-coordinates of the points of intersection of the two graphs. (3)

1991-1 Paper(2) Q.13

6 (2x+ 3)2 = A(x− 1)(x+ 4) +B.

(a) Find the values of A and B. (3)

g(x) =(2x+ 3)2

x− 1, x 6= 1.

(b) Show that g(x) = 4(x+ 4) +25

x− 1, x 6= 1. (2)

(c) Use result (b) to find the coordinates of the maximum and minimum points of the curve with

equation y = g(x), distinguishing between these points. (6)

The curve with equation y = g(x) has an asymptote which is parallel to the y-axis.

(d) State the equation of this asymptote. (1)

(e) Sketch the curve with equation y = g(x) for −2 6 x 6 4, indicating on your sketch the

coordinates of the points at which the curve meets the axes. (3)


1991-6 Paper(2) Q.10

7 The curve C has equation y =x+ 1

x− 2, x 6= 2.

(a) State the equations of the asymptotes of C. (2)

(b) Show that, for all points on C,dy

dxis negative. (2)

(c) Sketch C, giving the coordinates of the points where C crosses the axes. (3)

The line with equation 3y = x+ 9 cuts the curve at the points A and B.

(d) Find the coordinates of A and B. (4)

(e) Show that the line with equation 3y = x+ 9 is a normal to C at one of these points, but not

at the other point. (4)

2011-6 Paper(1) Q.9

8 A curve has equation

y =2x2 − 6

3x− 6x 6= 2

(a) Write down an equation of the asymptote to the curve which is parallel to the y-axis. (1)

(b) Find the coordinates of the stationary points on the curve. (7)

The curve crosses the y-axis at the point A.

(c) Find an equation of the normal to the curve at A. (3)

The normal at A meets the curve again at B.

(d) Find the x-coordinate of B. (4)


2012-6 Paper(2) Q.7

9 The curve G has equation y = 3− 1

x− 1, x 6= 1

(a) Find an equation of the asymptote to G which is parallel to

(i) the x-axis,

(ii) the y-axis. (2)

(b) Find the coordinates of the point where G crosses

(i) the x-axis,


(c) Sketch G, showing clearly the asymptotes and the coordinates of the points where the curve

crosses the coordinates axes. (3)

A straight line l intersects G at the points P and Q. The x-coordinate of P and the x-coordinate

of Q are roots of the equation 2x− 3 =1

x− 1

(d) Find an equation of l. (2)

2014-1 Paper(2) Q.5

10 A curve C has equation y =2x− 5

x+ 3, x 6= −3

(a) Find an equation of the asymptote to C which is parallel to

(i) the x-axis, (ii) the y-axis. (2)

(b) Find the coordinates of the point where C crosses


(c) Sketch the graph of C, showing clearly its asymptotes and the coordinates of the points where

the graph crosses the coordinate axes. (3)

(d) Find the gradient of C at the point on C where x = −1 (3)


2015-6 Paper(1) Q.9

11 A curve C has equation y =3x+ 1

2x+ 3x 6= −3

2

(a) Write down an equation of the asymptote of C which is parallel to

(i) the x-axis,


(b) Find the coordinates of the points where C crosses

(i) the x-axis,


(c) Using the axes opposite, sketch the curve C, showing clearly the asymptotes and the coordi-

nates of the points where C crosses the axes. (3)

The curve C intersects the x-axis at the point A.

The line l is the normal to C at A.

(d) Find an equation for l. (5)

The line l meets C again at the point B.

(e) Find the x-coordinate of B. (5)

O x

y


2016-6 Paper(2) Q.8


y =3x2 − 1

3x+ 2where x 6= −2

3

(a) Write down an equation of the asymptote to C which is parallel to the y-axis. (1)

(b) Find the coordinates of the stationary points on C. (8)


(c) Write down the coordinates of A. (1)

(d) On the axes on the opposite page, sketch C, showing clearly the asymptote parallel to the

y-axis, the coordinates of the stationary points and the coordinates of A.

O x

y

(3)

The line l is the normal to the curve at A.

(e) Find an equation of l. (3)


2017-1 Paper(1) Q.6

13

O(s, 0)

(0, 3.5)

x

y

y = 3

x = −2

Figure 3

Figure 3 shows a sketch of the curve with equation

y =bx+ c

x+ ax 6= −a,

where a, b and c are integers.

The equations of the asymptotes to the curve are x = −2 and y = 3

The curve crosses the y-axis at (0, 3.5)

(a) Write down the value of a and the value of b. (2)

(b) Find the value of c. (2)

Given that the curve crosses the x-axis at (s, 0)

(c) find the value of s. (2)


9 Binomial Expansion

1988-1 Paper(1) Q.9

1 Given that x+1

x= 3,

(a) Expand

(x+

1

x

)3

and use your expansion to show that

x3 +1

x3= 18,

(4)

(b) Expand

(x+

1

x

)5

. (5)

(c) Use your expansion and the previous result to find the value of

x5 +1

x5.

(1)

Given that

(1 + 3x)1.5 = 1 +Ax+Bx2 + Cx3 + ...,

(d) find the values of the constants A, B and C. (5)

1988-1 Paper(2) Q.2

2 Obtain the binomial expansion of (1−x)− 1

2 in ascending powers of x up to and including the term

in x3. (4)


1988-1 Paper(2) Q.12

3 The values of the function

y = (1 + 2x)14 at x = 0.25 and x = 0.251 are y1 and y2 respectively.

(a) Estimate the difference δy = y2 − y2 by using the small increment formula

δy ≈ dy

dxδx,

giving your answer to 2 significant figures. (5)

(b) Expand (1 + 2x)14 in ascending powers of x up to and including the term in x2. (3)

(c) Using your expansion to obtain approximations to y1 and y2 and hence estimate y2 − y1 to

2 significant figures. (4)

(d) Use your calculate to evaluate

(i) y1 and y2 to 7 decimal places,

(ii) y2 − y1 to 2 significant figures. (3)

1988-6 Paper(1) Q.6

4 Given that the coefficient of x in the expansion of (1 + ax)5 is equal to the coefficient of x4 in the

expansion of

(9 +

x

3

)6

, calculate the value of a. (5)

1988-6 Paper(2) Q.6

5 Obtain the binomial expansion of (1+5x)12 in ascending powers of x up to and including the term

in x2. State the range of values of x for which this expansion is valid. (5)


1989-1 Paper(1) Q.2

6 In the binomial expansion of (x+

3

x

)6

,

find the term which is independent of x. (4)

1989-1 Paper(1) Q.14

7

x

y

O (1, 0) Q

P (2, 1)

Fig. 4

Figure 4 shows a sketch of the graph of y = (x− 1)2 for x > 0. The line PQ is the normal to the

curve at the point P (2, 1).

(a) Find the equation of PQ. (5)

Given that PQ intersects the x-axis at Q,

(b) find the coordinates of the point Q. (1)

(c) Write down the binomial expansion of (x− 1)4 in descending powers of x. (2)

The finite region A is bounded by the curve y = (x− 1)2, the x-axis and the line PQ. The region

A is rotated through 2π about the x-axis. Using your answer to (c), or otherwise,

(d) find the volume of revolution so formed, leaving your answer in terms of π. (7)


1989-1 Paper(2) Q.3

8 Write down the first 4 terms of the expansion, in ascending powers of x, of

1

(1 + 3x)2

simplifying each term as far as possible.

State the range of values of x for which this expansion is valid. (5)

1989-1 Paper(2) Q.12

9 (a) Use your calculator to find the value, correct to 2 decimal places, of

1 + x√(1− 2x)

+1 + x√(1 + 2x)

when x = 0.1. (1)

(b) Given that −12 < x < 1

2 , expand

1 + x√(1− 2x)

in ascending powers of x up to and including the term in x3. Each term should be simplified

as far as possible. (6)

(c) Show that,

1 + x√(1− 2x)

+1 + x√(1 + 2x)

= 2 + 2x+ 3x2 + 3x3,

when x is so small that the terms in x4 and higher powers of x may be neglected. (5)

(d) By finding the value of 2 + 2x+ 3x2 + 3x3 when x = 0.1, show that the approximation in (c)

is accurate to 3 significant figures for this value of x. (3)


1989-6 Paper(1) Q.7

10 In the binomial expansion of

(1 +

x

k

)n, where k is a constant and n is a positive integer, the

coefficient of x and x2 are equal.

(a) Show that 2k = n− 1. (3)

For the case when n = 7

(b) deduce the value of k. (1)

(c) Hence find the first three terms in the expansion in ascending powers of x. (1)

1989-6 Paper(2) Q.2

11 Write down and simplify the first 3 terms, in ascending powers of x, of the expansion of√(1 + 3x).

(4)

1989-6 Paper(2) Q.11

12 (a) Use the binomial series to find the values of constants p, q, r, and s where

1

1− 2x− 1

(1 + x)3= p+ qx+ rx2 + sx3 + ... .

(8)

(b) State the range of values of x for which this expansion is valid. (2)

(c) Use the expansion in (a) as far as the term in x3 to estimate, to 3 significant figures, the

value of ∫ 0.1

0

(1

1− 2x− 1

(1 + x)3

)dx.

(5)


1990-1 Paper(1) Q.6

13 In the expansion, in ascending powers of x, of

(1 +

x

2

)nwhere n is a positive integer, the

coefficient of x2 is15

4. Find the value of n. (5)

1990-1 Paper(2) Q.5

14 (a) Write down and simplify the first three terms, in ascending powers of x, of the expansion of

(1 + 8x)12 . (3)

In the expansion of (1 + cx+ 4x2)(1 + 8x)12 the coefficient of x2 is zero.

(b) Determine the value of c. (2)

1990-6 Paper(1) Q.5

15 (1 + 3x)8 = 1 + px+ qx2 + ....

(a) Find the value of p and q. (3)

Given that in the expansion of (k+ x)(1 + 3x)8, where k is a constant, the coefficient of x is zero,

(b) calculate the value of k. (2)


1991-1 Paper(1) Q.7

16 Given that f(x) =

(x2 +

1

x

)6

,

(a) expand f(x) in descending powers of x. (2)

Given that g(x) =

(x2 − 1

x

)6

,

(b) expand g(x) in descending powers of x. (1)

(c) Use your results to find y = f(x) + g(x) and hence finddy

dx. (3)

1991-1 Paper(2) Q.1

17 Given that (1− 3x)12 = 1 + ax+ bx2 + ..., find the values of a and b. (3)

1991-6 Paper(1) Q.6

18 f(x) = x3 + px2 − x+ q, where p and q are constants.

(x+ 1) and (x− 2) are both factors of f(x).

(a) Find the values of p and q. (4)

(b) Hence, or otherwise, obtain the third factor of f(x). (2)


1991-6 Paper(1) Q.12

19 Given that

(x+

1

x

)= p,

(a) expand

(x+

1

x

)2

and use your expansion to show that x2 +1

x2= p2 − 2, (2)

(b) expand

(x+

1

x

)4

and hence find, in terms of p, the value of x4 +1

x4. (5)

(c) (1 + qx)7 = 1 + 75px+ (4− p)x2 + ...+ q7x7, where p and q are constants. Find the possible

values of p and q. (8)

1991-6 Paper(2) Q.1

20 Expand (1− 4x)− 5

2 in ascending powers of x as far as the term in x2. State the range of values of

x for which the series is valid. (4)

1991-6 Paper(2) Q.14

21 f(x) =(x− 3)2

(1− 2x), x 6= 1

2 .

(a) Show that f ′(x) =2(x+ 2)(3− x)

(1− 2x)2. (3)

(b) Find the coordinates of the points on the curve with equation y = f(x) at which y has

stationary values and determine their nature. (5)

(c) Expand f(x) as a series in ascending powers of x as far as the term is x2. (3)

When asked to calculate the gradient of the curve with equation y = f(x) at the point whose

x-coordinate is 0.1 a candidate mistakenly uses his answer to (c) as f(x).

(d) Calculate, to 3 significant figures, the percentage error made by the candidate in finding the

gradient using this. (4)


1992-1 Paper(1) Q.4

22 (1 + kx)8 = 1 + 2ax+ (a2 + 3)x2 + ...+ k8x8,

where k and a are positive constants. Find the values of k and a. (5)

1993-6 Paper(2) Q.9

23 (a) Expand (1 + 12x)16 and (1 + 5x)−1 in ascending powers of x up to and including the term in

x2. (8)

(b) Hence show that the expansion of (1 + 12x)16 is the same as the expansion of

1 + 7x

1 + 5xas far

as the term in x2. (3)

(c) Use the result in (b) to find an approximate value for 6√

1.12 as a rational number. (4)

1994-1 Paper(1) Q.7

24 In the binomial expansion of (1 + kx)n, where k is a constant and n is a positive integer, the

coefficients of x and x2 are equal.

(a) Show that k(n− 1) = 2. (3)

Given that nk = 213 , find

(b) the value of k, (2)

(c) the value of n. (1)


1994-1 Paper(2) Q.7

25 (a) Obtain the binomial expansion of (1− 4x)1.5, in ascending powers of x, up to and including

the term in x2. (3)

(b) By putting x = 0.05 in your answer to (a) obtain an approximation to (0.8)1.5. (1)

(c) Use your calculator to find (0.8)1.5. Write down all the digits on your calculator display. (1)

(d) Hence find, to 2 significant figures, the percentage error of the approximation in (b). (1)

1995-1 Paper(1) Q.7

26 Given that

(1 + 2x)5 + (1− 2x)5 = A+Bx2 + Cx4,

where A, B, and C are constants, find A, B and C. (7)

1995-1 Paper(2) Q.2

27 Expand (1 − 2x)12 , where −1

2 < x < 12 , as a series in ascending powers of x up to and including

the term in x3, simplifying each coefficient. (4)


1995-1 Paper(2) Q.10

28 f(x) = (2 + x)√

(1 + x),−1 < x < 1.

(a) Find and simplify the expansion of f(x) in ascending powers of x up to and including the

term inx2. (4)

g(x) =4

1 + x2− k

2 + x,−1 < x < 1, where k is a constant.

(b) Find and simplify the expansion of g(x) in ascending powers of x up to and including the

term in x2. (6)

Given that x is so small that terms in x3 and higher powers of x can be neglected,

(c) find values of h, j and k, where h and j are constants, such that g(x) + hx+ jx2 = f(x). (5)

1995-6 Paper(2) Q.3

291√

1− 3x= 1 + ax+ bx2 + ..., where a and b are constants.

Find the values of a and b. (4)


1995-6 Paper(2) Q.11

30 (a) Expand√

4− 3x in ascending powers of x up to and including the term x2. (4)

(b) Using a suitable value for x in your answer to (a), find an approximate value, to 5 significant

figures, for√

2.5. (4)

(c) Using your calculator, write down the value of√

2.5 to 5 significant figures. (1)

(d) Hence calculate, to 2 significant figures, the percentage error in taking your answer to (b) as

the value of√

2.5. (2)

Given that (8 + 3x)√

4− 3x = 16 + hx2 + ...,

(e) find the value of h. (4)

1996-1 Paper(1) Q.2

31 Given that (1− 2x)6 = 1− ax+ b2 − ..., where a and b are constants, find the values of a and b.

(3)

1996-1 Paper(2) Q.8

32 f(x) = (1 + 3x)n.

Given that the coefficient of x2 in the binomial expansion of f(x) is 5,

(a) find the possible values of n. (4)

Given also that n is positive, using your appropriate answer to (a),

(b) find the binomial expansion of f(x), in ascending powers of x, as far as the term in x3. (3)


1996-1 Paper(2) Q.12

33 f(x) =k + x

1− 2x, where k is a constant.

(a) Find f ′(x), in terms of x and k. (3)

(b) Given that f ′(3) = 0.52, show that k = 6. (2)

Using the value k = 6,

(c) state the equations of the asymptotes of the curve with equation y = f(x), (1)

(d) sketch the curve with equation y = f(x), giving the coordinates of the points where the curve

crosses the axes, (3)

(e) find the binomial expansion of f(x), in ascending powers of x, up to and including the term

in x3. (3)

(f) Using your answer to (e), obtain an approximation for f(0.1) and show that the error in this

approximation, to 2 significant figures, is 0.17%. (3)

1996-6 Paper(1) Q.7

34 f(x) =

(x− 2

x

)3

.

(a) Expand f(x) as a binomial series in descending powers of x, simplifying each term. (3)

Using your answer to (a), or otherwise,

(b) find f ′(x). (3)


1996-6 Paper(2) Q.12

35 (a) Given that (1− 4x)14 = 1− px− qx2 − rx3 + ..., find the values of p, q and r. (4)

(b) Show that 814 = 2(12)

14 and hence, using your answer to (a) with x = 1

8 and showing all

working to 5 significant figures, estimate the value of 814 (5)

(c) From your calculator write down the value of 814 to 5 significant figures. (1)

(d) Hence find the percentage error, to one significant figure, in your answer to (b). (1)

Given that(1− 4x)

14

(3− x)3=

1

27(1− sx2 + ...),

(e) find the value of s. (4)

1997-6 Paper(1) Q.6

36 In the binomial expansion of (k+ 2x)4, where k is a positive constant, the coefficient is x2 is 150.

Calculate


(b) the value of the coefficient of x in the expansion. (2)

1997-6 Paper(2) Q.1

37 Given that

(1− ax)12 = 1− 6x+ bx2,

where a and b are constants and x is sufficiently small that terms in x3 and higher powers of x

can be ignored, calculate the values of a and b. (3)


1997-6 Paper(2) Q.9

38 (1− kx)−7 = 1 + px+ qx2 + rx3,

where k, p, q and r are non-zero constants and x is sufficiently small that terms in x4 and higher

powers of x can be ignored.

(a) Find expressions, in terms of k, for p, q and r. (5)

Given that p, q, and r form respectively the first three terms of an arithmetic series,

(b) find the possible values of k. (5)

(c) Expand (2 − x)−7 as a series in ascending powers of x as far as the term in x3. Give each

coefficient as an exact fraction, simplified as far as possible. (5)

2007-1 Paper(1) Q.9

39 (a) Expand (1 + 5x)13 in ascending powers of x up to and including the term in x3, simplifying

each term as far as possible. (4)

(b) By substituting x = 18 into your expression, obtain an approximation, to 2 decimal places,

for 3√

13. (4)

(c) Calculate the percentage error, to 2 significant figures, in the approximation obtained in part

(b). (2)

Given that(1 + 5x)

13

(1 + x)4= a+ bx+ cx2 + ...

(d) find the exact values of a, b and c. (6)


2007-6 Paper(1) Q.6



(b) Show that (1 + 532)

15 = 1

25√

37. (2)

(c) Hence obtain an approximation, to 6 significant figures, for 5√

37. (2)

(d) Calculate the percentage error, to one significant figure, in the approximation obtained in

part (c). (3)

2008-1 Paper(2) Q.8

41 (a) Expand (1 + 14x)

13 in ascending powers of x, up to and including the term in x2, simplifying


(b) Expand (1− 14x)− 1

3 in ascending powers of x, up to and including the term in x2, simplifying


(c) State the range of values of x for which both of your expansions are valid. (1)

Using your answers to parts (a) and (b),

(d) expand

(4 + x

4− x

)13 in ascending powers of x, up to and including the term in x2, simplifying


(e) Hence obtain an estimate, to 3 significant figures, of

∫ 0.3

0

(4 + x

4− x

)13 dx. (4)


2008-6 Paper(2) Q.10

42 (a) Expand (1 +1

2x)

15 in ascending powers of x up to and including the term in x2, simplifying

each term. (3)

(b) Expand (1− 1

2x)− 1


each term. (3)

(c) State the range of values of x for which both expansions are valid. (1)

(d) Expand

(2 + x

2− x

)15 in ascending powers of x up to and including the term in x2, simplifying

each term. (3)

(e) Hence obtain an estimate, to 4 significant figures, of∫ 0.5

0

(2 + x

2− x

)15 dx. (4)

2009-6 Paper(1) Q.12

43 (a) Expand

(1− x

2


your terms as far as possible. (3)

(b) Expand

(1 +

x

2

)− 1



(c) State the range of values of x for which your expansions are valid. (1)

(d) Using your answer to parts (a) and (b) or otherwise, expand

(2− 3y

2 + 3y

)15 in ascending powers

of y up to and including the term in y2, simplifying your terms as far as possible. (5)

(e) Find the range of values of y for which your expansion is valid. (1)

(f) Use your expansion from part (d) to find an esitimate, to 3 significant figures, of∫ 0.5

0

(2− 3y

2 + 3y

)15 dy. (4)


2010-1 Paper(1) Q.8

44 (a) Expand fully (a+ bx)6, simplifying each term as far as possible. (4)

In the expansion of (a + bx)6, a 6= 0, b 6= 0, the coefficient of x3 is twice the coefficient of x4.

When x = 3 the value of (a+ bx)6 is 46656.

(b) Find the possible pairs of values of a and b. (6)

2010-6 Paper(1) Q.7

45 (a) Expand (1 + 3x2)− 1



(b) State the range of values of x for which your expansion is valid. (1)

f(x) =2 + kx2

(1 + 3x2)12

, k 6= 0

(c) Obtain a series expansion for f(x) in ascending powers of x up to and including the term in

x6. (3)

Given that the coefficient of x6 in the series expansion of f(x) is zero,

(d) show that k = 5 (2)

(e) Hence use your series expansion form part (c) to obtain an estimate, to 4 decimal places, of

∫ 0.30 f(x) dx.

(4)


2011-1 Paper(2) Q.6

46 (a) Expand fully (a+ bx)4, simplifying each term as far as possible. (2)

In the expansion of (a+ bx)4, a 6= 0, b 6= 0, the coefficient of x is equal to the coefficient of x2.

Also, when x = 2, the value of (a+ bx)4 is 2401

(b) Find the possible pairs of values of a and b. (6)

2011-6 Paper(2) Q.9

47 (a) Expand

(1− 3x

4

) 13

in ascending powers of x of up to and including the term in x3, simplifying


(b) Expand

(1 +

3x

4

)− 13

in ascending powers of x of up to and including the term in x3, sim-

plifying your terms as far as possible. (3)

(c) Write down the range of values of x for which both of your expansions are valid. (1)

(d) Expand

(4− 3x

4 + 3x

) 13

in ascending powers of x of up to and including the term in x3, simplifying


(e) Hence, obtain an estimate, to 3 significant figures, of∫ 0.5

0

(4− 3x

4 + 3x

) 13

dx (4)

2012-1 Paper(1) Q.4

48 Find the coefficient of x7 in the expansion of

(1 +

x√3

)10

, giving your answer in the form a√

3,

where a is a rational number. (4)


2012-1 Paper(2) Q.5



(b) By substituting x = −1

8into your expansion, obtain an approximation for 5

√20

Write down all the figures on your calculator display. (4)

(c) Explain why you cannot obtain an approximation for 5√

4 by substituting x = 1 into your

expansion. (1)

2012-6 Paper(1) Q.3

50 (a) Find the full binomial expansion of (1 + x)5, giving each coefficient as an integer. (2)

(b) Hence find the exact value of(

1− 2√

3)5

, giving your answer in the form a+ b√

3, where a

and b are integers. (3)

2013-1 Paper(2) Q.3

51 (a) Expand (1 + 3x2)− 1

4 in ascending powers of x up to and including the term in x6, giving each

coefficient as a fraction in its lowest terms. (3)

(b) Find the range of values of x for which your expansion is valid.

f(x) =3 + k2

(1 + 3x2)14

k ∈ R+

(1)


x6. (3)

Given that the coefficient of x4 in the series expansion of f(x) is zero

(d) find the exact value of k. (2)


2013-6 Paper(2) Q.9

52 (a) Expand, in ascending powers of x up to and including the term in x3, simplifying each term

as far as possible,

(i) (1 + x)−1

(ii) (1− 2x)−1 (4)

Given that2

1− 2x+

1

1 + x=

Ax+B

(1− 2x)(1 + x)

(b) find the value of A and the value of B. (2)

(c) (i) Obtain a series expansion for1

(1− 2x)(1 + x)in ascending powers of x up to and including

the term in x2

(ii) State the range of values of x for which this expansion is valid. (4)

(d) Use your series expansion from part (c) to obtain an estimate, to 3 decimal places, of∫ 0.2

0.1

1

(1− 2x)(1 + x)dx (4)


2014-1 Paper(2) Q.9

53 (a) Show that the first four terms of the expansion of (1− x)−k, k 6= 0, in ascending powers of x

can be written as

1 + kx+k(k + 1)

2x2 +

k(k + 1)(k + 2)

6x3

(3)

(b) Expand (1 + kx)12 , k 6= 0, in ascending powers of x, up to and including the term in x3,

simplifying your terms. (3)

Given that the coefficients of x2 in the two expansions are equal,

(c) find the value of k. (3)

Given that√

15 = λ

√3

5

(d) find the value of λ. (2)

(e) Hence, using your value of k and one of your expressions with a suitable value of x, obtain

an approximation for√

15 (4)

2014-6 Paper(1) Q.6

54 (a) Expand (1 + 4x2)− 1

5 in ascending powers of x up to and including the term in x6, expressing

each coefficient as an exact fraction in its lowest terms. (4)

(b) Find the range of values of x for which your expansion is valid. (2)

f(x) =1 + kx

(1 + 4x2)15

where k 6= 0


x5. (3)

Given that the coefficients of x2 and x5 in the expansion of f(x) are equal,

(d) find the value of k. (2)


2015-1 Paper(2) Q.8

55 (a) Find the full binomial expansion of (1− 2x)5, giving each coefficient as an integer. (3)

(b) Expand (1 + 2x)−5 in ascending powers of x up to and including the term in x3, giving each

coefficient as an integer. (3)

(c) Write down the range of values of x for which this expansion is valid. (1)

(d) Expand

(1− 2x

1 + 2x

)5

in ascending powers of x up to and including the term in x2, giving each


(e) Find the gradient of the curve with equation y =

(1− 2x

1 + 2x

)5

at the point (0, 1). (2)

2015-6 Paper(1) Q.7

56 (a) Expand

(1 +

x

3

) 14


coefficient as an exact fraction. (3)

(b) Expand

(1− x

3

)− 14



(c) Write down the range of values of x for which both of your expansions are valid. (1)

(d) Expand

(3 + x

3− x

) 14



(e) Hence obtain an estimate, to 3 significant figures, of

∫ 0.6

0

(3 + x

3− x

) 14

dx (4)


2016-1 Paper(1) Q.5

57 Given that1√

4− xcan be written as p(1− qx)

− 12

(a) find the value of p and the value of q. (2)

(b) (i) Find the first four terms in the expansion of1√

4− xin ascending powers of x, simplifying

each term.

(ii) State the range of values of x for which this expansion is valid. (4)

Given that the first three terms of the expansion of2(1 + x)√

4− xare a+ bx+ cx2

(c) find the exact value of

(i) a (ii) b (iii) c (3)

2016-6 Paper(1) Q.2

58 (a) Expand (1 + 3x2)−13 , 3x2 < 1, in ascending powers of x, up to and including the term in x6,

simplifying each term as far as possible. (3)

f(x) =1− kx2

(1 + 3x)13

where k is a constant

(b) Obtain a series expansion for f(x) in ascending powers of x up to and including the term in

x4. (3)

Given that the coefficient of x2 in the expansion f(x) is −5

(c) find the value of k. (1)


2017-1 Paper(1) Q.8

59 (a) (i) Expand

(1 +

x

2

)−3in ascending powers of x up to and including the term in x3, expressing

each coefficient as an exact fraction in its lowest terms.

(ii) Find the range of values for which your expression is valid. (4)

(b) Express (2+x)−3 in the form A(1+Bx)−3 where A and B are rational numbers whose values

should be stated. (2)

f(x) =(1 + 4x)

(2 + x)3


x2. (2)

(d) Hence obtain an estimate, to 3 significant figures, of

∫ 0.2

0

(1 + 4x)

(2 + x)3dx (3)

2017-6 Paper(2) Q.6

60 f(x) = (p+ qx)6 where p 6= 0 and q 6= 0

(a) Find the expansion of f(x) in ascending powers of x up to and including the term in x4,

simplifying each term as far as possible. (3)

In the expansion of f(x), 4 times the coefficient of x4 is equal to 9 times the coefficient of x2

Given that (p+ q) > 0 and f(1) = 15 625

(b) find the possible pairs of values of p and q. (6)


2018-1 Paper(1) Q.7

61 (a) Expand (1 − 4x2)− 1

2 in ascending powers of x, up to and including the term in x6, giving

each coefficient as in integer. (3)

(b) Write down the range of values of x for which your expansion is valid. (1)

(c) Expand3 + x√1− 4x2



(d) Hence, use algebraic integration to obtain an estimate, to 3 significant figures, of∫ 0.3

0

3 + x√1− 4x2

dx (4)


10 Chain Rule

1988-6 Paper(2) Q.3

1 Differentiate with respect to x

(a) x sinx, (2)

(b)√

(1 + x3). (3)

1989-1 Paper(2) Q.6

2 Differentiate, with respect to x,

(a) ex(x2 + 1), (3)

(b)sin 3x

x2. (3)

1989-6 Paper(2) Q.4

3 (a) Differentiate x2e3x with respect to x. (2)

(b) Evaluate

∫0

π8 cos (2x+

π

4) dx. (3)


1989-6 Paper(2) Q.10




cosxshow that

d


(3)





terms of π. (5)

1990-1 Paper(2) Q.2


(a) e−3x sinx, (2)

(b) (1 +√x)5. (2)

1991-1 Paper(2) Q.3

6 Differentiate with respect to x,

(a) x2 sin 7x, (2)

(b) cosx2. (2)


1991-6 Paper(2) Q.7




1994-1 Paper(2) Q.2


(a) e−x sinx, (2)

(b)1

cos 2x. (2)

1995-6 Paper(2) Q.2


(a) x3 cos 2x, (2)

(b)ex

1 + x. (2)


1995-6 Paper(2) Q.14

10 Given that y = xe3x, show that

(a) xdy

dx= y(3x+ 1). (4)

(b)1

y

dy

dx= 3 +

1

x. (3)

(c) Given that (h, k) are the coordinates of the stationary point of the curve with equation

y = e3x, find h and k. (4)

Given that h and k are the first two terms of an infinite geometric series,

(d) calculate, in terms of e, the sum to infinity of this series. (4)

1996-1 Paper(2) Q.4


(a) x2ex, (2)

(b) cos (x2 + 2x). (3)

1996-6 Paper(2) Q.6


(a) x3 cos 2x, (3)

(b)x3

sinx, 0 < x < π (2)


2007-6 Paper(2) Q.1

13 Differentiate with respect to x, y =cos 2x

x2 + 3. (3)

2011-1 Paper(1) Q.5

14 Given that y =ekx

x+ 2,

(a) show thatdy

dx=y(kx+ 2k − 1)

x+ 2(5)

Given also thatdy

dx=

5

4when x = 0,

(b) show that k = 3 (3)

(c) Find an equation, with integer coefficients, for the normal to the curve with equation y =e3x

x+ 2at the point where x = 0 (3)

2011-6 Paper(1) Q.3

15 Given that y = e2x sin 3x

(a) finddy

dx(3)

(b) show thatd2y

dx2= 2

dy

dx− 9y + 6e2x cos 3x (4)


2012-1 Paper(1) Q.5


(a) y = x2ex (2)

(b) y = (x3 + 2x2 + 3)5 (3)

2012-6 Paper(2) Q.4


(a)1

x2(2)

(b)1

(2x+ 1)2(2)

(c)1

1− cos2 x(3)

2013-1 Paper(2) Q.4


(a) 3x sin 5x (3)

(b)e2x

4− 3x2(3)


2014-1 Paper(1) Q.3


(a) e3x(5x− 7)2 (3)

(b)cos 2x

x+ 9(3)

2014-6 Paper(1) Q.3

20 Given that 2xy − 3y = e2x

(a) show thatdy

dx=

4e2x(x− 2)

(2x− 3)2(5)

(b) find the value ofdy

dxwhen x = 0 (1)

(c) find an equation, with integer coefficients, of the tangent to the curve with equation 2xy−3y =

e2x at the point on the curve where x = 0 (3)

2015-6 Paper(1) Q.2

21 Given that y = 4x2e2x

(a) finddy

dx(3)

(b) hence show that xdy

dx= 2y(1 + x) (2)

2016-1 Paper(2) Q.4

22 Given that y = e2x√x+ 1

show thatdy

dx=e2x(4x+ 5)

2√x+ 1

(6)


2016-6 Paper(2) Q.4


e2x cos 3x

(3)

2018-1 Paper(2) Q.5

24 Given that y = 2ex(3x2 − 6)

show thatd2y

dx2− 2

dy

dx+ y = 12ex (7)


11 Complete the Square

1988-6 Paper(2) Q.2

1 Given that

f(x) = x2 + 4x+ 7,

find the constants a and b such that

f(x) = (x+ a)2 + b.

State the least value of f(x). (4)

1990-1 Paper(2) Q.10

2 f(x) = (2x− 1)(2x− 3).

(a) Express f(x) in the form p(x+ q)2 + r, where p, q and r are constants. (4)

(b) Deduce the minimum value of f(x) and state the value of x for which it occurs. (2)

(c) Sketch the graph of y = f(x), showing the coordinates of the points of intersection with the

coordinate axes and the coordinates of the minimum point.

g(x) =1

(2x− 1)(2x− 3).

(2)

(d) State the equations of the asymptotes of the curve with equation y = g(x). (2)

(e) Show that1

2

(1

2x− 3− 1

2x− 1

)= g(x). (1)

(f) Using the result in part (e), or otherwise, find the gradient of the tangent to the curve with

equation y = g(x) at the point with coordinates (0, 13). (4)


1990-6 Paper(1) Q.1

3 x2 + 2x− 8 = (x+ 1)2 + k

(a) Find the value of the constant k. (1)

(b) Deduce the minimum value of x2 + 2x− 8. (1)

1991-1 Paper(2) Q.12

4 f(x) = x2 + 3x+ 5 = (x+A)2 +B.

(a) Find the values of the constants A and B and hence deduce the minimum value of f(x). (4)

g(x) = x2 + kx+ 2k − 3, where k is a constant.

(b) Find the range of values of k for which the equation g(x) = 0 has no real solutions. (4)

(c) Given that the equation g(x) = 0 has roots α and β, form a quadratic equation whose roots

are (2α+ β) and (α+ 2β). (7)


1994-1 Paper(2) Q.12

5 f(x) = 4x2 + 12x− 9 = A(x+B)2 + C,

where A, B, and C are constants.

(a) Find the values of A, B, and C and hence deduce the minimum value of f(x). (4)

Given that

g(x) =1

f(x),

(b) using your answers to (a), or otherwise, find the coordinates of the stationary point of the

curve with equation y = g(x) and state whether it is a maximum or minimum. (3)

(c) Find the range of values of k for which the equation

f(x) = k

has no real roots. (2)


(d) find a quadratic equation with roots(α+

2

β

)and

(β +

2

α

).

(6)


1995-1 Paper(2) Q.9

6 f(x) = 2x2 − 8x+ 3.

(a) Express f(x) in the form a(x− 2)2 − b, where a and b are constants to be found. (2)

(b) Hence state the minimum value of f(x) and the value of x which gives this minimum value.

(2)


(c) Find the value of α2 + β2. (3)

Given that g(x) = x2 − px + q, where p and q are constants, and that the roots of the equation

g(x) = 0 are 3α+ β and 3β + α,

(d) calculate the values of p and q. (4)

(e) For your values of p and q express g(x) in the form (x+ r)2− s, where r and s are constants

to be found. (2)

(f) Hence write down the maximum value of1

g(x)and the value of x which gives this maximum

value. (2)


1995-6 Paper(2) Q.12

7 f(x) = 2x2 + 3x+ 4.


Hence






(f) Find

∫f(x) dx. (2)




1996-6 Paper(2) Q.9

8 f(x) = 3x2 + 6x− 7.


(a) form a quadratic equation, with integer coefficients, whose roots are2

αand

2

β, (6)

(b) form a quadratic equation, with integer coefficients, whose roots are α(α+ 1) and β(β + 1).

(5)

Given that f(x) can be expressed in the form A(x+B)2 − C, where A, B, and C are constants,

(c) find the values of A, B and C. (2)

(d) Hence write down the minimum value of f(x) and the value of x for which it occurs. (2)

1997-6 Paper(2) Q.12

9 2x2 − 6x+ 13 = A(x+B)2 + C

where A, B and C are constants.

(a) Determine the values of A, B and C. (4)

f(x) =1

2x2 − 6x+ 13.

Using your answers to (a), or otherwise,

(b) find, giving your answer as an exact fraction, the maximum value of f(x), (1)

(c) state the value of x at which the maximum value occurs. (1)

Given that α and β are the roots of the equation 2x2−5x+p = 0, where p is a non-zero constant,

(d) obtain a quadratic equation with roots

(1 +

1

α

)and

(1 +

1

β

). (6)

Given that the equation found in (d) has two equal roots,

(e) find the value of p. (3)


2009-6 Paper(2) Q.8

10 f(x) = 3− 5x− 7x2

(a) Show that f(x) can be written in the form A−B(x+C)2, stating the values of A, B and C.

(4)

(b) Write down the maximum value of f(x) and the value of x for which this maximum occurs.

(2)


Without solving the equation find, as exact fractions,

(c) α2 + β2, (3)

(d)α

β+β

α. (3)

(e) Form a quadratic equation, with integer coefficients, which has rootsα

βand

β

α. (2)

2011-1 Paper(2) Q.10

11 f(x) = 3x2 − 6x− 2


Without solving the equation, form an equation with integer coefficients,

(a) with roots αβ and1

αβ, (6)

(b) with roots 2α+ β and α+ 2β. (5)

(c) Express f(x) in the form f(x) = A(x+B)2 +C, stating the values of the constants A, B and

C. (3)

(d) Hence write down




2011-6 Paper(2) Q.11

12 f(x) = x2 + 6x+ 8

Given that f(x) can be expressed in the form (x+A)2 +B where A and B are constants,

(a) find the value of A and the value of B. (3)

(b) Hence, or otherwise, find

(i) the value of x for which f(x) has its least value

(ii) the least value of f(x). (2)

The curve C has equation y = x2 + 6x+ 8

The line l, with equation y = 2− x, intersects C at two points.

(c) Find the x-coordinate of each of these two points. (4)

(d) Find the x-coordinate of the points where C crosses the x-axis. (2)

The curve C has equation y = x2 + 6x+ 8 and the line l has equation y = 2− xIn the space below

(e) sketch, on the sames axes, the curve C and the line l. (2)

(f) Find the area of the finite region bounded by the curve C and the line l. (5)

2013-1 Paper(1) Q.3

13 f(x) = 3x2 + 6x+ 7

Given that f(x) can be written in the form A(x+B)2+C, where A, B and C are rational numbers.

(a) find the value of A, the value of B and the value of C. (3)


(i) the value of x for which1

f(x)is a maximum,

(ii) the maximum value of1

f(x). (2)


2014-1 Paper(1) Q.2

14 f(x) = 2x2 − 8x+ 5

Given that f(x) can be written in the form a(x− b)2 + c

(a) find the value of a, the value of b and the value of c. (3)

(b) Write down


(ii) the value of x at which this minimum occurs. (2)

2015-6 Paper(1) Q.3

15 f(x) = 4x2 − 8x+ 7

Given that f(x) = l(x−m)2 + n, for all values of x,

(a) find the value of l, the value of m and the value of n. (3)





2016-6 Paper(1) Q.9

16 f(x) = 3x2 − 5x− 4



(i) rootsα

βand

β

α




2018-1 Paper(1) Q.1

17 f(x) = 6 + 5x− 2x2

Given that f(x) can be written in the form p(x+ q)2 + r, where p, q and r are rational numbers,

(a) find the value of p, the value of q and the value of r. (3)


(i) the maximum value of f(x),

(ii) the value of x for which this maximum occurs. (2)

g(x) = 6 + 5x3 − 2x6

(c) Write down

(i) the maximum value of g(x),

(ii) the exact value of x for which this maximum ocrrus. (3)


12 Coordinates

1990-6 Paper(1) Q.6

1 Two perpendicular lines L1, and L2 each pass through the point P with coordinates (2, 6). The

line L1 has gradient 3.

(a) Find the gradient of L2 and hence find an equation of the line L2. (2)

The line L2 crosses the x-axis at the point Q.

(b) Show that the coordinates of Q are (20, 0). (2)

The point R lies on PQ such that PR : RQ is 7 : 2, and R lies between P and Q.

(c) Calculate the coordinates of R. (2)

1991-1 Paper(1) Q.8

2 A is the point with coordinates (0, 9) and B is the point with coordinates (4, 1). The mid-point

of AB is M .

(a) Calculate the coordinates of M . (1)

The line through M , perpendicular to AB, meets the y-axis at C.

(b) Calculate the gradient of AB and hence find an equation of the line MC. (3)

(c) Find the coordinates of C. (1)

(d) Calculate the length of BC. (2)


1991-6 Paper(1) Q.3

3 The points A and B have coordinates (3, 7) and (6, 3) respectively.

(a) Calculate the distance AB. (2)

(b) Find the coordinates of the point which divides AB internally in the ratio 2 : 3. (2)

1991-6 Paper(1) Q.8

4 (a) By shading the region for which the following inequalities do not apply, show, on a single

sketch, the region R in which the inequalities y > 0, x + y 6 6, y + 3x 6 12 and x > 0 are

satisfied simultaneously. (5)

(b) Find the coordinates of the point in the region R which give the maximum value of 20x+10y.

(2)

1991-6 Paper(1) Q.9

5 The coordinate of the points A,B,C and D are (−1,−4), (3, 7), (9, 2) and (1,−6) respectively.

The points P,Q and R are the midpoints of AB,BC and CD respectively.

(a) State the coordinates of P,Q and R. (3)

(b) Find an equation of the line PQ. (2)

The line L passes through R and is parallel to PQ.

(c) Find an equation of the line L. (3)

The line L intersects the line AD at the point X.

(d) Calculate the coordinates of X and show that X is the mid-point of AD. (5)

(e) Show also that PQRX is a parallelogram. (2)


1991-6 Paper(2) Q.2

6 The line with equation y = 5x + 1 cuts the curve with equation y = 2x2 + kx + 3, where k is a

constant, at the points A and B. The x-coordinates of A and B are x1 and x2 respectively.

(a) Show that x1 + x2 = 12(5− k). (2)

(b) Given that the x-coordinate of the mid-point of the line segment AB is −1, find the value of

k. (2)

1992-1 Paper(1) Q.9

7 The straight line l passes through the points A and B with coordinates (2, 0) and (5, 6) respectively.

(a) Find an equation of l.

The straight line L is perpendicular to l and passes through the point C with coordinates (12 , 7).

The lines l and L intersect in the point D.

(b) Find an equation of L and show that the coordinates of D are (412 , 5).

Hence calculate

(c) the ratio in which D divides the line segment AB,

(d) the coordinates of the image of the point C when reflected in the line l,

(e) the area of 4ABC.


1993-6 Paper(1) Q.12

8

O x

y

A

D

B

C

Fig. 2

Figure 2 shows part of the curve with equation y = 7 + 6x − x2. The point A has coordinates

(0, 19) and AD is parallel to the tangent to the curve at the point on the curve at which x = 4.

AD cuts the curve at B and C.

(a) Show that an equation of the line AD is y + 2x = 19. (4)

(b) Find the coordinates of B and C. (4)



1994-1 Paper(1) Q.9

9 The line with equation y = x + 2 intersects the curve with equation x2 + y2 = 12 − 2x at the

points A and B. The coordinates of A are both positive. Calculate the coordinates of

(a) the point A, (4)

(b) the point B. (1)

The line through A and B meets the x-axis at C and the y-axis at D. The mid-point of CD is

M .

(c) Calculate the coordinates of M. (4)

(d) Show that the line through M perpendicular to AB passes through the origin O. (2)

(e) Calculate the area of 4ABO. (4)


1994-1 Paper(1) Q.11

10

OA C

B

x

y

Fig. 1

Figure 1 shows part of the curve with equation y = 3 + 2x− x2. The curve cuts the x-axis at Aand C and cuts the y-axis at B.

(a) calculate the coordinates of A, B and C. (4)

(b) Show that the line OB divides the area of the finite region bounded by the curve and thex-axis in the ratio of 5:27. (6)

0 5 10 x

y

Fig. 2

Figure 2 shows part of the curve with equation y = 10x− 3

2 .

(c) Calculate, in terms of π, the volume of the solid generated when the region bounded by thiscurve, the x-axis and the lines x = 5 and x = 10 is rotated through 360◦ about the x-axis.

(5)


1995-1 Paper(1) Q.8

11 The point A has coordinates (6, 3) and the line l has equation y = 2x− 4.

(a) Find an equation of the line l′ which passes through A and is perpendicular to l. (3)

(b) Using your answer to (a), find the coordinates of the point of intersection of l and l′. (2)

B is the reflection of A in l.

(c) Find the coordinates of B. (2)

1996-6 Paper(1) Q.8

12 The curve C has equation y = 6x− x3 and the line l has equation y = 3x+ 1.

(a) Find the coordinates of the points of C at which the tangents to C are parallel to l. (4)

(b) Find the equations of tangents to C which are parallel to l. (3)


1996-6 Paper(1) Q.10

13

O

A

x

y

P

Fig. 2

Figure 2 shows part of the curve C with equation y = 2x3 − 13x2 + 24x. The point P , on C, has

coordinates (1, 13) and the normal to the curve at P cuts the y-axis at A.

(a) Find an equation of the line AP . (6)


(c) Calculate, as an exact fraction, the finite area bounded by C, AP and the y-axis. (7)


1996-6 Paper(1) Q.12

14 The points A, B and C have coordinates (−4, 5), (18, 9) and (2,−3) respectively.

(a) Show, by calculation, that ∠ACB = 90◦. (3)

(b) Calculate, to 0.1◦, the size of ∠ABC. (4)

The point D is on CB such that CD : DB = 3 : 1.

(c) Obtain the coordinates of D. (2)

(d) Calculate the area of 4BAD. (3)

(e) Calculate, to 2 significant figures, the length of the perpendicular from B to AD produced.

(3)


1997-6 Paper(1) Q.9

15

O

A

B

x

y

Fig. 2

Figure 2 shows part of the curve, C, with equation y = 6x− 2x2. The curve C cuts the x-axis at

O and A. The normal, l, to C at A cuts C again at the point B.


(b) Obtain an equation for l. (5)

(c) Find, as an exact fraction, the x-coordinate of B. (4)

The shaded region is bounded by C, l and the y-axis.

(d) Show that the area of the shaded region is 934 . (5)


1997-6 Paper(2) Q.11

16 The curve C has equation y = ex + 2e−x.

(a) Show that C has a stationary point, P , when x = 12 ln 2. (5)

(b) Determine whether P is a maximum or a minimum point. (2)

(c) Find the y-coordinate of P , giving your answer in the form k√

2, stating the value of the

constant k. (3)

The finite region bounded by C, the y-axis, the x-axis and the line x = 1 is rotated through 2π

radians about the x-axis.

(d) Calculate, giving your answer in terms of e and π, the volume of the solid generated. (5)

2007-1 Paper(2) Q.7

17 The point A and B have coordinates (−2, 4) and (5, 5) respectively.

(a) Show that an equation of the perpendicular bisector of AB is y + 7x = 15. (5)

The point C has coordinates (6, 4).

(b) Write down an equation for the perpendicular bisector of AC. (1)

A circle passes through the points A,B and C.

(c) Find

(i) the coordinates of the centre of the circle,

(ii) the radius of the circle. (4)


2007-6 Paper(1) Q.4

18 The point A has coordinates (3, 2). The line l has gradient 724 and passes through A.

(a) Find an equation, with integer coefficient, for l. (3)

The point B, with coordinates (b, 9), lies on l.

(b) Find the value of b. (2)

(c) Calculate the length of AB. (3)


2007-6 Paper(1) Q.10

19

O RP Q

S

T

x

yl

Figure 2

Figure 2 shows the curve with equation y = k+ 7x− x3, where k is a constant. The curve crosses

the x-axis at the points P , Q, and R. Given that R has coordinates (3, 0), find


(b) the coordinates of P and the coordinates of Q. (3)

The curve crosses the y-axis at the point S. The line l passes through P and S.

(c) Find an equation for l. (3)

The line l meets the curve again at the point T .

(d) Find the coordinates of T . (3)

(e) Calculate the area of the region shown in Figure 2. (7)


2008-1 Paper(1) Q.8

20

O x

yP

Q

Figure 1

Figure 1 shows the curve with equation y = f(x) where f ′(x) = 3x2− 4x− 4. Given that the curve

passes through the point with coordinates (1, 0).

(a) find f(x). (3)

The curve has a maximum point at P and a minimum point Q.

(b) Find the exact value of the coordinates of

(i) P , (ii) Q. (3)

(c) Write down an equation for

(i) the tangent at P ,

(ii) the normal at Q. (2)

(d) Find the exact value of the finite area enclosed by the curve between the points P and Q,

the tangent at P and the normal Q. (7)


2008-1 Paper(2) Q.7

21 The points A, B and C have coordinates (2, 6), (6, 8), and (4, 2) respectively.

(a) Find the exact lengths of

(i) AB, (ii) BC, (iii) AC. (4)

(b) Find the size of each angle of 4ABC. (3)

A circle is drawn to pass through the points A, B and C. Find

(c) the coordinates of the centre of the circle, (2)

(d) the exact length of the radius of the circle. (2)


2009-6 Paper(1) Q.11

22 (a) By writing tanx =sinx

cosx, show that

d

dx= (tanx) =

1

cos2 x. (3)

0

A

B C x

y

−π2

3π

2

Figure 4

Figure 4 shows the curve with equation y = 1 + tanx,−π2< x <

3π

2.

The curve crosses the y-axis at the point A and the x-axis at the points B and C.


(c) Find the x-coordinate of

(i) B,

(ii) C. (2)

The point D on the curve has x-coordinateπ

6.

The normal to the curve at D meets the curve again at the point E and crosses the x-axis at G.

(d) Find the exact value of the x-coordinate of G. (6)

Given that the coordinates of E are (e, f),π

2< e <

3π

2,

(e) hence or otherwise show that f > 0. (2)


2009-6 Paper(2) Q.7

23 A curve has equation a32 y = x

52 , where x > 0 and a is a positive constant.

(a) Show that an equation of the normal to the curve at the point with coordinates(a, a) is

5y + 2x = 7a. (6)

(b) Find the coordinates of the point where this normal meets the x-axis. (1)

The finite region bounded by the curve, the normal to the curve at the point (a, a) and the x-axis

is rotated 360◦ about the x-axis.

(c) Find, in terms of π, the volume of the solid generated. (6)


2010-1 Paper(1) Q.6

24

O

C1

C2

x

y

A

B

Figure 1

Figure 1 shows the curve C1 with equation y2 = 8x+4 and the curve C2 with equation y2 = 8−4x.

The curve C1 and C2 intersect at the points A and B.

(a) Find the exact coordinates of A. (3)

The shaded region enclosed by C1, C2 and the x-axis is rotated through 360◦ about the x-axis.

(b) Find, in terms of π, the volume of the solid generated. (6)


2010-1 Paper(1) Q.10

25 The curve C1, with equation y = x2, meets the curve C2, with equation y =x2

x− 1, at the origin

and at the point A.

Find

(a) the coordinates of A, (4)

(b) an equation of the tangent to C1 at A, (4)

(c) an equation of the tangeth to C2 at A. (4)

The tangent to C1 at A meets the y-axis at the point B and the tangent to C2 at A meets the

y-axis at the point D.

(d) Find the area of 4BAD. (3)


2010-6 Paper(1) Q.9

26

O x

y

P Q R

C

S

Figure 1

f(x) = x3 + ax2 + bx+ d, a, b, d ∈ Z

Figure 2 shows the curve C with equation y = f(x).

The curve C crosses the x-axis at the points P , Q, and R.

The x-coordinates of P , Q and R are −3,−1 and 2 respectively.

The point S on C has x-coordinate −2.

(a) Find the value of a, the value of b and the value of d. (4)

The line l is the tangent to C at S.

(b) Find an equation for l giving your answer in the form y = px+ q. (5)

(c) Hence show that l passes through R. (1)

(d) Find the area of the region enclosed by C and l. (5)


2010-6 Paper(1) Q.10

27 The points P , Q and R have coordinates (1, 3), (4, 5), and (6, 2) respectively.

(a) Find the gradient of (i) PQ, (ii) QR. (3)

(b) Show, by calculation, that PQ is perpendicular to QR. (2)

(c) Find the exact length of PQ. (2)

The line l is the perpendicular bisector of PR.


(e) Show that Q lies on l. (1)

The line l meets the x-axis at the point S.

(f) Show that PQRS is a square. (4)

(g) Find the area of 4PQR. (2)


2011-1 Paper(1) Q.6


The line l passes through A and B.

(a) Find an equation for l. (2)

O

A

B

D

l

C

x

y

Figure 1

f(x) = x3 − px2 − qx+ r, p, q, r ∈ Z+

Figure 1 shows the curve C with equation y = f(x) and the line l.

The point D has coordinates (1,−3).

The curve C passes through the points A, D and B.

(b) Show that r = 2 (1)

(c) Find the value of p and the value of q. (4)

(d) Find the area of the shaded region shown in Figure 1. (5)


2011-1 Paper(1) Q.9

29 The points P , Q and R have coordinates (3, 9), (−2, 4) and (0, 8) respectively.

The line l1 is the perpendicular bisector of PQ and the line l2 is the perpendicular bisector of QR.

(a) Find an equation for l1. (4)

(b) Find an equation for l2. (3)

The lines l1 and l2 meet at the point S.

(c) Show that the coordinates of S are (3, 4). (3)

The circle C has centre S and radius PS.

(d) Find, as a multiple of π, the area enclosed by C. (2)

(e) Explain briefly why C also passes through Q and R. (2)

The line l3 passes through S and is perpendicular to PR.

(f) Find the coordinates of the point where l3 crosses PR. (2)

2012-1 Paper(2) Q.7

30 The points A, B and C have coordinates (3, 5), (7, 8) and (6, 1) respectively.

(a) Show, by calculation, that AB is perpendicular to AC. (4)

(b) Find an equation for AC in the form ax + by + c = 0, where a, b and c are integers whose

values must be stated. (3)

The point D is on AC produced and AC : CD = 1 : 2

(c) Find the coordinates of D. (2)

(d) Calculate the area of triangle ABD. (4)


2012-6 Paper(1) Q.9

31 The point P with coordinates (4, 4) lies on the curve C with equation y =1

4x2

(a) Find an equation of

(i) the tangent to C at P ,

(ii) the normal to C at P . (6)

The point Q lies on the curve C. The normal to C at Q and the normal to C at P intersect at

the point R. The line RQ is perpendicular to the line RP .

(b) Find the coordinates of Q. (2)

(c) Find the x-coordinate of R. (4)

The tangent to C at P and the tangent to C at Q intersect at the point S.

(d) Show that the line RS is parallel to the y-axis. (5)

2012-6 Paper(1) Q.10

32 The point A has coordinates (−3, 4) and the point C has coordinates (5, 2). The mid-point of AC

is M . The line l is the perpendicular bisector of AC.

(a) Find an equation of l. (4)

(b) Find the exact length of AC. (2)

The point B lies on the line l. The area of triangle ABC is 17√

2

(c) Find the exact length of BM . (2)

(d) Find the exact length of AB. (2)

(e) Find the coordinates of each of the two possible positions of B. (6)


2013-1 Paper(1) Q.7

33 The point C with coordinates (2, 1) is the centre of a circle which passes through the point A with

coordinates (3, 3).

(a) Find the radius of the circle. (2)

The line AB is a diameter of the circle.

(b) Find the coordinates of B. (2)

The points D with coordinates (0, 2) and E with coordinates (4, 0) lie on the circle.

(c) Show that DE is a diameter of the circle. (2)

The point P has coordinates (x, y).

(d) Find an expression, in terms of x and y, for the length of CP . (2)

Given that the point P lies on the circle,

(e) show that x2 + y2 − 4x− 2y = 0 (2)

2013-1 Paper(2) Q.7

34 The line l passes through the points with coordinates (1, 6) and (3, 2).

(a) Show that an equation of l is y + 2x = 8 (3)

The curve C has equation xy = 8

(b) Show that l is a tangent to C. (3)

Given that l is the tangent to C at the point A,

(c) find the coordinates of A. (2)

(d) Find an equation, with integer coefficients, of the normal to C at A. (3)


2013-6 Paper(1) Q.10

35 The curve C has equation y = x4 − 4x3 − 2x2 + 13x+ 5 and the line l1 is the tangent to C at the

point R(1, 13).

(a) Find an equation for l1 (4)

The points P and Q lie on C. The x-coordinates of P and Q are p and q respectively, where

p < q. The tangent to C at P is parallel to l1 and the tangent to C at Q is parallel to l1

(b) Find the coordinates of P and the coordinates of Q. (4)

The line l2 passes through P and Q.

(c) Find an equation for l2 (2)

(d) Show that l2 is tangent to C at P and a tangent to C at Q. (1)

The normal to C at R(1, 13) intersects l2 at the point S.

(e) Find the exact length of RS. (5)

(f) Find the area of the triangle PQR. (2)


2013-6 Paper(2) Q.8

36 The equation of the line l1 is 2x+ 3y + 6 = 0

(a) Find the gradient of l1 (1)

The line l2 is perpendicular to l1 and passes through the point P with coordinates (7, 2).

(b) Find an equation for l2 (3)

The lines l1 and l2 intersect at the point Q.

(c) Find the coordinates of Q. (3)

The line l3 is parallel to l1 and passes through the point P .

(d) Find an equation for l3 (2)

The line l1 crosses the x-axis at the point R.

(e) Show that PQ = QR. (3)

The point S lies on l3

The line PR is perpendicular to QS.

(f) Find the exact area of the quadrilateral PQRS. (3)


2014-1 Paper(1) Q.11

37 The curve C has equation 5y = 4(x2 + 1). The coordinates of the point P on the curve are

(p, 8), p > 0

The line l with equation 5y − 24x+ q = 0 is the tangent to C at P .

(a) (i) Show that p = 3

(ii) Find the value of q (4)

(b) Find an equation, with integer coefficients, for the normal to C at P . (5)

(c) Find the exact value of the area of the triangle formed by the tangent to C at P , the normal

to C at P and the x-axis. (3)

The finite region bounded by C, the tangent to C at P , the x-axis and the y-axis is rotated

through 360◦ about the x-axis.

(d) Find, to 2 significant figures, the volume of the solid generated. (6)

2014-1 Paper(2) Q.1

38 The points A and B have coordinates (5, 9) and (9, 3) respectively. The line l is the perpendicular

bisector of AB.

Find an equation for l in the form ax+ by + c = 0, where a, b and c are integers. (5)


2014-6 Paper(1) Q.9

39 The points A and B have coordinates (2, 5) and (16, 12) respectively. The point D with coordinates

(8, 8) lies on AB.

(a) Find, in the form p : q, the ratio in which D divides AB internally. (3)

The line l passes through D and is perpendicular to AB.


The point E with coordinates (e, 6) lies on l.

(c) Find the value of e. (1)

The line ED is produced to F so that ED = DF .

(d) Find the coordinates of F . (2)

(e) Find the area of the kite AEBF . (3)


2015-1 Paper(1) Q.10

40 The points A, B and C have coordinates (−2, 3), (2, 5) and (4, 1) respectively.

(a) Show, by calculation, that AB is perpendicular to BC. (3)

(b) Show that the length of AB = the length of BC. (3)

The mid-point of AC is M .

(c) Find the coordinates of M . (1)

(d) Find the exact length of the radius of the circle which passes through the points A, B and

C. (3)

The point P lies on BM such that BP : PM = 2 : 1

(e) Find the coordinates of P . (2)

The point Q lies on AP produced such that AP : PQ = 2 : 1

(f) Find the coordinates of Q. (3)

(g) Show that Q lies on BC. (3)


2015-1 Paper(2) Q.10

41

O

P (0, 24)

Q

R

x

y

C

Figure 3

Figure 3 shows the curve C with equation y = 9x3 − 18x2 − 8x+ 24

The curve cuts the y-axis at the point P with coordinates (0, 24).

The point Q lies on C and the line PQ is the tangent to C at P .

(a) Find an equation of PQ. (4)

(b) Find the coordinates of Q. (5)

The point R lies on C and S is the point such that PQRS is a parallelogram.

Given that RS is the tangent to C at R.

(c) find the coordinates of R, (4)

(d) find the coordinates of S. (2)

(e) Show that S lies on C. (2)


2015-6 Paper(2) Q.9


The point M is the midpoint of AB.

(a) Find the coordinates of M . (2)

(b) Find the length of AB. (2)

The line l is the perpendicular bisector of AB.

(c) Find an equation for l giving your answer in the form ay = bx + c, where a, b and c are

integers. (4)

The point D lies on l and has coordinates (d, 2).

(d) Find the value of d. (2)

The point E lies on l and is such that DM : ME = 1 : 2

(e) Find the coordinates of E. (2)

(f) Find the area of the kite AEBD. (4)


2016-1 Paper(1) Q.11

43 f(x) = 4 + 3x− x2

(a) Write f(x) in the form P −Q(x+R)2, where P , Q and R are rational numbers. (2)

The curve C has equation y = 4 + 3x− x2

(b) Find the coordinates of the maximum point of C. (1)

The line l1 is a tangent to C at the point where x = 1

(c) Find an equation for l1 (5)

Another l2 is perpendicular to l1 and is also a tangent to C.

The lines l1 and l2 intersect at the point A.

(d) Find the coordinates of A. (5)

The point B with coordinates (−3, 2) lies on l1

(e) Find the exact length of AB. (2)

The point D with coordinates (8, 0) lies on l2

(f) Find the exact area of triangle ABD. (3)


2016-6 Paper(1) Q.10

44 The points A and B have coordinates (2, 4) and (5,−2) respectively.

The point C divides AB in the ratio 1:2

(a) Find the coordinates of C. (2)

The point D has coordinates (1, 1)

(b) Show that DC is perpendicular to AB. (3)

(c) Find the equation of DC in the form py = x+ q (2)

The point E is such that DCE is a straight line and DC = CE.

(d) Find the coordinates of E. (2)

(e) Calculate the area of quadrilateral ADBE. (4)


2017-1 Paper(2) Q.9

45 The points P and Q have coordinates (−2, 5) and (2,−3) respectively.

(a) Find an equation for the line PQ. (2)

The point N is such that PNQ is a straight line and PN : NQ = 3 : 1

The straight line l passes through N and is perpendicular to PQ.

(b) Find

(i) the coordinates of N ,

(ii) an equation for l. (5)

The points S and T lie on l and have coordinates (3, s) and (t,−2) respectively.

(c) Find

(i) the value of s,

(ii) the value of t. (2)

(d) Find the area of the quadrilateral PSQT . (4)


2017-6 Paper(1) Q.8


(a) Find the exact length of AB. (2)

The point C divides AB in the ratio 1 : 2

(b) Find the coordinates of C. (2)

The line l passes through C and is perpendicular to AB.

(c) Find an equation of l, giving your answer in the form y = ax+ b where a and b are integers.

(4)

The point D with coordinates (9, d) lies on l.

(d) Find the value of d. (1)

The point E is the midpoint of CD.

(e) Find the exact value of the area of the quadrilateral ADBE. (5)


2018-1 Paper(2) Q.10

47 The point A has coordinates (−6,−4) and the point B has coordinates (4, 1)

The line l passes through the point A and the point B.

(a) Find an equation of l. (2)

The point P lies on l such that AP : PB = 3 : 2

(b) Find the coordinates of P . (2)

The point Q with coordinates (m,n) lies on the line through P that is perpendicular to l.

Given that m < 0 and the length of PQ is 3√

5

(c) find the coordinates of Q. (5)

The point R has coordinates (−13, 0)

(d) Show that

(i) AB and RQ are equal in length,

(ii) AB and RQ are parallel. (4)

(e) Find the area of the quadrilateral ABQR. (2)


13 Cosine Rule

1988-1 Paper(1) Q.4

1 In 4ABC, AB = 8 cm, BC = 9 cm and angle ABC = 42◦.

Calculate

(a) the length of AC, in cm, to 3 significant figures, (3)

(b) the magnitude of ∠ACB, to the nearest one tenth of a degree. (2)

1988-6 Paper(1) Q.8

2 In 4ABC, AB = 2x cm, AC = x cm, BC = 14 cm and ∠BAC = 120◦.

(a) Calculate, to 3 significant figures, the value of x. (3)

Given also that ∠ABC = y◦,

(b) without evaluating y, show that 2 sin y◦ = sin (60− y)◦. (3)

1990-1 Paper(1) Q.1

3 In 4ABC, AB = 6 cm, ∠CAB = 105◦ and ∠ABC = 40◦.

Calculate

(a) the length, in cm to 3 significant figures, of BC, (2)

(b) the area, in cm2 to 2 significant figures, of 4ABC. (2)


1990-1 Paper(2) Q.12




cos 2x = 1− 2 sin2 x.

(3)

(c) Show that

∫2 sin2 x dx = x− 1


O x

y

P

Q

R

π

Fig. 2









1990-6 Paper(1) Q.8

5

A

B C

D4 cm

3 cm

2 cm

x◦x◦

Fig. 2

Figure 2 shows a quadrilateral ABCD in which AB = 4 cm, BC = 3 cm, CD = 2 cm and

∠ABC = 90◦. The diagonal AC bisects ∠BCD and ∠ACB = ∠ACD = x◦.

(a) State the values of cosx◦ and sinx◦. (1)

(b) Calculate, in cm2, the exact value of AD2. (2)

(c) Calculate, in cm2, the exact value of the area of the quadrilateral ABCD. (4)

1991-6 Paper(1) Q.1

6 In 4ABC, AB = 12 cm, BC = 15 cm and CA = 10 cm. Calculate ∠ABC, in degrees to 1

decimal place. (3)


1991-6 Paper(2) Q.11






(3)



(ii) sin

(x+

π

3

)+ sin

(x− π

3

)= 1. (7)

(c) Find

∫ √(1 + cos 2x) dx. (2)



1992-1 Paper(1) Q.7

8 In 4ABC, AB = 6 cm, BC = 8√

3 cm and ∠ABC is obtuse. The area of the triangle is 36 cm2.

(a) Show that ∠ABC = 120◦.

(b) Calculate, in cm to 3 significant figures, the length of the side AC.

(c) Calculate, in cm to 2 significant figures, the distance of B from the side AC.


1996-1 Paper(1) Q.7

9

A

C

B

x◦ 6 cm

y cm

Fig. 1

Figure 1 shows 4ABC in which AB = y cm, ∠BAC = x◦ and AC = 6 cm. Given that the area

of the triangle is 15 cm2,

(a) find, in terms of y, the value of sinx◦. (3)

Given that y2 cos2 x◦ = y2 − k2,

(b) find the value of the positive constant k. (3)

Given that x = 30,

(c) find the value of y. (1)

1997-6 Paper(1) Q.1

10 In 4ABC,AB = 4 cm, AC = 6 cm, ∠ACB = 40◦ and ∠ABC is obtuse.

Calculate, to the nearest 0.1◦, the size of ∠ABC. (4)

2007-1 Paper(2) Q.1

11 A triangle has sides of length 4.6 cm, 5.3 cm and 6.5 cm. Find, to the nearest degree, the size of

the largest angle of the triangle. (3)


2007-6 Paper(2) Q.3

12 In 4LMN,LM = 5.6 cm, LN = 8.2 cm and ∠MLN = 57◦. Find, to 3 significant figures,

(a) the length of MN , (3)

(b) the size of ∠LNM . (3)

2008-1 Paper(1) Q.1

13 Triangle LMN has LM = 5 cm, LN = 8.2 cm and MN = 6.4 cm. Calculate, in degrees to the

nearest 0.1◦, the size of ∠LMN . (3)

2009-6 Paper(1) Q.9

14

B C

D

A

P

12 cm

9 cm

8 cm

Figure 3

Figure 3 shows 4ABC with AB = 12 cm, AC = 8 cm and BC = 9 cm.

The point D is on BA produced and the bisector of ∠DAC meets BC produced at P .

(a) Find, to the nearest 0.1◦, the size of each of the three angles of 4ABC. (6)

(b) Find, to the nearest cm, the length of BP . (5)


2010-1 Paper(1) Q.1

15 In 4ABC,AB = 5.7 cm, BC = 8.4 cm and ∠ACB is 42◦.

Find, to the nearest 0.1◦, the two possible sizes of ∠BAC. (4)

2010-6 Paper(1) Q.4

16 The lengths of the sides of a triangle are 5 cm, 6 cm and 8 cm.

(a) Find, in degrees to one decimal place, the size of the smallest of the triangle. (4)

(b) Find, to the nearest cm2, the area of the triangle. (3)

2011-1 Paper(1) Q.2

17 A triangle ABC has AB = 4.6 cm, AC = 5.7 cm and ∠C = 52◦

Angle B is acute.

Calculate, to the nearest 0.1◦, the size of ∠B. (3)

2011-6 Paper(2) Q.3

18 In triangle ABC, AB = 5 cm, AC = 3 cm, angle B = 25◦ and angle C is obtuse.

(a) Find, to the nearest degree, the size of angle C. (3)

The point D lies on BC produced and AD = 3 cm.

(b) Find, to 3 significant figures, the length of CD. (3)


2012-6 Paper(1) Q.2

19 In triangle ABC, AB = 8 cm, BC = 5 cm and CA = 7 cm.

(a) Find, to the nearest 0.1◦, the size of angle BAC. (3)

(b) Find, to 3 significant figures, the area of triangle ABC. (2)

2013-1 Paper(1) Q.6

20

A CD

B

6 cm10 cm

6 cm

28◦

Figure 1

Figure 1 shows triangle ABC with AB = 10 cm, BC = 6 cm and ∠BAC = 28◦. The point D lies

on AC such that BD = 6 cm.

(a) Find, to the nearest 0.1◦, the size of ∠DBC. (4)

(b) Find, to 3 significant figures, the length of AD. (3)

(c) Find, to 3 significant figures, the area of the triangle ABC. (3)


2014-1 Paper(1) Q.6

21 In triangle ABC, AB = x cm, BC = 7 cm, AC = (5x− 6) cm and ∠BAC = 60◦

(a) Find, to 3 significant figures, the value of x. (5)

Using your value of x

(b) find, in degrees to 1 decimal place, the size of ∠ACB. (3)

2014-6 Paper(2) Q.11

22 In triangle ABC, ∠BAC = 60◦, AB = (3x− 1) cm, AC = (3x+ 1) cm and BC = 2√

7x cm.

(a) Show that (9x− 1)(x− 3) = 0 (3)

(b) Hence find the value of x, justifying your answer. (2)

(c) Find, to the nearest 0.1◦, the size of angle ABC. (3)

(d) Find the exact value, in cm2, of the area of triangle ABC. (2)


2015-6 Paper(1) Q.6

23

A C

B

22 cm 20 cm

14 cm

Figure 1


(a) Find, to 3 decimal places, the size of each of the three angles of 4ABC. (5)

The bisector of angle BAC meets BC at P .

(b) Find, in cm to 3 significant figures, the length of AP . (3)

(c) Find, to the nearest cm2, the area of 4ABC. (2)


2016-1 Paper(1) Q.7

24

B C

A

D4 cm

5 cm

4 cm

30◦

Figure 1

Figure 1 shows the triangle ABC with AB = 4 cm, BC = 5 cm and angle BCA = 30◦

The point D lies on AC such that BD = 4 cm and angle BDC is obtuse.

Find

(a) the size of angle BDC, giving your answer in degrees correct to 1 decimal place, (3)

(b) the length, in cm, of AD, giving your answer correct to 3 significant figures, (3)

(c) the area, in cm2, of triangle ABD, giving your answer correct to 3 significant figures. (2)

2016-6 Paper(2) Q.1

25 A triangle has sides of length 10 cm, 8 cm and 9 cm.

(a) Calculate, in degrees to the nearest 0.1◦, the size of the largest angle of this triangle. (3)

(b) Find, to 3 significant figures, the area of this triangle. (2)


2017-1 Paper(1) Q.5

26

B

A

C

D

8 cm

12 cm

120◦

35◦

Figure 2

Figure 2 shows the quadrilateral ABCD in which AB = BC.

DC = 8 cm AC = 12 cm ∠ABC = 120◦ ∠CAD = 35◦

Find

(a) the exact length, in cm, of AB. (2)

Given that angle ADC is obtuse, find

(b) the size, in degrees to 1 decimal place, of angle ADC, (3)

(c) the area, in cm2 to 3 significant figures, of the quadrilateral ABCD. (6)


2018-1 Paper(1) Q.6

27

B

A

C

θ◦

x cm (x+ 4) cm

(2x− 2) cm

Figure 1

Figure 1 shows the triangle ABC with AB = x cm, BC = (2x − 2) cm, AC = (x + 4) cm and

∠BAC = θ◦

Given that tan θ◦ =√

255 and without finding the value of θ,

(a) show that cos θ◦ =1

16(2)

Hence find


(c) the size, in degrees to 1 decimal place, of ∠ABC, (2)

(d) the area, in cm2 to 3 significant figures, of triangle ABC. (2)


14 Definite Integral

1989-1 Paper(2) Q.5

1 Evaluate

∫π4

π2 (sin 2x− cos 2x) dx. (5)

1989-1 Paper(2) Q.13

2 (a) By expanding each side of the equation

cos (x◦ + 30◦) = 2 sin (x◦ + 60◦),

solve the equation for 0 < x < 180. (6)

(b) Show that

cos

(x+

π

4

)+ cos

(x− π

4

)= 2 cosx cos

π

4.

(3)

(c) Hence, or otherwise, find the maximum value of

cos

(x+

π

4

)+ cos

(x− π

4

).

(2)

(d) Find

∫0

π2

[cos

(x+

π

4

)+ cos

(x− π

4

)]dx. (4)

1989-6 Paper(1) Q.4

3 (a) Differentiate 5x− 1

xwith respect to x. (2)

(b) Evaluate

∫ 1

0(2 +

√x)

2dx. (3)


1989-6 Paper(2) Q.4


(b) Evaluate

∫0

π8 cos (2x+

π

4) dx. (3)

1990-1 Paper(1) Q.12

5 (a) Evaluate

∫ 4

2

(3x− 2

x2

)dx. (4)

A B

M CD

y m

4x m

y m3x m

4x m

Fig. 3

In Fig. 3, ABCD is a trapezium. The point M is such that ABCM is a rectangle.

(b) Find AD in terms of x. (1)

The perimeter of ABC is 180 m.

(c) Find y in terms of x. (2)

(d) Write down and simplify an expression in terms of x for the area, in m2, of ABCD. (4)

(e) Find the maximum value of this area in m2. (4)

1991-1 Paper(2) Q.5

6 Given that

∫ 2

0eax dx =

1

a, where a is a constant, calculate, to 2 significant figures, the value of

a. (5)


1991-1 Paper(2) Q.9

7 (a) Given that cos (x+ α) = 2 sin (x− α) and that both cosx and cosα are non-zero, show that

tanx =1 + 2 tanα

2 + tanα.

(4)

(b) Starting with the formula for cos (A+B), show that cos 2A = 1− 2 sin2A.

Hence show that

∫0

π12 sin2 3x dx =

1

24(π − 2). (5)

Solve, for x in the interval 0 6 x 6 π, giving your answers to 3 significant figures, the equations

(c) 3 sin 2x = cosx, (3)

(d) cos 2x = sinx. (3)

1991-6 Paper(1) Q.2

8 Evaluate ∫ 2

−2(2x+ 5)(x2 + 1) dx.

(4)

1992-1 Paper(1) Q.3

9 Evaluate

∫ 9

1(3x

12 + 5) dx. (4)

1993-6 Paper(2) Q.3

10 (a) Evaluate

∫(sinx+ cos 3x) dx. (2)

(b) Hence calculate, to 3 significant figures.

∫ π3

π6

(sinx+ cos 3x) dx. (2)


1993-6 Paper(2) Q.5

11 The volume, V cm3, of a sphere of radius r cm, is giving by the formula V =4

3πr3.

An x% error, where x is small, is made in measuring the radius of the sphere.

Find an estimate for the percentage error in the value of V . (5)

1993-6 Paper(2) Q.14

12 (a) Using the formula for cos (A+B), show that

cos 2x = 2 cos2 x− 1.

(4)

(b) Evaluate

∫0

π4 2 cos2 2x dx, giving your answer in terms of π. (4)

(c) Sketch for −π < x 6 π the curve with equation y = cos 2x. (2)

The line l has equation y = 1− 4x

π.

(d) Draw the line l on your sketch. (1)

The finite region bounded by the curve with equation y = cos 2x, and the line l, for 0 6 x 6 14π,

is rotate through 360◦ about the x-axis.

(e) Calculate, in terms of π, the volume of the solid generated. (4)

1994-1 Paper(1) Q.8

13 Given that y = (3x+ 1x)2

(a) finddy

dx, (2)

(b) evaluate

∫ 3

2y dx. (5)


1994-1 Paper(2) Q.5

14 Find the exact values of

(a)

∫ 9

1x

12 dx, (3)

(b)

∫0

π6 cos 3x dx. (2)

1994-1 Paper(2) Q.13

15 Using the formula

sinA+B = sinA cosB + cosA sinB,

(a) show that

cos (A−B)− cos (A+B) = 2 sinA sinB.

. (3)

(b) Hence show that

cos 2x− cos 4x = 2 sin 3x sinx.

(1)

(c) Find all solutions in the range 0 ≤ x ≤ π of the equation

cos 2x− cos 4x = sinx,

giving your solution in multiples of π radians. (7)

(d) Evaluate ∫0

π4 sin 3x sinx dx.

(4)


1995-1 Paper(1) Q.3

16 (a) Find

∫4

x3dx. (2)

(b) Hence evaluate

∫ 4

1

4

x3dx. (2)

1995-1 Paper(2) Q.4

17 Evaluate

∫ 1

04e3x dx, giving your answer in terms of e. (4)

1995-1 Paper(2) Q.14

18 Using the identity cos (A+B) = cosA cosB − sinA sinB show that

(a) cos 2x = 2 cos2 x− 1, (3)

(b) cos 6θ = 1− 2 sin2 3θ. (2)

Hence show that

(c) cos 2x+ cos 6θ = 2(cosx+ sin 3θ)(cosx− sin 3θ). (2)

I =

∫0

π12 (cosx+ sin 3θ)(cosx− sin 3θ) dx.

Given that θ is a constant

(d) show that I =a+ π cos 6θ

b, where a and b are constants to be found.

Given 0 6 θ 6π

3find the values of θ, in radians to 3 significant fgures, for which I = 0. (8)

1995-6 Paper(2) Q.6

19 Find in its simplest form, in terms of a, the value of

∫0

πa sin axdx, a 6= 0. (5)


1995-6 Paper(2) Q.13


(3)

(b) Hence show that

∫4 sin2 3x dx =

1


(c) Evalute

∫ π4




1996-1 Paper(2) Q.6

21 Evaluate, giving your asnwers in terms of e.∫ 1

0(ex + 1)2 dx.

(5)


1996-1 Paper(2) Q.11

22 (a) Use the formula


to show that

cos2 θ = 12(1 + cos 2θ).

(3)

(b) Express sin2 θ in terms of cos 2θ. (2)

f(θ) = 3 sin2 θ + 4 cos2 θ.

(c) Using your answer to (a) and (b), or otherwise, express f(θ) in terms of cos 2θ. (2)

(d) Hence evaluate, giving your answer in terms of π, (4)

∫0

π4 f(θ) dθ.

(e) Solve, in radians to 2 decimal places, in the range 0 6 θ < 2π, the equation f(θ) = 3.8. (4)


1997-6 Paper(2) Q.10

23 Using the identity cos (A+B) = cosA cosB − sinA sinB,

(a) solve, in radians to 3 decimal places, for 0 6 x < 2π, the equation

(4)

cos (x+ π3 ) = 3 sinx,

(b) show that cos 2x = 1− 2 sin2 x, (3)

(c) solve, in radians to 3 decimal places, for 0 6 x < 2π, the equation

cos 2x = 2 sinx.

(4)

(d) Find∫

sin2 xdx. (2)

(e) Use your answer to (d) to evaluate ∫0

π3 sin2 x dx,

giving your answer in terms of π. (2)

2007-1 Paper(1) Q.8

24 cos (A+B) = cosA cosB − sinA sinB.

f(θ) = 5 cos θ − 12 sin θ.

Given that f(θ) = p cos (θ + β), p > 0, 0 < α < π2 ,

(a) (i) Show that p = 13,

(ii) find in radians to 3 significant figures, the value of α. (5)

(b) Hence solve, to 2 significant figures, for 0 6 θ < 2π, 5 cos θ − 12 sin θ = 9. (4)

(c) Evaluate∫0

π3 f(θ) dθ, giving your answer in the form c + d

√3, where c and d are rational

numbers. (5)


2007-6 Paper(2) Q.10

25 cos (A+B) = cosA cosB − sinA sinB

sin (A+B) = sinA cosB + cosA sinB.

(a) Write down an expression for sin 2θ in terms of sin θ and cos θ. (1)

Show that

(b) sin2 θ = 12(1− cos 2θ). (2)

(c) sin2 (A+B)− sin2 (A−B) = sin 2A sin 2B. (5)

(d) Hence show that

(i) sin2 3θ − sin2 θ = sin 4θ sin 2θ,

(ii) sin2 3θ − sin2 θ = 12(cos 2θ − cos 6θ). (4)

(e) find the exact value of∫ π

30 (6 sin 4θ sin 2θ + 2) dθ. (5)

2008-1 Paper(1) Q.9


cos (A+B) = cosA cosB − sinA sinB.

(a) Obtain an expression for cos 2θ in terms of cos2 θ. (2)

(b) Write down an expression for sin 2θ in terms of sin θ and cos θ. (1)

(c) Show that cos 3θ = 4 cos3 θ − 3 cos θ. (4)

(d) Solve, for 0 6 θ 6 π, the equation 9 cos θ− 12 cos3 θ = 2, giving your answers to 3 significant

figures. (4)

(e) Find∫ π

20 (3 cos3 θ + 2 sin θ) dθ. (5)


2008-6 Paper(2) Q.10

27 (a) Expand (1 +1

2x)


each term. (3)

(b) Expand (1− 1

2x)− 1


each term. (3)

(c) State the range of values of x for which both expansions are valid. (1)

(d) Expand

(2 + x

2− x


each term. (3)

(e) Hence obtain an estimate, to 4 significant figures, of∫ 0.5

0

(2 + x

2− x

)15 dx. (4)

2009-6 Paper(2) Q.10

28 cos (A+B) = cosA cosB − sinA sinB,

cos (A−B) = cosA cosB + sinA sinB.

(a) Prove that cos 2A = 2 cos2A− 1 (2)

f(θ) = cos 5θ + cos 3θ + 2 cos θ

(b) Show that

(i) cos 5θ + cos 3θ = 2 cos 4θ cos θ,

(ii) f(θ) = 16 cos5 θ − 16 cos3 θ + 4 cos θ. (6)

(c) Hence or otherwise solve, for −π 6 θ 6 π, giving the values of θ in terms of π, the equation

cos 5θ + cos 3θ − 2 cos θ = 0 (5)

(d) Find, to 3 significant figures, the value of∫ π

30 (cos5 θ − cos3 θ) dθ. (5)


2010-6 Paper(1) Q.7

29 (a) Expand (1 + 3x2)− 1



(b) State the range of values of x for which your expansion is valid. (1)

f(x) =2 + kx2

(1 + 3x2)12

, k 6= 0


x6. (3)

Given that the coefficient of x6 in the series expansion of f(x) is zero,

(d) show that k = 5 (2)

(e) Hence use your series expansion form part (c) to obtain an estimate, to 4 decimal places, of

∫ 0.30 f(x) dx.

(4)

2010-6 Paper(2) Q.9


Show that

(a) cos (A+B) + cos (A−B) = 2 cosA cosB, (1)

(b) cos 2A = 2 cos2A− 1, (2)

(c) cosP + cosQ = 2 cos

(P +Q

2

)cos

(P −Q

2

). (2)

(d) Hence show that cos 8x+ 2 cos 6x+ cos 4x = 4 cos 6x cos2 x. (4)

(e) Find the exact value of∫ π

40 cos 6x cos2 x dx. (6)


2011-1 Paper(1) Q.8


(a) Show that

(i) sin2 θ = 12(1− cos 2θ)

(ii) cos2 θ = 12(cos 2θ + 1) (3)

f(θ) = 8 sin4 θ + 4 sin2 θ − 5

(b) Show that f(θ) = cos 4θ − 6 cos 2θ (4)

(c) Solve, for 0 6 θ 6 π2 , the equation 4 sin4 θ + 2 sin2 θ + 3 cos 2θ = 2.4

Give your solutions to 3 significant figures. (4)

Given that 4

∫ π4

π8

f(θ) dθ = m+ n√

2

(d) find the value of m and the value of n. (5)

2013-6 Paper(2) Q.3

32 (a) (i) Find

∫ (1 + 3x− 2

x2

)dx


∫ 2

1

(1 + 3x− 2

x2

)dx = 4

1

2(4)

(b) (i) Find

∫3 sin 2x dx


∫ π6

03 sin 2x dx =

3

4(4)


2014-6 Paper(2) Q.10








(d) Find

∫(24 sin3 θ + 6 cos θ) dθ (2)

(e) Hence evaluate

∫ π3


√c, where a,



2015-6 Paper(1) Q.8








(4)

(d) (i) Find

∫sin3 θ dθ

(ii) Evaluate

∫ π4


a− b√

2


integers. (5)


2016-6 Paper(2) Q.9








6 cos θ − 8 cos3 θ + 1 = 0

(4)

(e) find

(i)



∫ π3

0(8 cos3 θ + 4 sin θ) dθ (4)


2017-6 Paper(1) Q.9

36 Using



2(cos 2θ + 1) (2)

f(θ) = 8 cos4 θ + 4 cos2 θ − 5


Hence


8 cos4 x+ 4 cos2 x− 6 cos 2x = 4.5

(4)

(d) find

(i)∫

f(θ) dθ


30 f(θ) dθ (5)


2018-1 Paper(1) Q.10


(a) Show that cos2 θ =1

2(cos 2θ + 1) (3)

Given that f(θ) = 8 cos4 θ + 8 sin2 θ − 7

(b) show that f(θ) = cos 4θ (5)

(c) Solve, for 0 6 θ 6π

2, the equation

16 cos4(θ − π

6

)+ 16 sin2

(θ − π

6

)− 15 = 0

(4)

(d) Using calculus, find the exact value of∫ π2

0(8 cos4 θ + 8 sin2 θ + 2 sin 2θ) dθ

(4)


15 Differentiation

1988-1 Paper(1) Q.3

1 (a) Differentiate with the respect to x,

3x2 − 2

x.

(2)

(b) Evaluate

∫ 2

1(x− 3)2 dx. (3)

1988-1 Paper(2) Q.1


x

sinx.

(3)


1988-1 Paper(2) Q.11

3 The function f is defined by

f: x 7→ x

(x− 1)(x+ 3)where x ∈ R and x 6= −3, x 6= 1.

(a) Verify that f ′(x) =−3− x2

(x− 1)2(x+ 3)2. (5)

(b) Give a reason why f ′(x) < 0. (2)

(c) Determine the number of roots of the equation

x =x

(x− 1)(x+ 3).

(4)

(d) The region enclosed by the curve whose equation is y = ex, the x-axis and the lines x = −1

and x = 2 is completely rotated about the x-axis. Calculate, to 2 significant figures, the

volume of the solid so formed. (4)

1988-1 Paper(2) Q.13

4 f(x) = x2 cos 2x where −π46 x 6

π

4.

(a) Find f ′(x). (3)

(b) Show that f ′(x) = 0 when x = 0. (1)

(c) Show that the other values of x, in the interval −π4

6 x 6π

4, for which f ′(x) = 0 may be

obtained by solving the equation tan 2x =1

x. (3)

(d) Sketch, for −π46 x 6

π

4, on the same diagram the graphs of y = tan 2x and y =

1

x.

Hence, find the number of points on the graph of y = f(x) at which f ′(x) = 0 in the interval

−π46 x 6

π

4. (8)


1988-6 Paper(1) Q.5

5 (a) Differentiate

(1

x+ 3x

)with respect to x. (2)

(b) Evaluate

∫ 2

1x(x+ 2) dx. (3)

1988-6 Paper(2) Q.3


(a) x sinx, (2)

(b)√

(1 + x3). (3)

1989-1 Paper(1) Q.7

7 Find the coordinates of the turning points of the curve whose equation is

y =

(x+

1

x

)2

.

(6)

1989-1 Paper(2) Q.6


(a) ex(x2 + 1), (3)

(b)sin 3x

x2. (3)


1989-1 Paper(2) Q.14

9 y = xex

(a) Finddy

dx. (2)

(b) Find the coordinates of the turning point of the curve y = xex and determine whether this

point is a maximum or minimum. (6)

(c) Using the result of (a), find

∫xex dx. (4)

(d) Hence, or otherwise, find the area enclosed by the curve y = xex, the ordinate x = 2 and the

x-axis, leaving your answer in terms of e. (3)

1989-6 Paper(1) Q.4

10 (a) Differentiate 5x− 1

xwith respect to x. (2)

(b) Evaluate

∫ 1

0(2 +

√x)

2dx. (3)

1989-6 Paper(2) Q.4


(b) Evaluate

∫0

π8 cos (2x+

π

4) dx. (3)


1989-6 Paper(2) Q.10




cosxshow that

d


(3)





terms of π. (5)

1990-1 Paper(2) Q.2


(a) e−3x sinx, (2)

(b) (1 +√x)5. (2)

1990-6 Paper(1) Q.2

14 (a) Differentiate 4 + 3√x with respect to x. (2)

(b) Find

∫ (5 +

1

x2

)dx. (2)


1991-1 Paper(2) Q.3


(a) x2 sin 7x, (2)

(b) cosx2. (2)

1991-6 Paper(2) Q.7




1991-6 Paper(2) Q.14

17 f(x) =(x− 3)2

(1− 2x), x 6= 1

2 .

(a) Show that f ′(x) =2(x+ 2)(3− x)

(1− 2x)2. (3)









1993-6 Paper(2) Q.2


(a) e2x cos 3x, (2)

(b)2x

e3x. (2)

1993-6 Paper(2) Q.11

19 f(x) = 8e3x + 27e−3x.

(a) Find the rate of change of f(x) with respect to x when x = 12 , giving your answer to 2 decimal

places. (4)

(b) Find the area of the finite region bounded by the curve with equation y = f(x), the x-axis

and the lines x = 13 and x = −1

3 , giving your answer in terms of e. (5)

(c) By using the substitution z = e3x solve, for x > 0, the equation f(x) = 35, giving your answer

to 2 decimal places. (6)

1994-1 Paper(2) Q.2


(a) e−x sinx, (2)

(b)1

cos 2x. (2)


1995-1 Paper(2) Q.6


(a)2x

ex, (3)

(b) 3x sin 2x. (3)

1995-6 Paper(2) Q.2


(a) x3 cos 2x, (2)

(b)ex

1 + x. (2)

1995-6 Paper(2) Q.14


(a) xdy

dx= y(3x+ 1). (4)

(b)1

y

dy

dx= 3 +

1

x. (3)






1996-1 Paper(1) Q.6

24 (a) Differentiate 2 +3√x

with respect to x. (3)

(b) Find

∫(2− x)3 dx. (3)

1996-1 Paper(2) Q.4


(a) x2ex, (2)

(b) cos (x2 + 2x). (3)

1996-6 Paper(2) Q.6


(a) x3 cos 2x, (3)

(b)x3

sinx, 0 < x < π (2)


1997-6 Paper(2) Q.14

27 f(x) =2x2

x− 3, x ∈ R, x 6= 3.

(a) Find f ′(x). (3)

(b) Find the set of values of x for which f ′(x) > 0. (4)

g(x) = ex sinx, 0 6 x 6 2π.

(c) Solve the equation g(x) = 0. (2)

(d) Solve the equation g′(x) = 0. (2)

(e) Sketch the graph of the curve with equation y = g(x), showing the coordinates of the points

where the curve intersects the axes. (2)

(f) State the set of values of x for which g′(x) < 0. (2)

2007-1 Paper(1) Q.1

28 Differentiate with respect to x, (x+ 2)e3x. (3)

2007-6 Paper(2) Q.1


x2 + 3. (3)


2008-1 Paper(1) Q.4

30 Given that y = (3x− 2)e2x.

(a) finddy

dx, (3)

(b) show that (3x− 2)dy

dx= (6x− 1)y. (3)

2011-1 Paper(1) Q.1

31 Differentiate x2 cos 3x (3)

2011-1 Paper(1) Q.5


x+ 2,

(a) show thatdy

dx=y(kx+ 2k − 1)

x+ 2(5)

Given also thatdy

dx=

5

4when x = 0,





2011-6 Paper(1) Q.3


(a) finddy

dx(3)

(b) show thatd2y

dx2= 2

dy

dx− 9y + 6e2x cos 3x (4)

2012-1 Paper(1) Q.5


(a) y = x2ex (2)

(b) y = (x3 + 2x2 + 3)5 (3)


2012-1 Paper(1) Q.7

35 The curve C with equation y =2x− 3

x− 3, x 6= 3, crosses the x-axis at the point A and the y-axis

at the point B.

(a) Find the coordinates of A and the coordinates of B. (2)

(b) Write down an equation of the asymptote to C which is

(i) parallel to the y-axis,

(ii) parallel to the x-axis. (2)

(c) Sketch C showing clearly the asymptotes and the coordinates of the points A and B. (3)

(d) Find an equation of the normal to C at the point B. (5)

The normal to C at the point B crosses the curve again at the point D.

(e) Find the x-coordinate of D. (4)

2012-1 Paper(2) Q.9

36 The curve C, with equation y = f(x), passes through the point with coordinates (0, 4).

Given that f ′(x) = x3 − 3x2 − x+ 3

(a) find f ′(x). (3)

(b) Show that C has a minimum point at x = −1 and a minimum point at x = 3 (6)

(c) (i) Find the coordinates of the maximum point on C.

(ii) Show that the point found in (i) is a maximum point. (3)

(d) State the ranges of values of x for which f ′(x) > 0 (2)


2012-6 Paper(2) Q.4


(a)1

x2(2)

(b)1

(2x+ 1)2(2)

(c)1

1− cos2 x(3)

2013-1 Paper(2) Q.4


(a) 3x sin 5x (3)

(b)e2x

4− 3x2(3)

2014-1 Paper(1) Q.3


(a) e3x(5x− 7)2 (3)

(b)cos 2x

x+ 9(3)


2014-6 Paper(2) Q.8

40 A curve has equation y =3x− 2

4x+ 5, x 6= −5

4

(a) Write down an equation of the asymptote to the curve which is parallel to

(i) the x-axis (ii) y-axis. (2)

(b) Find the coordinates of the points where the curve crosses

(i) the x-axis (ii) the y-axis. (2)

(c) Sketch the curve, showing clearly the asymptotes and the coordinates of the points where

the curve crosses the coordinate axes. (3)

(d) Find an equation of the normal to the curve at the point where x = −1

Give your answer in the form ax+ by + c = 0 where a, b and c are integers. (7)

2015-6 Paper(1) Q.2


(a) finddy

dx(3)


dx= 2y(1 + x) (2)

2016-1 Paper(1) Q.1

42 f(x) = 3x3 + 2 sinx− 4

x2where x 6= 0

(a) Find f ′(x) (3)

(b) Find∫

f(x) dx (4)


2016-1 Paper(2) Q.4

43 Given that y = e2x√x+ 1

show thatdy

dx=e2x(4x+ 5)

2√x+ 1

(6)

2016-1 Paper(2) Q.7

44

O x

y

Figure 1

Figure 1 shows the curve with equation y =x2 − 2

2x− 3where x 6= 3

2


(b) Finddy

dx(3)

(c) Find the coordinates of the stationary points on the curve. (5)

2016-6 Paper(2) Q.4


e2x cos 3x

(3)


2016-6 Paper(2) Q.8


y =3x2 − 1

3x+ 2where x 6= −2

3







O x

y

(3)




2017-1 Paper(2) Q.5

47 Given that y = 3x√

2x− 1 x >1

2

(a) show thatdy

dx=

3(3x− 1)√2x− 1

(5)

The straight line l is the normal to the curve with equation y = 3x√

2x− 1 at the point on the

curve where x = 1

(b) Find an equation, with integer coefficients, for l. (6)

2018-1 Paper(2) Q.5


show thatd2y

dx2− 2

dy

dx+ y = 12ex (7)


16 Discriminant

1988-1 Paper(2) Q.10

1 (a) Given that 2x2 + 5x− 7 = k, find the value of k for which this quadratic equation has equal

roots. (5)

(b) Given that

f(x) = 2x2 + 5x− 7 = A(x+B)2 + C,

find the values of the constants A, B and C. (5)

(c) Hence, or otherwise, find the least value of f(x) and state the value of x for which this occurs.

(2)

(d) Sketch the curve with equation

y = f(x).

(3)


1988-6 Paper(2) Q.10

2 Given that

f(x) = 2x2 + px+ 18,

(a) write down, in terms of p, the two solutions of f(x) = 0, (2)

(b) find the range of values of p for which the equation f(x) = 0 has unreal roots. (3)

Given that p = 16,

(c) solve f(x) = 7, giving your answers to 2 decimal places. (4)

Given instead that one root of the equation f(x) = 0 is four times the other,

(d) find the two possible values of p, (4)

(e) find the two pairs of solutions of the equation corresponding to these values of p. (2)

1989-1 Paper(2) Q.1

3 Given that p is a positive integer, find the smallest value of p for which the equation

x2 + px+ 3 = 0

has real roots. (3)


1989-1 Paper(2) Q.10

4 Given that the equation x2 − 5x+ 7 = 0 has roots α and β,

(a) find a quadratic equation, with integer coefficients, whose roots are1

αand

1

β. (6)

(b) Find constant p and q such that

x2 + 3x+ 8 = (x+ p)2 + q

Hence show that x2 + 3x+ 8 > 0 for all real values of x. (4)

(c) Use your answer to (a) to find the minimum value of the x2 + 3x+ 8. (2)

(d) Confirm your answer to (c) by using a calculus method. (2)

(e) Sketch the curve whose equation is y = x2 + 3x+ 8. (1)

1991-1 Paper(2) Q.2

5 Find the values of the constants p for which the equation

x2 − 2px+ 5p = 4

has equal roots. (4)

1996-6 Paper(2) Q.1

6 Find the set of values of the constant k for which the equation 2x2 − kx+ 2k = 6 has real roots.

(4)

2008-6 Paper(2) Q.1

7 Find the set of values of p for which the equation x2 + 2px+ (10−3p) = 0 has real, unequal roots.

(4)


2009-6 Paper(2) Q.1

8 Find the set of values of the constant p for which the equation

4x2 + 4(2− p)x+ (3p− 8) = 0 has no real roots. (4)

2011-1 Paper(2) Q.1

9 Find the set of values of m for which the equation x2 +mx+ 9 = 0 has real roots. (3)

2013-1 Paper(1) Q.2

10 The equation x2 + 4px+ 9 = 0 has unequal real roots. Find the set of possible values of p. (4)

2015-1 Paper(2) Q.3

11 The equation 2x2 + 3x+ c = 0, where c is a constant, has two equal roots.

(a) Find the value of c. (2)

(b) Solve the equation. (2)


2016-6 Paper(1) Q.9

12 f(x) = 3x2 − 5x− 4



(i) rootsα

βand

β

α




2017-6 Paper(2) Q.3

13 (a) Find the set of possible values of p for which the equation 3x2 + px+ 3 = 0 has no real roots.

(3)

(b) Find the integer values of q for which the equation x2 + 7x+ q2 = 0 has no real roots. (3)

2018-1 Paper(2) Q.4

14 Here is a quadratic equation 3x2 + px+ 4 = 0 where p is a constant.

(a) Find the set of values of p for which the equation has two real distinct roots. (5)

(b) List all the possible integer values of p for which the equation has no real roots. (1)


17 Disguised Quadratic

1988-1 Paper(2) Q.9

1 (a) Use the substitution y = 3x to find the solution, to 2 decimal places, of the equation

32x − 3(3x)− 4 = 0.

Give a reason why there is only one root of this equation. (6)

(b) Sketch the curve with equation

y = log3 x.

(3)

Given that

y = log3 x and y = 12 [1 + log3 9x],

(c) find the value of x and the corresponding value of y which satisfy these simultaneous equa-

tions. (6)

1989-6 Paper(2) Q.7

2 Given that 3x = y,

(a) express 32x and 3x+1 in terms of y. (2)

(b) Re-write the equation 32x − 2(3x+1) + 9 = 0 as a quadratic equation in y. (1)

(c) Solve this equation for y and, hence, obtain x. (3)


1991-1 Paper(2) Q.11

3 (a) Sketch the graph of y = 3x. (1)

(b) Solve the equation 3x = 5, giving your answer to 3 significant figures. (2)

(c) Use the substitution y = 3x to solve, to 2 decimal places, the equation

2(32x)− 7(3x) + 6 = 0.

(5)

(d) The tangent to the curve with equation y = e−2x, at the point whose x-coordinate is 1,

crosses the x-axis at P and the y-axis at Q. Show that the area of 4POQ, where O is the

origin, is9

4e2. (7)

1993-6 Paper(2) Q.10

4 (a) Show that roots of the equation x2 + 4x+ 2 = 0 are −2 +√

2 and −2−√

2. (4)

(b) Hence solve the equation sin2 y + 4 sin y + 2 = 0, where 0◦ < y 6 360◦, giving your answers

to the nearest degree. (3)

(c) Given that t1 and t2 are the roots of the equation e2t + 4et + 2 = 0, show that t1 + t2 = ln 2.

(4)

(d) Find in the form px2 +qx+r = 0 the quadratic equation with roots −4+2√

2 and −4−2√

2.

(4)


1994-1 Paper(2) Q.10

5 (a) Solve the equation

32x+1 = 7,


(b) Solve the equation

4x2 − 17x+ 4− 0.

(2)

(c) Using the substitution y = log2 x, solve, giving your answers to 2 decimal places where

appropriate, the equation

4(log2 x)2 − 17 log2 x+ 4 = 0.

(4)

(d) Using the substitution z = sin2 θ◦, or otherwise, solve, giving all solutions in the range

0 ≤ θ < 360, the equation

4 sin4 θ◦ − 17 sin2 θ◦ + 4 = 0.

(3)

(e) Solve the equation

4(22x)− 17(2x) + 4 = 0.

(2)


1995-1 Paper(2) Q.13

6 (a) Solve 2x2 − 7x+ 5 = 0. (2)

(b) Using the substitution x = 2y solve 22y+1−7(2y)+5 = 0, giving your answers to 3 significant

figures where appropriate. (4)

(c) Given that log16 y =log2 y

k, that log8 x =

log2 x

l, and that log4 y =

log2 y

m, find k, l, and m.

(3)

(d) Hence solve the simultaneous equations

log2 x+ 28 log16 y = 17,

9 log8 x+ 8 log4 y = 17.

(6)

2011-1 Paper(2) Q.8

7 (a) Solve

5p2 − 13p+ 6 = 0

(2)

(b) Hence solve 52x+1 − 13(5x) + 6 = 0, giving your answers to 3 significant figures. (5)

The curve with ewuation y = 52x+1 − 3(5x) + 2 meets the curve with ewuation y = 10(5x)− 4 at

two points.

(c) Find the exact value of the y-coordinate of each of these two points. (4)


2011-6 Paper(1) Q.7

8 (a) Solve

5p2 − 11p+ 2 = 0

(2)

(b) Hence solve 5(32x)− 11(3x) + 2 = 0 giving your answers to 3 significant figures. (4)

The curve with equation y = 5(32x) − 6(3x) intersects the curve with equation y = 5(3x) − 2 at

two points.

(c) Find the coordinates of each of these two points, giving your answers to 3 significant figures

where appropriate. (4)

2015-1 Paper(1) Q.6

9 (a) Solve, giving your answer to 3 significant figures,

3z − 4 = 0

(3)

Solve, giving your answer to 3 significant figures where appropriate,

(b) 9y − 13(3y) + 36 = 0 (4)

(c) 6x − 4(2x)− 3x + 4 = 0 (5)

2017-6 Paper(1) Q.1

10 Find the exact solution of the equation

16

ex− ex = 6

(5)


18 Geometric Series

1988-1 Paper(1) Q.7

1 Of the three series

(i) 1− 13 + 1

9 −127 + ...,

(ii) 1− 13 − 1− 4

3 − ...,(iii) 1− 1

3 − 123 − 3− ...,

one is an arithmetic series and one is a geometric series.

(a) Find the tenth term of the arithmetic series. (3)

(b) Find the sixth term of the geometric series. (3)

1988-6 Paper(1) Q.10

2 (a) In a geometric series, (x − 1), (x + 1) and (x + 9) are consecutive terms. Find the value of

the common ratio of the series. (5)

The sum of the first n terms of an arithmetic series is 6014. Given that the sum of the first term

and the nth term is 124.

(b) calculate the value of n, (3)

(c) show that the value of the 49th term is 62. (2)

Given also that the value of the 61st term is 77, find

(d) the value of the first term of the series, (2)

(e) the common difference, (1)

(f) the sum of the first 80 terms. (2)


1988-6 Paper(2) Q.9

3 (a) Write down the common ratio of the geometric series G,

e+ e12 + 1 + ... .

(2)

(b) Calculate, to 3 significant figures, the sum of the first six terms of the series. (3)

(c) Write down, in its simplest form, the common difference of the arithmetic series A,

log3 2 + log3 6 + log3 18 + ... .

(3)

(d) Show that the sum of the first ten terms of A is 10 log3 2 + 45 and evaluate this to 2 decimal

places. (4)

(e) One of these two series has a sum to infinity. Calculate, to 2 decimal places, this sum. (3)


1989-1 Paper(1) Q.11

4 The first three terms of a series, S, are (m− 4), (m+ 2) and (3m+ 1).

Given, also, that S is an arithmetic series,

(a) find m. (2)

Using your value of m,

(b) write down the first four terms of the arithmetic series. (1)

Given, instead, that S is a geometric series,

(c) find the two possible values of m, (5)

(d) write down the first four terms of each of the two geometric series obtained with your values

of m, (2)

(e) state the value of the common ratio of each of the series. (2)

One of these two geometric series has a sum to infinity.

(f) Find the sum to infinity of that series. (3)

1989-6 Paper(1) Q.10

5 (a) The second term of an arithmetic series is −2. The sum of the first and seventh terms of

the series is equal to the sum of the first eight terms. Find the value of the first term of the

series. (7)

The numbers1

t,

1

t− 1and

1

t+ 2are the first three terms of a geometric series.

(b) Find the value of t. (4)

(c) Show that the common ratio of this series if −13 . (2)

(d) Deduce the sum to infinity of the series. (2)


1990-1 Paper(1) Q.13

6 The sixth term of an arithmetic series is 25 and the eighteenth term is −11.

(a) Find the first term and the common difference of the series. (5)

(b) Find the smallest value of n for which the sum of the first n terms of the series is negative.

(4)

(c) After n years £300 invested at 8% per annum compound interest amounts to £300(1.08)n.

A man invests £300 on January 1st each year for 10 consecutive years at this rate, leaving

his money to accumulate. Find, to the nearest £, the total sum due to him to January 1st

of the eleventh year. (6)

1990-6 Paper(1) Q.11

7 The first term of an arithmetic series is a and the common difference is d. The first, fourth and

sixth terms of this series are also the first three terms of a geometric series.

(a) Show that a+ 9d = 0. (4)

The sixth term of the arithmetic series is −6.

(b) Calculate the values of a and d. (4)

(c) Find the sum of the first 50 terms of the arithmetic series. (3)

(d) Calculate the value of the common ratio of the geometric series. (2)

(e) Find the sum to infinity of the geometric series. (2)


1991-1 Paper(1) Q.9

8 The rth term of an arithmetic series is giving by

2r − 5, r = 1, 2, 3, ... .

(a) Write down the first four terms of this series and find

Sn =n∑r=1

(2r − 5)

in terms of n. (7)

Given that Sn = 165 find the value of n.

(b) In a geometric series, (x + 1), (x + 3) and (x + 4) are the first, second and third terms

respectively. Calculate the value of x and hence write down the numerical values of the

common ratio and the first term of the series.

Calculate the numerical value of the sum to infinity of the series. (8)

1991-1 Paper(2) Q.10

9 The first three terms of an arithmetic series are lg x, lg 2(x+ 1) and lg 4(x+ 6) respectively.

(a) Find the value of x. (4)

(b) Find the value of the common difference of this series. (2)

The equation of a curve is y = e−x sinx.

(c) Show that the values of the x-coordinates of the turning points of the curve are the solutions

of tanx = 1. (3)

The points A, B and C on the curve have x-coordinatesπ

4,

5π

4and

9π

4respectively.

(d) Show that the y-coordinates of the points A, B and C respectively are consecutive terms of

a geometric series. (4)

(e) State the value of the common ratio of this series. (2)


1991-6 Paper(1) Q.4

10 The first and third terms of a geometric series of positive terms are 36 and 4 respectively. For

this series find

(a) the common ratio (3)

(b) the sum to infinity. (2)

1991-6 Paper(1) Q.10

11 The first three terms of an arithmetic series of positive terms have sum 24 and product 440. Find

(a) the values of each of the first three terms, (6)


The first three terms of a geometric series have sum 21 and product 216. Given that the terms

are increasing, find

(c) the value of each of the first three terms, (6)



1991-6 Paper(2) Q.8

12 (a) Show that x = 2 is a solution of the equation

27x3 − 42x2 − 28x+ 8 = 0.

(1)

(b) Find the other two solutions. (3)

An infinite geometric series has the three solutions as its first three terms, 2 being the first term.

(c) Find the sum to infinity of this series. (2)

1992-1 Paper(1) Q.11

13 The sum of the first two terms of a geometric series of positive terms is 825 , and the sum to infinity

of the series if 834 . For this series find the value of

(a) the first term,

(b) the common ratio.

(c) An arithmetic series and a geometric series each have a second term equal to 6 and a third

term equal to q. The first term of the arithmetic series is (2p − 15); the first term of the

geometric series is p. Given that the first term of both series is positive, find the values of p

and q.


1995-1 Paper(1) Q.13

14 The sum of the first four terms of an arithmetic series is 20 and the fifth term is five terms the

second term. Given that the first term of the series is a and the common difference of the series

is d.

(a) form a pair of simultaneous equation in a and d, (2)

(b) solve these equations to find a and d, (3)

(c) find the sum of the first 40 terms of the series. (2)

Given that (p2 − 4), (5p− 2) and 27 are, respectively, the first three terms of a geometric series,

(d) find the two possible values of p. (4)

(e) For each of your values of p, write down the first three terms of the series. (2)

Given also that the series has a sum to infinity, find

(f) the common ratio of the series. (2)

1995-6 Paper(2) Q.14


(a) xdy

dx= y(3x+ 1). (4)

(b)1

y

dy

dx= 3 +

1

x. (3)






1996-1 Paper(1) Q.9

16 Given that x, (x + y), and (2x + 2) are respectively the first three terms of an arithmetic series

and that 2x, x, and 2(x − y), x 6= 0, are respectively the first three terms of a geometric series,

show that

(a) the common ratio of the geometric series is 0.5. (2)

Find


(c) the value of y, (2)

(d) the sum to infinity of the geometric series, (2)

(e) the sum of the first 121 terms of the arithmetic series, (3)

(f) the value of the 87th term of the arithmetic series. (2)

1996-6 Paper(1) Q.11

17 The first three terms of a geometric series are 25p, (3t + 4)p and t2p respectively, where p and t

are non-zero constants.

(a) Find the possible values of t. (6)

(b) Calculate the possible values of the common ratio of the series. (3)

Given also that the value of t is positive and that the sum to infinity of the series is 50,

(c) show that p = 1.2, (3)

(d) find the difference, to 2 significant figures, between the sum to infinity of the series and the

sum of the first 8 terms of the series. (3)


1997-6 Paper(1) Q.4

18 The first term of a geometric series is 2 and the sum of the first three terms is 912 . Find the two

possible values of the common ratio of the series. (5)

1997-6 Paper(1) Q.14

19 Sn =n∑r=1

(50− 4r).

(a) Write down the first three terms of the series. (1)

(b) Calculate the value of S20. (3)

(c) Find n such that Sn = 0. (3)

The sum of the first four terms of a geometric series of positive terms is 78.336 and the sum to

infinity of the series is 90. Calculate

(d) the common ratio of the series, (4)

(e) the first term of the series, (2)

(f) the difference, to 3 significant figures, between the sum to infinity of the series and the sum

of the first thirty terms. Give your answer in standard form. (2)


2007-1 Paper(2) Q.6

20 The first three terms of a geometric series are non-identical and are given by (x + 2), 3x and

(7x− 4) respectively. Find


(b) the common ratio of the series, (1)

(c) the sum of the first 17 terms of the series. (2)

2007-6 Paper(2) Q.8

21 The sum to infinity of a convergent geometric series is 243 and the sum of the first four terms of

the series is 240.

(a) Find the two possible values of the common ratio of the series, giving your answers as exact

fractions. (5)

For each value of the common ratio

(b) find the first term of the series. (4)

Given that the sum of the second and third terms is negative,

(c) find the eighth term of the series. (2)

(d) Find, to 2 decimal place, the sum of the first 8 terms of the series. (2)


2008-1 Paper(1) Q.7

22 The third, fourth and fifth terms of a geometric series are (5x − 9), (7x − 3) and (12x + 4)

respectively.

(a) Determine the two possible values of x. (5)

Given that all the terms of the series are positive, find, for the series,

(b) the common ratio, (2)

(c) the first term, (2)


2009-6 Paper(2) Q.6

23 A geometric series is such that the difference between the third and second terms is 12.

Also, the difference between the fifth and fourth terms is 27.

The common ratio of the series is r, where r > 1.

Find

(a) the value of r, (4)


A new geometric series starts with the second term of the original series.

The common ratio of this series is r2.

(c) Find, to the nearest whole number, the sum of the first 10 terms of this new series. (4)


2010-6 Paper(2) Q.11

24 S and T are two geometric series.

The first, third and fifth terms of both series are (x− 4), (2x− 1) and (16x+ 1) respectively.

(a) Find the two possible values of x. (5)

The terms of S are all positive.

For S, find

(b) the first term, (2)

(c) the common ratio. (3)

The terms of T are all negative.

(d) Find the first term of T . (2)

(e) Find the sum to infinity of T .

Give your answer to 3 significant figures. (4)


2011-1 Paper(2) Q.9

25 The third and fifth terms of a geometric series G are 360 and 90 respectively.

Find

(a) the two possible values of the common ratio of G. (3)

(b) the first term of G. (1)

The sum of the first three terms of G is greater than the sum of the first four terms of G.

(c) Find the sum of the first 12 terms of G. Write down all the figures on your calculator display.

(4)

(d) Find the sum to infinity of G. (3)

(e) Find, to 3 significant figures, the percentage error when the sum of the first 12 terms of G is

used as an approximation for the sum to infinity of G. (3)

2011-6 Paper(2) Q.8

26 The sum of the first and third terms of a geometric series is 100

The sum of the second and third term is 60

(a) Find the two possible values of the common ratio of the series. (5)

Given that the series is convergent, find


(c) the least number of terms for which the sum is greater than 159.9 (4)


2012-1 Paper(2) Q.10

27 The sum of the first and third terms of a geometric series G is 104

The sum of the second and third terms of G is 24

Given that G is convergent and that the sum to infinity is S, find

(a) the common ratio of G (4)

(b) the value of S (4)

The sum of the first and third terms of another geometric series H is also 104 and the sum of the

second and third terms of H is 24

The sum of the first n terms of H is Sn

(c) Write down the common ratio of H (1)

(d) Find the least value of n for which Sn > S (6)

2012-6 Paper(2) Q.6

28 The first term of a geometric series S is√

2

The second term of S is√

2− 2

(a) (i) Find the exact value of the common ratio of S.

(ii) Find the third term of S, giving your answer in the form a√

2 + b, where a and b are

integers. (5)

(b) (i) Explain why the series is convergent.

(ii) Find the sum to infinity of S. (3)


2013-1 Paper(2) Q.9

29 The third and fifth terms of a geometric series S are 48 and 768 respectively. Find

(a) the two possible values of the common ratio of S, (3)

(b) the first term of S. (1)

Given that the sum of the first 5 terms of S is 615

(c) find the sum of the first 9 terms of S. (4)

Another geometric series T has the same first term as S. The common ratio of T is1

rwhere r is

one of the values obtained in part (a). The nth term of T is tn

Given that t2 > t3

(d) find the common ratio of T . (1)


(e) Writing down all the numbers on your calculator display, find T9 (2)

The sum to infinity of T is Tn

Given that T∞ − Tn > 0.002

(f) find the greatest value of n. (5)


2013-6 Paper(2) Q.4

30 The nth term of a geometric series is tn and the common ratio is r, where r > 0

Given that t1 = 1

(a) write down an expression in terms of r and n for tn (1)

Given also that tn = tn+1 = tn+2

(b) show that r =1 +√

5

2(4)

(c) find the exact value of tn giving your answer in the form f + g√h, where f , g, and h are

integers. (3)

2014-1 Paper(2) Q.10

31 The sum of the second and third terms of a convergent geometric series is 7.5

The sum to infinity, S, of the series is 20

The common ratio of the series is r.

(a) Show that r is a root of the equation

8r3 − 8r + 3 = 0

(4)

(b) Show that r =1

2is a root of this equation. (1)

Given that r < 0.6

(c) show that1

2is the only possible value of r. (4)

(d) Find the first term of the series. (2)


(e) Find the least value of n for which Sn exceeds 99% of S. (6)


2014-6 Paper(2) Q.6

32 The sum to infinity of a convergent geometric series with common ratio r is S.

Given that S = 200 and that the sum of the first 3 terms is 175

(a) find the value of r, (4)

(b) find the first term of the series. (1)


Given also thatSnS

=255

256

(c) find the value of n. (4)

2015-6 Paper(2) Q.3

33 Every term of a convergent geometric series is positive. The difference between the third term

and the fourth term is twice the fifth term.

(a) Show that the common ratio of the series is1

2(3)

The sum to infinity of this convergent series is 400

Find


(c) the sum of the first 10 terms of the series, writing down all the digits on your calculator

display. (2)


2016-1 Paper(1) Q.4

34 An arithmetic series has first term p and common difference p where p 6= 0

A geometric series also has first term p. The common ratio of this geometric series is r.

The sum of the first three terms of the arithmetic series is equal to the sum of the first three terms

of the geometric series.

Given that r > 0

show that r =−1 +

√21

2(5)

2016-6 Paper(2) Q.3

35 A geometric series has first term (11x− 3), second term (5x+ 3) and third term (3x− 3).

(a) Find two possible values of x. (4)

For each of your values of x,

(b) find the corresponding value of the common ratio of the series. (3)

Given that the series is convergent,

(c) find the sum to infinity of the series. (3)

2017-1 Paper(1) Q.4

36 The nth term of a geometric series is tn and the common ratio is r.

Given that t2 + t5 =28

81and t2 − t5 =

76

405

(a) (i) show that r =2

3(ii) find the first term of the series. (6)

(b) Find the sum to infinity of this geometric series. (2)


2017-6 Paper(1) Q.6

37 The sum of the first term and the third term of a geometric series is 250

The sum of the second term and the third term of the series is 150

The common ratio of the series is r.

(a) Find the two possible values of r. (5)


Given that r > 0 and that Sn > 399.99

(b) find the least value of n. (6)

2018-1 Paper(1) Q.8

38 The sixth term of a geometric series G, with common ratio r(r 6= 0), is four times the second

term.

(a) Find the two possible exact values of r. (2)

The sum of the third and seventh term of G is 30

(b) Find the first term of the series. (3)

Given that r > 0

(c) find the sum of the first 10 terms of G. (2)

Given that tn is the nth term of G,

(d) find the least value of n for which tn > 2400 (3)


19 Integration

1988-1 Paper(1) Q.3

1 (a) Differentiate with the respect to x,

3x2 − 2

x.

(2)

(b) Evaluate

∫ 2

1(x− 3)2 dx. (3)

1988-6 Paper(1) Q.5

2 (a) Differentiate

(1

x+ 3x

)with respect to x. (2)

(b) Evaluate

∫ 2

1x(x+ 2) dx. (3)

1989-1 Paper(2) Q.14

3 y = xex

(a) Finddy

dx. (2)

(b) Find the coordinates of the turning point of the curve y = xex and determine whether this

point is a maximum or minimum. (6)

(c) Using the result of (a), find

∫xex dx. (4)

(d) Hence, or otherwise, find the area enclosed by the curve y = xex, the ordinate x = 2 and the

x-axis, leaving your answer in terms of e. (3)


20 Kinematics

1988-1 Paper(1) Q.12

1 A particle moves along the x-axis so that at time t seconds its velocity is v m/s and its displacement

from O is x metres, where

x = 5 + 9t+ 3t2 − t3, t > 0.

(a) Find an expression for v in terms of t. (2)

(b) Find an expression for the acceleration, a m/s2, of the particle, in terms of t. (1)

(c) Find the maximum velocity of the particle. (2)

(d) Given that maximum velocity occurs at the point A, find the displacement OA. (1)

(e) Given that the particle is instantaneously at rest at the point B, show that the time taken

to reach B is 3 s and find the displacement OB. (5)

(f) Show that, at time t = 5, the particle passes through O. (2)

(g) Find the velocity of the particle when it passes through O. (2)

1988-6 Paper(1) Q.1

2 A particle P moves in a straight line so that its velocity v m/s at time t seconds, where t > 0, is

given by

v = t2 − t− 12.

Find the acceleration, in m/s2, at the instant when P is instantaneously at rest. (4)


1989-6 Paper(1) Q.3

3 A particle P passes through the point A with speed u m/s at time t = 0 and moves in a straight

line. The displacement, s metres, of P from A after t seconds is given by

s = t3 − 3t2 + 2t.

(a) Find the value of u. (3)

(b) Calculate, in m/s2, the acceleration of P when t = 2. (2)

1990-1 Paper(1) Q.7

4 A particle P moves along the x-axis in such a way that, t seconds after it leaves the origin O, its

velocity, v m/s, is given by

v = 24t− 2t3, t > 0.

Given that the acceleration of P is zero at the point A, calculate

(a) the speed, in m/s, of P at the point A, (3)

(b) the distance OA in m. (3)


1990-6 Paper(1) Q.13

5 A particle P pass through a point O and moves in a straight line. The displacement, s metres, of

P from O, t seconds after passing through O, is given by

s = −t3 + 11t2 − 24t.

(a) Find, using a calculus method, an expression, in terms of t, for the velocity, v m/s, of P at

time t seconds. (2)

(b) Hence find the values of t at which P is instantaneously at rest. (3)

(c) Find the values of t at which the acceleration of P is zero. (2)

(d) Sketch the velocity-time graph for P , in the interval 0 6 t 6 6, indicating on your sketch the

coordinates of the points at which the graph crosses the axes. (4)

(e) Calculate the values of t between which the speed of P is greater than 16 m/s. (4)


1991-1 Paper(1) Q.13

6 A particle P moves along the x-axis so that at time t seconds its displacement from O is x metres

and its velocity is v m/s where

v = 16t− t3, t > 0.

The particle starts at time t = 0 from the point A where x = 36.

(a) Find an expression for x in terms of t. (2)

(b) Find the acceleration of the particle when t = 2. (3)

Given that the particle first comes instantaneously to rest at the point B, find

(c) the time taken from the start for P to reach B, (2)

(d) the distance OB. (1)

(e) Find the total distance travelled by P in the first five seconds. (3)

(f) Find the time taken from the start for P to reach O. (4)


1992-1 Paper(1) Q.13

7 A particle P passes through a fixed point O at the time t = 0 and moves in a straight line. At

time t seconds after passing through O, the velocity v m/s, of P is given by

v = 2t2 + kt+ 5, t > 0,

where k is a constant.

(a) Given that, at time t = 4, the acceleration of P is 5 m/s2, show that k = −11.

(b) Find the values of t for which v = 0.

(c) Find the value of t for which the acceleration of P is zero.

(d) Sketch, for 0 6 t 6 6, the velocity-time graph of P .

(e) Calculate, in m to 3 significant figures, the distance of P from O at time t = 12 .

(f) Deduce the average speed, in m/s to 3 significant figures, of P in the interval 0 6 t 6 12 .

1993-6 Paper(1) Q.6

8 A particle P moves along the x-axis so that at time t seconds its displacement from O is x metres

and its velocity is v m/s, where

v = 4t+ t3.

(a) Find its acceleration, in m/s2, when t = 3. (3)

Given also that x = −2 when t = 0,

(b) calculate the distance OP , in m, when t = 4. (3)


1994-1 Paper(1) Q.4

9 At time t = 0 a particle P , moving in a straight line, passes through the point A with speed 6

m/s. After t seconds, the displacement, s metres, of P from A is given by s = 3t4 + kt Calculate


(b) the acceleration of P , in m/s2, when t = 4. (2)

1994-1 Paper(2) Q.11

10 A small stone is falling vertically through water. After time t seconds (t ≥ 0), the stone has fallen

a distance of s metres and has speed v m/s, given by

v = 20(1−Ae−0.5t),

where A is a constant.

Given that the intital speed of the stone is 2 m/s.

(a) show that A = 0.9. (2)

Hence

(b) find the value of t, to 3 significant figures, when v = 10, (3)

(c) sketch the graph of v against t, stating the equation of the asymptote of the curve, (3)

(d) find an equation of s in terms of t, (4)

Given also that after time t seconds, the acceleration of the stone is a m/s2,

(e) show that a = 10− 12v (3)


1995-1 Paper(1) Q.10

11 A particle P , moving in a straight line, passes through a fixed point O at time t = 0. At time t

seconds after passing through O, the velocity, v m/s, of P is given by

v = t3 − 10t2 + kt, t > 0,

where k is a constant.

(a) Find an expression, in terms of t and k, for the acceleration of P . (2)

Given that the acceleration is zero when t = 5,

(b) show that k = 25, (2)

(c) find the maximum and minimum values of v, distinguishing between them. (4)

(d) Sketch the velocity-time graph of P for 0 6 t 6 6. (2)

At time t = 5, P is at the point A.

(e) Find, in metres, the distance OA. (3)

(f) Use your answer to (e) to find the average speed, in m/s, of P in the interval 0 6 t 6 5. (2)

1996-1 Paper(1) Q.4

12 A particle moves in a straight line such that its velocity v m/s at time t seconds is given by

v = 3t2 − t− 10, t > 0.

Calculate

(a) the time, in s, when the particle is at rest, (2)

(b) the acceleration, in m/s2, when t = 1. (2)


1996-6 Paper(1) Q.1

13 A particle P moves in a horizontal straight line. At time t seconds (t > 0), the velocity, v m/s, of

P is given by v = 12− 4t. Find

(a) the value of t when P is instantaneously at rest, (2)

(b) the distance, in m, travelled by P between the time when t = 0 and the time when P is

instantaneously at rest. (2)

1997-6 Paper(2) Q.7

14 A particle, P , is moving in a straight line with velocity v m/s. At time t seconds, v is given by

v = 4(5− 6e−0.5t), t > 0.

(a) Calculate, to 3 significant figures, the value of t when v = 0. (3)

(b) Calculate, in m/s2 to 3 significant figures, the acceleration of P when t = 2. (3)

2007-1 Paper(1) Q.6

15 A particle P moves in a straight line such that at time t seconds its displacement, s metres, from

a fixed point O on the line is given by

s = t3 − 7t2 + 10, t > 0.

(a) Find the values of t(t > 0) at which P passes through O. (3)

(b) Find the speed of P each time it passes through O. (4)

(c) Find the greatest speed of P in the interval 0 6 t 6 5. (5)


2007-6 Paper(1) Q.2

16 A particle P moves in a straight line. At time t seconds, the displacement, s metres, of P from

a fixed point O of the line is given by s = 2t cos t + t2. Find, in m/s to 3 significant figures, the

velocity of P when t = 3. (5)

2008-1 Paper(1) Q.3

17 A particle P moves in a straight line. At time t seconds, the velocity, v m/s, of P is given by

v = 5− 2t+ t2. Find

(a) the accelerate, in m/s2, of P when t = 3. (3)

(b) the distance, in metres, travelled by P in the interval 0 6 t 6 4. (3)

2008-6 Paper(2) Q.5

18 A particle P moves in a straight line. At time t seconds, the velocity, v m/s, of P is given by

v = t2 − 2t+ 9. Find

(a) the acceleration of P , in m/s2, when t = 3, (2)

(b) the distance P travels in the interval 0 6 t 6 6. (4)

2009-6 Paper(1) Q.6

19 A particle is moving along a straight line. At time t seconds, the displacement, s metres, of the

particle from a fixed point of the line is given by s = 4t3 − 22t2 + 24t+ 31.

Find

(a) the values of t when the particle is instantaneously at rest, (4)

(b) the acceleration of the particle when it is instantaneously at rest for the first time. (3)


2010-6 Paper(1) Q.2

20 A particle P is moving in a straight line.

At time t seconds, t > 0, the velocity of P is v m/s, where v = 4 + 6t− t2.

(a) Find an expression, in terms of t, for the acceleration of P at time t. (2)

(b) Find the maximum velocity of P . (2)

2011-6 Paper(2) Q.2

21 A particle is moving along a straight line. At time t seconds, t > 0, the displacement, s metres,

of the particle from a fixed point of the line is given by s = t3 + 2t2 − 3t+ 6

Find the value of t for which the particle is moving with the velocity 12 m/s. (4)

2012-6 Paper(2) Q.9

22 The particle M is moving along the straight line PQ with a constant acceleration of 2 m/s2.

At time t = 0, M is at the point P moving with velocity 6 m/s towards Q.

(a) Find an expression for the velocity of M at time t seconds. (2)

(b) Show that the displacement of M from P at time t seconds is (t2 + 6t) metres. (2)

A second particle N is moving along PQ. The acceleration of N at time t seconds is 6t m/s2. At

time t = 0, N is stationary at the point P .

(c) Find an expression for the velocity of N at time t seconds. (2)

(d) Find an expression for the displacement of N from P at time t seconds. (2)

(e) Find the distance between M and N at time t = 5 seconds. (2)

(f) Find the value of t, t > 0, when two particles meet. (3)


2013-1 Paper(1) Q.5

23 A particle P moves along the x-axis. At time t seconds (t > 0) the velocity, v m/s, of P is given

by v = 5 cos 2t. Find

(a) the least value of t for which P is instantaneous at rest, (2)

(b) the magnitude of the maximum acceleration of P . (3)

When t = 0, P is at the point (2, 0).

(c) Find the distance of P from the origin when P first comes to instantaneous rest. (4)

2013-6 Paper(2) Q.10

24 tan θ =sin θ

cos θ


A particle P is moving along a straight line. At time t seconds (t > 0) the displacement, s metres,

of P from a fixed point O on the line is given s =√

3 sin1

2t+ cos

1

2t

(a) Find the value exact of s when t =π

3(2)

(b) Find the exact value of t when P first passes through O. (4)

The velocity of P at time t seconds is v m/s.

(c) Find an expression for v in terms of t. (2)

(d) Show that v = cos

(π

6+

1

2t

)(2)

(e) Find the exact value of t for which v =1

2when

(i) 0 6 t < 2π

(ii) 2π 6 t < 4π (4)


2014-1 Paper(1) Q.9

25 A particle P moves in a straight line such that, at time t seconds, its displacement, s metres, from

a fixed point O of the line is given by s = t3 − 6t2 + 5t

Find

(a) the values of t for which P passes through O (3)

(b) the speed of P each time it passes through O (5)

(c) the greatest speed of P in the interval 0 6 t 6 5 (4)

2014-6 Paper(1) Q.7

26 [In this question all distances are measured in metres.]

A particle P is moving along the x-axis. At time t seconds, P is at the point with coordinates

(xp, 0) where xp = 8− 10t+1

3t3

Find, in terms of t,

(a) the velocity of P at time t seconds. (2)

(b) the acceleration of P at time t seconds. (2)

A second particle Q is also moving along the x-axis. At time t seconds, the velocity of Q is vQ

m/s, where vQ = t2 − 3t+ 4

At time t = 0, Q is at the origin and at time t seconds Q is at the point with coordinates (xQ, 0).

(c) Find xQ in terms of t. (3)

The particles P and Q collide at time T seconds, where T < 5

(d) Find the value of T . (4)


2015-1 Paper(1) Q.2

27 A small stone is thrown vertically upwards from a point A above the ground. At time t seconds

after being thrown from A, the height of the stone above the ground is s metres. Until the stone

hits the ground, s = 1.4 + 19.6t− 4.9t2

(a) Write down the height of A above the ground. (1)

(b) Find the speed with which the stone was thrown from A. (2)

(c) Find the acceleration of the stone until it hits the ground. (1)

(d) Find the greatest height of the stone above the ground. (3)

2015-6 Paper(2) Q.5

28 A particle P moves in a straight line such that at time t seconds, the displacement, s metres, of

P from a fixed point O on the line is given by

s = t3 − 5t2 + 6t t > 0

(a) Find the values of t(t > 0) when P passes through O. (3)

(b) Find the speed of P when t = 1 (4)

(c) Find the magnitude of the acceleration of P at each of the times when it passes through O.

(3)


2016-1 Paper(1) Q.8

29 A particle P is moving along the positive x-axis. At time t seconds (t > 0), the acceleration a

m/s2 of P is given by a = 6− 4t

When t = 0, P is at rest and the displacement of P from the origin O is 5 metres.

At time t seconds, the velocity of P is v m/s and the displacement of P from O is s metres.

(a) Find, in terms of t, an expression for

(i) v

(ii) s (6)

For t > 0, P comes to instantaneous rest at the point A.

(b) Find

(i) the value of t when P reaches A.

(ii) the distance OA. (5)

2016-6 Paper(2) Q.7

30 A particle P moves in a straight line so that, at time t seconds (t > 0), its velocity, v m/s, is given

by v = 3t2 − 4t+ 7

Find

(a) the acceleration of P at time t = 2 (2)

(b) the minimum speed of P . (3)

When t = 0, P is a the point A and has velocity V m/s.

(c) Write down the value of V . (1)

When P reaches the point B, the velocity of P is also V m/s.

(d) Find the distance AB. (6)


2017-1 Paper(1) Q.10

31 A particle P moves along the positive x-axis. At time t seconds (t > 0) the velocity, v m/s, of P

is given by v = t3 − 4t2 + 5t+ 1

The acceleration of P at time t seconds is a m/s2

(a) Find an expression for a in terms of t. (2)

(b) Find the values of t for which the magnitude of the acceleration of P is instantaneously zero.

(2)

With t = 0, the displacement of P from the origin is 3 m.

(c) Find the displacement of P from the origin when t = 2 (5)

2017-6 Paper(2) Q.4

32 A particle P is moving along a straight line which passes through the point O.

At time t = 0 the particle P is at the point O.

At time t seconds the velocity, v m/s, of P is given by v = 3t2 + 2t+ 5

(a) Find the acceleration of P when t = 2 (3)

(b) Find the displacement of P from O when t = 3 (3)


2018-1 Paper(1) Q.4

33 A particle P moves along the x-axis. At time t seconds (t > 0), the displacement of P from the

origin x metres and the velocity, v m/s, of P is given by v = 2t2 − 16t+ 30

(a) Find the times at which P is instantaneously at rest. (2)

(b) Find the acceleration of P at each of these times. (3)

When t = 0, P is at the point where x = −4

(c) Find the distance of P from the origin when P first comes to instantaneous rest. (3)


21 Linear Programming

1988-1 Paper(1) Q.14

1 (a) Find an equation of the normal to the curve y = 12x

2 at the point A(2, 2). (4)

This normal cuts the x-axis at the point B. The curve y = 12x

2, the x-axis and the line segment

AB bound a finite region R. The region includes its boundaries.

(b) Draw a diagram, clearly labelled, to show the curve, the normal and the region R. (4)

(c) Write down 3 inequalities which define the region R. (3)

(d) Find the point in R for which 3x+ 2y has its greatest value. (4)


1989-1 Paper(1) Q.9

2

O x

y

P (1, 4)

Q

R

Fig. 2

Figure 2 shows the lines PQ and PR intersecting at right angles at the point P (1, 4).

(a) Given that the equation of PR is y = x+ 3, find the equation of PQ. (4)

Q is on the x-axis.

(b) Show that the coordinates of Q are (5, 0). (1)

The line QR has a gradient of 2.

(c) Find the equation of QR. (2)

(d) Find the coordinates of R. (3)

The region S consists of all those points lying within or on the boundary of 4PQR.

(e) Write down three linear inequalities which define the region S. (3)

(f) State the coordinates of the point of the region S at which 3x− 2y has its largest value. (2)


1990-1 Paper(1) Q.10

3

x

y

O B

C A

y = 16x

2

Fig. 2

In Fig. 2, A is the point (3, 112) on the curve with equation y = 1

6x2. The tangent to the curve at

A cuts the y-axis at B. The line through A parallel to the x-axis cuts the y-axis at C.

(a) Find an equation of AB. (3)

(b) Find the coordinates of the points B and C. (2)

(c) Write down the three inequalities which define completely the shaded region, including its

boundaries. (6)

(d) Calculate, to 3 significant figures, the greatest and least possible values of OP where P is

any point of the shaded region. (4)


1991-1 Paper(1) Q.6

4 (a) On a sheet of graph paper using a scale of 2 cm to represent 1 unit on each axis, draw, for

values of x in the interval −2 6 x 6 3, the lines with equations y = x + 2 and y = 2x − 1.

(3)

The region R is defined by

y 6 x+ 2, y > 2x− 1, y > 0.

(b) On your graph shade and label the region R. (2)

(c) From your sketch obtain the largest value of (x+ y) for points whose coordinates satisfy the

given inequalities. (1)

1991-6 Paper(1) Q.7

5 In the binomial expansion of (1− px)5 in ascending powers of x, the coefficient of x3 is 80.

(a) Find the value of p. (4)

(b) Evaluate the coefficient of x4 in the expansion. (2)

1995-1 Paper(1) Q.5

6 (a) Sketch, for −1 6 x 6 5, the graph of the line with equation 3x + 4y = 12, showing the

coordinates of the points where the line cuts the axes. (3)

The region A consists of those points whose coordinates (x, y) satisfy the inequalities

x > 1, y > 0, 3x+ 4y 6 12.

(b) Shade in and label the region A on your sketch. (2)


1996-1 Paper(1) Q.5

7 (a) On a sheet of graph paper, using a scale 2 cm to represent 1 unit on the x-axis and 1 cm to

represent 1 unit on the y-axis, draw, for values of x in the interval −1 6 x 6 2, the lines with

equations y = x+ 1 and y = 3(x− 1). (2)

The region R is defined by the inequalities y 6 x+ 1, y > 3(x− 1) and y > 0.

(b) On your graph shade and label clearly the region R. (2)

(c) Obtain the largest value of (x+y) for points whose coordinates satisfy the given inequalities.

(1)


1997-6 Paper(1) Q.12

8 (a) Find the set of values of x for which

2x(x− 2) < (x+ 1)(x− 2).

(5)

(b) On the graph paper, using the same axes and a scale of 1 cm for 1 unit on each axis, draw

the four lines with equations

3x+ 2y = 6, y = x+ 3, 3y + 2x+ 6 = 0, y = −1,

for −6 6 x 6 6,−6 6 y 6 6. (4)

The region A consists of those points whose coordinates (x, y) satisfy the inequalities

3x+ 2y 6 6, y 6 x+ 3, 3y + 2x+ 6 > 0, y > −1.

(c) Indicate clearly the region A on your graph. (2)

(d) List the coordinates of all points (m,n), where m and n are integers, which lie inside, but

not on the border, of A. (4)

2007-1 Paper(1) Q.2

9 (a) On the same axes, sketch the lines with equations x = 6, y = 3x and y = 15− 2x. (3)

(b) Show, by shading, the region for which x 6 6, y 6 3x and y > 15− 2x. (1)


2010-6 Paper(2) Q.1

10 (a) On the axes below sketch the lines with equations y = 2x+ 1 and y + 3x = 9. (2)

(b) Show, by shading, the region R defined by the inequalities

y 6 2x+ 1, y + 3x 6 9, x > 0 and y > 0

x

y

(1)


2013-1 Paper(1) Q.1

11 (a) On the axes below sketch the lines with equations

(i) y = 8 (ii) y + x = 6 (iii) y = 3x− 4

Show the coordinates of the points where each line crosses the coordinate axes. (3)

(b) Show, by shading, the region R which satisfies y > 3x− 4, y + x > 6, x > 0 and y 6 8

O x

y

(1)


2014-6 Paper(1) Q.1

12 (a) On the axes below, sketch the lines with equations y = x+ 3 and y + 2x = 7

On your sketch mark the coordinates of the points where the lines cross the y-axis. (2)

(b) Show, by shading on your sketch, the region R defined by the inequalities

y 6 x+ 3, y + 2x 6 7, x > 0 and y > 0

O x

y

(1)

(c) Determine, by calculation, whether or not the point with coordinates (2, 2) lies in R. (2)


2015-1 Paper(1) Q.5

13 (a) On the axes opposite, draw the lines with equations

(i) y = −x− 1 (ii) y = 3x− 9 (iii) 2y = x+ 7

(4)


y > −x− 1, y > 3x− 9 and 2y 6 x+ 7

(1)

For all points in R, with coordinates (x, y),

P = y − 2x

(c) Find

(i) the greatest value of P ,

(ii) the least value of P . (4)

y

xO−4 −2 2 4 6

−4

−2

2

4

6

8


2017-1 Paper(2) Q.1

14 (a) On the axes below, sketch the lines with equations x = 3, y = x+ 1 and 2y + x = 5

On your sketch, mark the coordinates of any points where the lines cross the axes. (3)

(b) Show, by shading on your sketch, the region R defined by the inequalities

x 6 3, y 6 x+ 1 and 2y + x > 5

O x

y

(1)

2017-6 Paper(2) Q.1

15 (a) On the grid opposite, draw the graphs of the lines with equations

(i) y = 2x (ii) y = 6− x (iii) 2y = x− 2 (3)

(b) Show, by shading on the grid, the region R defined by the inequalities

y 6 2x, y 6 6− x, 2y > x− 2, y > 0

(1)

For all points in R, with coordinates (x, y),

P = y + 2x

(c) Find the greatest value of P . (1)


y

xO−2 −1 1 2 3 4 5 6 7 8

−4

−3

−2

−1

1

2

3

4

5

6

7

8

2018-1 Paper(1) Q.2

16 (a) On the grid opposite, draw

(i) the line with equation y = 3x− 3

(ii) the line with equation 3x+ 2y = 12 (2)


y 6 3x− 3 3x+ 2y 6 12 y > −1

(2)

For all points in R with coordinates (x, y)

P = 4x− y

(c) Find the greatest value of P . (4)


22 Logarithm

1988-1 Paper(2) Q.4

1 Solve the equation

5x = 21+x,

giving your answer to 2 decimal places. (5)

1988-1 Paper(2) Q.9


32x − 3(3x)− 4 = 0.



y = log3 x.

(3)

Given that

y = log3 x and y = 12 [1 + log3 9x],


tions. (6)

1988-6 Paper(2) Q.1

3 Solve the equations

(a)

(1

3

)x= 81, (1)

(b) log5 x = −2. (2)


1989-1 Paper(2) Q.2

4 Find the exact value of

(a) log4 64, (2)

(b) 12523 −

(1

8

)− 1

3 . (2)

1989-6 Paper(2) Q.3

5 Given that log3 x = z, find , in terms of z,

(a) log3 (3x2), (3)

(b) log9 x. (2)

1990-1 Paper(2) Q.13

6 (a) Given that log3 x = 2, determine the value of x. (1)

(b) Calculate the value of y for which

2 log3 y − log3 (y + 4) = 2.

(6)

(c) Calculate the values of z for which

log3 z = 4 logz 3.

(4)

(d) Express loga (p2q) in terms of loga p and loga q. (2)

(e) Given that loga (pq) = 5 and loga (p2q) = 9, find the values of loga p and loga q. (2)


1991-1 Paper(2) Q.6

7 (a) Solve the equation log3 (x+ 1) = 2. (2)

(b) Calculate the value of log7 (15) + log7 ( 549). (3)

1991-6 Paper(2) Q.6


(a) 3 + log2 x = log2 (2x+ 1), (3)

(b) (e2x + 1)2 = 9. (3)

1993-6 Paper(2) Q.6


(a) log9 x = −12 , (2)

(b) logx 27− logx 8 = 3. (3)

1994-1 Paper(2) Q.1


log3 1 +√y = 2.

(3)


1994-1 Paper(2) Q.10

11 (a) Solve the equation

32x+1 = 7,


(b) Solve the equation

4x2 − 17x+ 4− 0.

(2)

(c) Using the substitution y = log2 x, solve, giving your answers to 2 decimal places where

appropriate, the equation

4(log2 x)2 − 17 log2 x+ 4 = 0.

(4)

(d) Using the substitution z = sin2 θ◦, or otherwise, solve, giving all solutions in the range

0 ≤ θ < 360, the equation

4 sin4 θ◦ − 17 sin2 θ◦ + 4 = 0.

(3)


4(22x)− 17(2x) + 4 = 0.

(2)


1995-1 Paper(2) Q.3

12 Given that a is a constant greater than one,

(a) sketch the graph of y = ax, x ∈ R, (2)

(b) sketch the graph of y = loga x.

Label the points at which the curves cross the coordinate axes. (2)

1995-1 Paper(2) Q.8

13 (a) Find the value of log3 9. (1)

(b) Solve log3 x+ 2 logx 9 = 5. (6)

1995-1 Paper(2) Q.13

14 (a) Solve 2x2 − 7x+ 5 = 0. (2)

(b) Using the substitution x = 2y solve 22y+1−7(2y)+5 = 0, giving your answers to 3 significant

figures where appropriate. (4)

(c) Given that log16 y =log2 y

k, that log8 x =

log2 x

l, and that log4 y =

log2 y

m, find k, l, and m.

(3)

(d) Hence solve the simultaneous equations

log2 x+ 28 log16 y = 17,

9 log8 x+ 8 log4 y = 17.

(6)


1995-6 Paper(2) Q.7

15 (a) Evaluate log51

125. (2)

(b) Solve the equation log2 (2x− 1) = 3. (3)

1996-1 Paper(2) Q.2

16 Given that 82x−1 = 4y,

(a) find y in terms of x. (2)

(b) Hence, or otherwise, solve 82x−1 = 16x. (2)


1996-1 Paper(2) Q.10

17 (a) Show that

logx 16 =4

log2 x.

(2)

(b) Use the result in (a), to solve

3 log2 x+ logx 16 = 8,

giving your anwer to 2 decimal places where appropriate. (6)

f : x 7→ log3 x, x ∈ R, x > 0,

g : x 7→ (x− 1)2, x ∈ R, x > 1.

(c) Evaluate (i) fg(10), (ii) fg(2). (2)

Given that h = fg,

(d) find h(x) and state the domain of h, (2)

(e) find h−1(x) and state the domain of h−1. (3)

1996-6 Paper(2) Q.8


(a) logx 81 = 4, (2)

(b) log5 (2y − 1) = 3, (2)

(c) 2 log4 (3z) = log2 (z + 3). (3)


1996-6 Paper(2) Q.10

19 (a) Sketch the graph of y = e2x. Indicate clearly on your sketch the coordinates of any point

where the curve cuts the coordinates axes. (2)

(b) Solve the equation e2x = 8. (2)

The curves with equations y = e2x and y2 = 9e2x − 8 meet at the points A and B.

(c) Find the coordinates of A and B. Give your answers to 3 significant figures where appropriate.

(4)

The tangents to the curve with equation y = e2x at the points A and B meet at the point P .

(d) Show that the x-coordinate of the point P is8 ln 8− 7

14. (7)

1997-6 Paper(2) Q.3


log 4(x+ 2) + log2 4 = 0.

(4)

2007-1 Paper(2) Q.3

21 Solve the equation log3 (5x+ 12) + log3 x = 2. (5)


2007-6 Paper(2) Q.5

22 (a) Solve the equation log4 2 = p. (1)

Given that log2 3 = k log4 3

(b) find the value of k. (2)

(c) Show that 5x log4 x− 2 log4 x− 10x log2 3 + 4 log2 3 = log4

(x5x−2

320x−8

). (4)

(d) Hence solve the equation 5x log4 x− 2 log4 x− 10x log2 3 + 4 log2 3 = 0. (4)

2008-1 Paper(2) Q.9

23 Solve

(a) logq 343 = 3, (2)

(b) log4 (5n+ 9) = 3, (3)

(c) logm 4 + 8 log4m = 6, (6)

(d) 2 log3 x− 3x log3 x+ 6x = 4. (5)

2009-6 Paper(1) Q.2

24 Solve the equation log3 (3x+ 19) = 4. (3)


2010-1 Paper(1) Q.9


(a) logx 125 = 3 (2)

(b) log4 (9y + 4) = 4 (3)

(c) 3− log3 p = logp 9 (6)

2010-6 Paper(1) Q.8

26 (a) Solve the equation log5 625 = x (2)

(b) Solve the equation log3 (5y + 3) = 5 (2)

(c) (i) Factorise 5x lnx+ 3 lnx− 10x− 6

(ii) Hence find the exact solution of the equation

5x lnx+ 3 lnx− 10x− 6 = 0 (5)

(d) Given that p 6= q, solve the simultaneous equations

logp q + 3 logq p = 4

pq = 81

(5)


2011-1 Paper(1) Q.7

27 (a) Solve logm 16807 = 5 (2)

(b) Solve log6 (7n− 1) = 3 (2)

(c) Solve 12 logp 5 = 3 log5 p (4)

(d) Show that

log4 3 + log4 6 + log4 12 + log4 24 = 3 + 4 log4 3

(3)


log4 3 + log4 6 + log4 12 + log4 24 = 3 + log4 x+ log4 x3

(3)

2011-1 Paper(2) Q.8

28 (a) Solve

5p2 − 13p+ 6 = 0

(2)

(b) Hence solve 52x+1 − 13(5x) + 6 = 0, giving your answers to 3 significant figures. (5)

The curve with ewuation y = 52x+1 − 3(5x) + 2 meets the curve with ewuation y = 10(5x)− 4 at

two points.

(c) Find the exact value of the y-coordinate of each of these two points. (4)


2011-6 Paper(1) Q.2

29 (a) Given that loga x =logb x

logb ashow that loga b =

1

logb a(2)

(b) Hence solve the equation

logx 8− 6 log8 x = 1 x ∈ Z+

(5)

2012-6 Paper(1) Q.5

30 The first four terms of an arithmetic series, S, are


(a) Write down an expression for the rth term of S. (1)

(b) Find an expression for the common difference of S. (2)

The sum of the first n terms of S is Sn

(c) Show that Sn =1

2n(n+ 1) loga 2 (2)

The first four terms of a second arithmetic series, T , are



(d) Find Tn − Sn and simplify your answer. (4)

2012-6 Paper(2) Q.1


5x+1 = 120



2013-1 Paper(2) Q.10


(a) logx 1024 = 5 (2)

(b) log5 (6y + 11) = 3 (3)

(c) 2 log3 t+ logt 9 = 5 (6)

2013-6 Paper(2) Q.2

33 Given that 2 log4 x− log2 y = 3

(a) show that x = 8y (4)

Given also that log5 (3x+ y) = 4

(b) find the value of x and the value of y (3)

2014-1 Paper(1) Q.5

34 (a) Solve the equation log7 (2x− 3) = 2 (2)

(b) (i) Factorise 2x ln 3x− 4x− 4 ln 3x+ 8

(ii) Hence find the exact roots of the equation 2x ln 3x− 4x− 4 ln 3x+ 8 = 0 (5)


2014-6 Paper(2) Q.5


(a) logx 243 = 5 (3)

(b) log6 (2y + 4) = 2 (2)

(c) log4 p+ logp 64 = 4 (5)

2015-1 Paper(1) Q.6

36 (a) Solve, giving your answer to 3 significant figures,

3z − 4 = 0

(3)

Solve, giving your answer to 3 significant figures where appropriate,

(b) 9y − 13(3y) + 36 = 0 (4)

(c) 6x − 4(2x)− 3x + 4 = 0 (5)


2015-6 Paper(2) Q.10

37 (a) Find the value of log3 9 (1)

Given that log9 4 = k log3 4

(b) find the value of k (2)

(c) Show that

2x log3 x− 3 log3 x− 4x log9 4 + 6 log9 4 = log3

(x

4

)(2x−3)

(5)

(d) Hence solve the equation 2x log3 x− 3 log3 x− 4x log9 3 + 6 log9 4 = 0 (3)

2016-1 Paper(1) Q.10

38 Given that 2 logy x+ 2 logx y = 5

(a) show that logy x =1

2or logy x = 2 (5)

(b) Hence, or otherwise, solve the equations

xy = 27

2 logy x+ 2 logx y = 5

(6)

2016-1 Paper(2) Q.1

39 Find the exact solution

4(x−2) = 8(3x−1)

(4)


2016-6 Paper(1) Q.6

40 Solve

(a) logx 1024 = 5 (2)

(b) log3 (7y − 3) = 4 (2)

(c) loga 25 + 2 loga 625 = 10 (3)

(d) logb 7− 2 log7 b+ 1 = 0 (5)

2017-1 Paper(2) Q.7

41 (a) Given that k is a constant such that27(x+2) − 3(3x+5)

3x × 9(x+2)= k

find the value of k. (5)

(b) Find the exact roots of the equation 2 log2 y + 3 logy 2 = 7 (6)

2017-6 Paper(1) Q.7

42 (a) Solve loga 1024 = 5 (1)

(b) Solve log3 (6c+ 9) = 4 (2)

(c) Solve 2(logb 25 + logb 125) = 5 (4)

(d) Solve the equations, giving the values of x and y to 3 significant figures,

3 log2 x+ 4 log3 y = 10

log2 x− 2 log3 y = 1

(6)


2017-6 Paper(2) Q.11

43 (a) Show that log pq4 − log pq2 = log pq6 − log pq4 (3)

Given that log pq2 and log pq4 are the second and third terms of an arithmetic series, find


(c) the sum of the first n terms of the series.

Give your answers in the form n log pqs, expressing s in terms of n. (4)

2018-1 Paper(2) Q.7

44 (i) Solve the equation(8x)x

32x= 4

(ii) Solve the equation logx 64 + 3 log4 x− logx 4 = 5 (11)


23 Optimization

1990-1 Paper(1) Q.12

1 (a) Evaluate

∫ 4

2

(3x− 2

x2

)dx. (4)

A B

M CD

y m

4x m

y m3x m

4x m

Fig. 3

In Fig. 3, ABCD is a trapezium. The point M is such that ABCM is a rectangle.

(b) Find AD in terms of x. (1)

The perimeter of ABC is 180 m.

(c) Find y in terms of x. (2)

(d) Write down and simplify an expression in terms of x for the area, in m2, of ABCD. (4)

(e) Find the maximum value of this area in m2. (4)


1990-6 Paper(1) Q.7

2

x cm

x cm

x cm

x cm

2x cm

y cm

Fig. 1

Figure 1 shows a framework of 8 rods for a child’s table. The four legs each have length x cm and

the rectangular top is of length y cm and width 2x cm. A thin rectangular piece of wood, cut to

fit the top exactly, is fitted to the framework. Given that the total length of the 8 rods is 100 cm

and that the area of top is A cm2,

(a) show that A = 100x− 8x2, (3)

(b) calculate the values of x and y for which the area of the table top is a maximum. (3)


2007-1 Paper(1) Q.7

3

5w cm

2w cm

h cm

Figure 1

Figure 1 shows a closed rectangular box of height h cm. The width of the box is 2w cm and the

length is 5w cm. The volume of the box is 540 cm3 and the total external surface area of the box

is A cm2.

(a) Show that A = 20w2 +756

w. (4)

(b) Find, to 3 significant figures, the value of w for whichdA

dw= 0. (3)

(c) Prove that the value of w obtained in part (b) gives a minimum value for A. (4)

(d) Find, to the nearest whole number, the minimum value of A. (2)

2007-6 Paper(2) Q.6

4 A solid rectangular block has width x cm, length 3x cm and height h cm. The volume of the

block is 450 cm3. The total surface area of the block is A cm2.

(a) Show that A = 6x2 +1200

x. (4)

(b) Find, to 3 significant figures, the value of x for which A is a minimum. Verify that the value

you have found does give a minimum value for A. (5)

(c) Find, to the nearest whole number, the minimum value of A. (2)


2008-1 Paper(1) Q.5

5 A water tank is in the shape of a right circular cylinder with no lid. The base of the cylinder is a

circle of radius r cm and the height is h cm. The total external surface area of the tank is A cm2.

The capacity of the tank is 50 000π cm3.

(a) Show that A =

(100 000

r+ r2

)π. (4)

(b) Find, to the nearest whole number, the minimum value of A. Verify that the value you have

found is a minimum. (6)

2009-6 Paper(1) Q.8

6

h cm

3x cm

x cm

Figure 2

Figure 2 shows a box in the shape of a cuboid of height h cm.

The base of the box is a rectangle of length 3x cm and width x cm.

The top of the box is open.

The volume of the box if V cm3 and the total external surface area of the box is 25 cm2.

(a) Show that V =3

8x(25− 3x2). (4)

Given that x can vary.

(b) find the maximum value of V . (5)


2010-6 Paper(2) Q.7

7

E

H

F

G

A B

CD

Figure 1

Figure 1 shows a rectangular box ABCDEFGH which is open at the top EFGH.

The volume of the box is 400 cm3.

The length, x cm, of the base is twice the width of the base and the height of the box is h cm.

(a) Write down an expressions, in terms of x and h, for the volume of the box. (1)

The total surface area of the outside of the box is S cm2.

(b) Show that

S =2400

x+

1

2x2

(3)

(c) Find, to 3 significant figures, the minimum value of S. (5)

(d) Prove that the value of S found in (c) is the minimum value. (2)


2011-6 Paper(2) Q.7

8

A B

D C

h cm

x cm

3x cm

Figure 1

A rectangular box has length 3x cm, width x cm and height h cm, as shown in Figure 2. The top

of the box, ABCD, is open. The volume of the box is 30 cm3 and the total external surface area

of the box is S cm2.

(a) Show that S = 3x2 +80

x(4)

Given that x can vary,

(b) find, to 3 significant figures, the minimum value of S. (5)

(c) Verify that your answer to part (b) does give the minimum value for S. (2)

2013-1 Paper(2) Q.6

9 A solid paperweight in the shape of a cuboid has volume 15 cm3. The paperweight has a rectan-

gular base of length 5x cm and width x cm and a height of h cm. The total surface area of the

paperweight is A cm2.


x(3)

(b) Find, to 3 significant figures, the value of x for which A is a minimum, justifying that this

value of x gives a minimum value of A. (6)

(c) Find, to 3 significant figures, the minimum value of A. (2)


2013-6 Paper(2) Q.5

10

A C

DF

E

2x cm

y cm y cmA cm2

Figure 2

Figure 2 shows a shape BCDEF of a rea A cm2. In the shape, BCDF is a rectangle and DEF

is a semicircle with FD as diameter.

BF = CD = y cm and BC = FD = 2x cm. The perimeter of the shape BCDEF is 30 cm.

(a) Find an expression for y in terms of x. (2)

(b) Show that A = 30x− 2x2 − 1

2πx2 (2)

(c) Find, to 2 significant figures, the maximum value of A, justifying that the value you have

found is a maximum. (7)


2014-1 Paper(2) Q.6

11

x cm

x cm

80 cm

40 cm

Figure 2

A rectangular sheet of card measures 80 cm by 40 cm. A square of side x cm is cut away from

each corner of the card as shown in Figure 2. The card is then folded along the dotted lines to

form an open box.

The volume of the box is V cm3.

(a) Show that V = 3200x− 240x2 + 4x3 (3)

(b) Find, to 3 significant figures, the value of x for which V is a maximum, justifying that this

value of x gives a maximum value of V . (6)

(c) Find, to 3 significant figures, the maximum value of V . (2)

2015-1 Paper(2) Q.2

12 A solid right circular cylinder has height h cm and base radius r cm. The total surface area of

the cylinder is S cm2 and the volume of the cylinder is V cm3

(a) Show that S =2V

r+ 2πr2 (2)

Given that V = 1600

(b) find, to 3 significant figures, the minimum value of S.

Verify that the value you have found is a minimum. (7)


2015-6 Paper(1) Q.10

13 A solid right circular cylinder has base radius r cm and height h cm. The volume of the cylinder

is 50 cm3 and the total surface area is A cm2.

(a) Show that A = 2πr2 +100

r(3)

(b) Use calculus to find, to 4 significant figures, the value of r for which A is a minimum. (3)

(c) Use calculus to verify that the value of r found in part (b) does give a minimum value of A.

(3)

(d) Find, to the nearest whole number, the minimum value of A. (2)

2016-6 Paper(2) Q.5

14 A solid cuboid has volume 772 cm3

The cuboid has width x cm, length 4x cm and height h cm.

The total surface area of the cuboid is A cm2


x(3)

(b) Find, to 3 significant figures, the value of x for which A is a minimum, justifying that this

value of x gives a minimum value of A. (5)

(c) Find, to 3 significant figures, the minimum value of A. (2)


2017-6 Paper(2) Q.7

15 A solid cuboid has width x cm, length 5x cm and height h cm. The total surface area of the block

is 480 cm2. The volume of the block is V cm3.

(a) Show that V = 200x− 25

6x3 (4)

(b) Find the maximum value of V . (5)

2018-1 Paper(2) Q.8

16

h cm

r cm

Figure 4

A solid right circular cylinder has radius r cm and height h cm, as shown in Figure 4.

The cylinder has a volume of 355 cm3 and a total surface area of S cm2

(a) Show that S = 2πr2 +710

r(4)

Given that r can vary.

(b) using calculus find, to 3 significant figures, the minimum value of S. (5)

(c) Verify that your answer to part (b) does give the minimum value of S. (2)


24 Polynomial Roots

1988-1 Paper(1) Q.10

1 f(x) = x3 − 5x2 + 3x+ 9.

(a) Show that (x− 3) is a factor of f(x) and find the remaining factors. (4)

(b) Find the maximum and minimum values of f(x), distinguishing between them. (5)

(c) Sketch the curve y = f(x), showing, in particular, the turning points and the points of

intersection with the coordinate axes. (3)

(d) Calculate the area of the finite region bounded by the curve, the axes and the line x = 2. (3)

1988-6 Paper(1) Q.12

2 The polynomial f(x), defined by

f(x) = 2x3 + px2 + qx− 12,

has a factor (x+ 2) and has remainder −9 when divided by (x− 1).

(a) Show that p = 5, and q = −4. (5)

(b) Hence factorise f(x) completely. (3)

(c) For the curve whose equation is y = f(x), find the coordinates of the points at which the

curve meets the coordinate axes. (2)

(d) Find the coordinates of the points on the curve at whichdy

dx= 0. (3)

(e) Sketch the graph of y = f(x). (2)


1989-1 Paper(1) Q.10

3 (a) Show that x− 1 is a factor

f(x) = x3 − 7x+ 6.

(2)

(b) Hence find the three solutions of the equation f(x) = 0. (5)

(c) Sketch the graph of y = f(x) showing clearly where it crosses the coordinate axes. There is

no need to find the stationary values of f(x). (3)

(d) Find the area of the finite region above the x-axis which is bounded by the curve and the

x-axis. (5)

1989-6 Paper(1) Q.14

4 The curve with equation y = f(x) is such thatdy

dx= 3x2 − 10x+ 3. The curve passes through the

point with coordinates (−1, 0).

(a) Find f(x) and show that the curve passes through the point with coordinates (0, 9). (4)

(b) Show that f(x) can be written in the form

f(x) = (x+ p)(x− q)2

where p and q are positive numbers. (3)

(c) Deduce the solutions f(x) = 0. (1)

(d) Determine the coordinates of any turning points on the curve. (4)

(e) Hence sketch the graph of y = f(x), showing

(i) the coordinates of the points where the curve meets the coordinate axes,



1989-6 Paper(2) Q.9

5 (a) f(x) = 2x2 − px+ q + 4, where p and q are constants.

Given that f(2) = 12 and that the equation f(x) = 0 has equal roots, find the two possible

values of p and write down the corresponding values of q.

Find, also, the value of the equal roots of the equation f(x) = 0 in each case. (10)

(b) Express 2x2 − 4x+ 9 in the form a(x− b)2 + c, where a, b and c are constants.

Hence find the minimum value of 2x2 − 4x + 9 and state the value of x for which it occurs.

(5)

1990-1 Paper(1) Q.11

6 (a) Show that (x− 1) is a factor of f(x) where

f(x) = 2x3 + 3x2 − 12x+ 7

and hence factorize f(x) completely. (5)

(b) Find all the solutions of the equation

f(x) = 0.

(2)

(c) Find the coordinates of the two turning points of the curve with equation y = f(x). (4)

(d) Hence make a sketch of the curve with equation y = f(x), showing

(i) the coordinates of the points where the curve meets the coordinates axes,



1990-6 Paper(1) Q.10

7 f(x) = px3 + qx2 + rx+ s.

The curve with equation y = f(x) has gradient 4 at the point with coordinates (0,−5).

(a) Find the values of r and s. (4)

The remainder when f(x) is divided by (x − 1) is 12, and the remainder when f(x) is divided by

(x+ 2) is 15.

(b) Calculate the values of p and q. (6)

g(x) = 2x3 − x2 − 23x− 20

(c) Show that (x+ 1) is a factor of g(x). (2)

(d) Factorise g(x) completely. (2)

(e) Solve the equation g(x) = 0. (1)

1991-1 Paper(1) Q.1

8 The remainder when (x3 + 2x2 + ax+ 17) is divided by (x+ 2) is 1. Find the value of a. (3)

1991-6 Paper(1) Q.5

9 (a) Find the coordinates of the point P , at which the line with equation x = 3 cuts the curve

with equation y = 3x2 + 2x. (2)

(b) Find also an equation of the tangent to the curve at P . (3)


1992-1 Paper(1) Q.8

10 f(x) = px3 + 11x2 + 2px− 5.

The remainder when f(x) is divided by (x+ 2) is 15.

(a) Show that p = 2.

(b) Factorise f(x) completely.

(c) Deduce the solutions of the equation

f(x) = (x+ 1)(x+ 5).

1992-1 Paper(1) Q.12

11 f(x) = x3 + 3x2 − 24x+ 28.

(a) Show that (x− 2) is a factor of f(x).

(b) Show further that f(x) = (x− 2)2(x+ 7).

(c) Find the coordinates of the maximum and minimum points on the curve with equation

y = f(x), distingusing between them.

(d) Sketch the graph of y = f(x) showing clearly the coordinates of

(i) the turning points,

(ii) the points in which the curve meets the axes.

(e) Calculate the area of the finite region bounded by the curve and the x-axis.


1993-6 Paper(1) Q.8

12 f(x) = 2x3 + px2 + 2x+ 3,

where p is a constant. Given that (x− 3) is a factor of f(x),

(a) find p, (2)

(b) solve the equation f(x) = 0. (5)

1994-1 Paper(1) Q.14


dy

dx= 6x(x+ 1).


(a) Find f(x). (4)









1995-1 Paper(1) Q.11

14 The functions f and g are defined by

f: x 7−→ x3 − 7x+ 6, x ∈ R,

g: x 7−→ x− 1, x ∈ R.

(a) Show that fg(x) = x3 − 3x2 − 4x+ 12. (3)

(b) Write down an expression for gf(x). (1)

(c) Solve the equation fg(x)− gf(x) = 7. (3)

(d) Show that (x− 2) is a factor of fg(x). (1)

(e) Using the result of (d), solve fg(x) = 0. (3)

(f) Sketch the graph of the curve with equation y = fg(x), showing the coordinates of the points

where the curve cuts the axes. (2)

(g) Using your sketch in (f), or otherwise, solve the inequality fg(x) < 0. (2)


1996-1 Paper(1) Q.12

15 f(x) = px3 − qx2 + 9, where p and q are positive constants.

Given that f(x) has (x− 3) as a factor and has remainder 4 when divided by (x− 1),

(a) find p and q. (4)

(b) Hence factorise f(x) completely. (3)

The curve with equation y = f(x) crosses the x-axis at the points A, B and C.

(c) Find the coordinates of A, B and C. (2)

At the points D and E on the curve the value ofdy

dxis 0.

(d) Find the coordinates of D and E. (4)

(e) Sketch the curve with equation y = f(x) showing clearly the points A, B, C, D and E. (2)

1996-6 Paper(1) Q.6

16 f(x) = kx3 + (3k − 2)x2 − 4, where k is a constant.

Given that (x+ 2) is a factor of f(x),

(a) find the value of k. (3)

With the value of k found in (a),

(b) find the remainder when f(x) is divided by (x− 1). (2)


1997-6 Paper(1) Q.8

17 f(x) = 2x3 + px2 − 12x+ q, where p and q are constants.

Given that (x− 2) is a factor of f(x) and that, when f(x) is divided by (x + 1), the remainder is

−3,

(a) form a pair of simultaneous equations satisfied by p and q, (2)

(b) show that p = 7, (1)

(c) find the value of q. (1)

With p = 7,

(d) show that f(x) has a stationary value when x = 23 . (2)

2007-1 Paper(2) Q.10

18 f(x) = x3 + px2 − 11x+ q, p, q ∈ RGiven that (x+ 5) and (x− 3) are factors of f(x),


(b) find the value of p and the value of q, (3)

(c) factorise f(x) completely, (1)

(d) sketch the curve with equation y = f(x), showing on the diagram the coordinates of the points

of intersection with the axes. (3)

The minimum point on the curve is A.

(e) Find the coordinates of the point where the tangent at A meets the curve again. (6)


2009-6 Paper(1) Q.1

19 f(x) = 2x3 + px2 − 5x+ 6

Given that (x− 3) is a factor of f(x), find the value of p. (3)

2010-1 Paper(1) Q.2

20 f(x) = x3 + px2 + qx− 36, p and q ∈ Z+

The three roots of the equation f(x) = 0 are α, α and 4, where α ∈ Z+.

(a) Show that α = 3 (2)

(b) Hence find the value of p and the value of q. (3)

2010-6 Paper(2) Q.2

21 f(x) = x3 + 2x2 − 5x− 6

(a) Factorise x2 − x− 2 (1)

(b) Hence, or otherwise, show that (x2 − x− 2) is a factor of f(x). (3)


2011-1 Paper(1) Q.6


The line l passes through A and B.

(a) Find an equation for l. (2)

O

A

B

D

l

C

x

y

Figure 1

f(x) = x3 − px2 − qx+ r, p, q, r ∈ Z+

Figure 1 shows the curve C with equation y = f(x) and the line l.

The point D has coordinates (1,−3).

The curve C passes through the points A, D and B.

(b) Show that r = 2 (1)

(c) Find the value of p and the value of q. (4)

(d) Find the area of the shaded region shown in Figure 1. (5)


2011-1 Paper(2) Q.2

23 f(x) = x3 + ax2 + bx+ 6, a, b ∈ Z(x− 2) is a factor of f(x).

When f(x) is divided by (x+ 3) the remainder is −15.

Find the value of a and the value of b. (4)

2011-6 Paper(2) Q.4

24 A curve has equation y = x3 + 2x2 − 11x −m, where m is a positive integer. The curve crosses

the x-axis at the point with coordinates (−4, 0).

(a) Show that m = 12 (2)

(b) Factorise x3 + 2x2 − 11x− 12 completely. (3)

The curve also crosses the x-axis at two other points.

(c) Write down the x-coordinate of each of these points. (1)


2012-6 Paper(1) Q.8

25 f(x) = ax3 + bx2 + cx+ d, where a, b, c, and d are integers.

Given that f(0) = 6

(a) show that d = 6 (1)

When f(x) is divided by (x− 1) the remainder is −6

When f(x) is divided by (x+ 1) the remainder is 12

(b) Find the value of b. (4)

Given also that (x− 3) is a factor of f(x),

(c) find the value of a and the value of c, (5)

(d) express f(x) as a product of linear factors. (3)


2013-1 Paper(1) Q.11

26 f(x) = x3 + px2 + qx+ 6 p, q ∈ ZGiven that f(x) = (x− 1)(x− 3)(x+ r)

(a) find the value of r. (1)


(b) find the value of p and the value of q. (3)

O

A

1 3

B

x

y

C

Figure 2

Figure 2 shows the curve C with equation y = f(x) which crosses the x-axis at the points with

coordinates (3, 0) and (1, 0) and at the point A. The point B on C has x-coordinate 2

(c) Find an equation of the tangent to C at B. (5)

(d) Show that the tangent at B passes through A. (2)

(e) Use calculus to find the area of the finite region bounded by C and the tangent at B. (5)


2013-6 Paper(2) Q.6

27 p(x) = 2x3 + 13x2 − 17x− 70

(a) Show that p(−2) = 0 (2)

(b) Solve the equation p(x) = 0 (4)

2014-6 Paper(2) Q.9

28 f(x) = x3 + 5x2 + px− q p, q ∈ ZGiven that (x+ 2) and (x− 1) are factors of f(x),


(b) show that p = 2 and find the value of q, (3)

(c) factorise f(x) completely. (1)

(d) Sketch the curve with equation y = f(x) showing the coordinates of the points where the

curve crosses the x-axis. (2)

The curve with equation y = x3 + 2x2 + 4x meets the curve with equation y = f(x) at two points

A and B. The x-coordinate of A is −4

3and the x-coordinate of B is 2

(e) Use algebraic integration to find, to 3 significant figures, the area of the finite region bounded

by the two curves. (5)


2015-1 Paper(1) Q.9

29 f(x) = 2x3 + ax2 + bx+ 15 where a and b are constants.

The remainder when f(x) is divided by (x− 1) is −12

The remainder when f(x) is divided by (x+ 1) is 48


(b) Show that f

(1

2

)= 0 (1)

(c) Express f(x) as a product of linear factors. (4)

(d) Solve the equation f(x) = 0 (1)


2015-6 Paper(2) Q.8

30

O

C

2 4x

y

l

R

Figure 3

Figure 3 shows part of the curve C with equation y = x3 + ax2 + bx+ c

The curve passes through the origin O and the points with coordinates (2, 0) and (4, 0).

(a) Show that c = 0 (1)

(b) Find the value of a and the value of b. (3)

The point P with x-coordinate 3 lies on C. The line l passes through O and meets C at P .

(c) Show that l is the tangent to C at P . (4)

The finite region R, shown shaded in Figure 3, is bounded by C and by l.

(d) Use algebraic integration to find the area of R. (5)


2016-1 Paper(2) Q.10

31 f(x) = 2x3 − px2 − 13x− qWhen f(x) is divided by (x− 2) the remainder is −20

Given that (x− 3) is a factor of f(x)

(a) find the value of p and the value of q (7)

(b) Hence use algebra to solve the equation f(x) = 0 (5)

2016-6 Paper(1) Q.1

32 f(x) = x3 − 7x+ 6

(a) Show that (x− 2) is a factor of f(x) (2)

(b) Hence, or otherwise, factorise f(x) completely. (3)

2017-1 Paper(1) Q.2

33 f(x) = 2x3 − 3px2 + x+ 4p where p is an integer.

Given that (x− 4) is a factor of f(x)

(a) show that the value of p is 3 (2)

Using this value of p,

(b) find the remainder when f(x) is divided by (x+ 2) (2)

(c) factorise f(x) completely (3)

(d) solve the equation 2x3 − 3px2 + x+ 4p = 0 (2)


2017-6 Paper(2) Q.9

34

OA B(2, 0) x

y

C

Figure 1

The curve C with equation y = x3−4x2−4x+16 crosses the x-axis at the point with coordinates

(2, 0) and at the points A and B, as shown in Figure 1. The coordinates of the points A and B

are (a, 0) and (b, 0) respectively.


The point D lies on C and has x coordinate 0

The line l is the tangent to C at the point D.


(c) Show that l passes through B. (1)

(d) Use algebraic integration to find the area of the finite region bounded by l and C. (5)


2018-1 Paper(2) Q.9

35

O (a, 0) (b, 0)(−2, 0) x

y

C

P

Figure 5

Figure 5 shows the curve C with equation y = x3 − 2x2 − 5x+ 6

The curve C crosses the x-axis at the points with coordinates (−2, 0), (a, 0) and (b, 0)

(a) (i) Show that a = 1

(ii) Find the value of b. (4)

The point P on C has x-coordinate 2 and the line l is the tangent to C at P .

(b) Show that l crosses the x-axis at the point with coordinates (−2, 0) (6)

(c) Use algebraic integration to find the exact area of the finite region bounded by C and l. (4)


25 Product Rule

1988-1 Paper(2) Q.13


π

4.

(a) Find f ′(x). (3)



6 x 6π



x. (3)


π


1

x.


−π46 x 6

π

4. (8)

1988-6 Paper(2) Q.3


(a) x sinx, (2)

(b)√

(1 + x3). (3)

1989-1 Paper(2) Q.6


(a) ex(x2 + 1), (3)

(b)sin 3x

x2. (3)


1989-6 Paper(2) Q.4


(b) Evaluate

∫0

π8 cos (2x+

π

4) dx. (3)

1989-6 Paper(2) Q.10




cosxshow that

d


(3)





terms of π. (5)

1990-1 Paper(2) Q.2


(a) e−3x sinx, (2)

(b) (1 +√x)5. (2)


1991-1 Paper(2) Q.3


(a) x2 sin 7x, (2)

(b) cosx2. (2)

1991-6 Paper(2) Q.7




1993-6 Paper(2) Q.2


(a) e2x cos 3x, (2)

(b)2x

e3x. (2)

1994-1 Paper(2) Q.2


(a) e−x sinx, (2)

(b)1

cos 2x. (2)


1995-1 Paper(2) Q.6


(a)2x

ex, (3)

(b) 3x sin 2x. (3)

1995-6 Paper(2) Q.2


(a) x3 cos 2x, (2)

(b)ex

1 + x. (2)

1995-6 Paper(2) Q.14


(a) xdy

dx= y(3x+ 1). (4)

(b)1

y

dy

dx= 3 +

1

x. (3)






1996-1 Paper(2) Q.4


(a) x2ex, (2)

(b) cos (x2 + 2x). (3)

1996-1 Paper(2) Q.13

15 (a) Find, in terms of e, the coordinates of the stationary points of the curve withe equation

y = xe2x, determining whether the point is a minimum or maximum. (8)

A formula which could be used to predict the value of N , the number of minutes spent in a certain

queue for tickets, from the value of r, the number of people who can be served in one minute, is

N =4

r2 − 4r, r > 4.

(b) Show that

dN

dr=

8(2− r)(r2 − 4r)2

.

(3)

At time t minutes (t > 0) after the ticket seller starts selling tickets, he speeds up so thatdr

dt= 0.4.

(c) FinddN

dtwhen r = 5. (4)

1996-6 Paper(2) Q.6


(a) x3 cos 2x, (3)

(b)x3

sinx, 0 < x < π (2)


1997-6 Paper(2) Q.14

17 f(x) =2x2

x− 3, x ∈ R, x 6= 3.

(a) Find f ′(x). (3)

(b) Find the set of values of x for which f ′(x) > 0. (4)

g(x) = ex sinx, 0 6 x 6 2π.

(c) Solve the equation g(x) = 0. (2)

(d) Solve the equation g′(x) = 0. (2)

(e) Sketch the graph of the curve with equation y = g(x), showing the coordinates of the points

where the curve intersects the axes. (2)

(f) State the set of values of x for which g′(x) < 0. (2)

2007-1 Paper(1) Q.1

18 Differentiate with respect to x, (x+ 2)e3x. (3)

2008-1 Paper(1) Q.4

19 Given that y = (3x− 2)e2x.

(a) finddy

dx, (3)

(b) show that (3x− 2)dy

dx= (6x− 1)y. (3)


2009-6 Paper(1) Q.11

20 (a) By writing tanx =sinx

cosx, show that

d

dx= (tanx) =

1

cos2 x. (3)

0

A

B C x

y

−π2

3π

2

Figure 4

Figure 4 shows the curve with equation y = 1 + tanx,−π2< x <

3π

2.

The curve crosses the y-axis at the point A and the x-axis at the points B and C.


(c) Find the x-coordinate of

(i) B,

(ii) C. (2)

The point D on the curve has x-coordinateπ

6.

The normal to the curve at D meets the curve again at the point E and crosses the x-axis at G.

(d) Find the exact value of the x-coordinate of G. (6)

Given that the coordinates of E are (e, f),π

2< e <

3π

2,

(e) hence or otherwise show that f > 0. (2)


2010-1 Paper(1) Q.10

21 The curve C1, with equation y = x2, meets the curve C2, with equation y =x2

x− 1, at the origin

and at the point A.

Find

(a) the coordinates of A, (4)

(b) an equation of the tangent to C1 at A, (4)

(c) an equation of the tangeth to C2 at A. (4)

The tangent to C1 at A meets the y-axis at the point B and the tangent to C2 at A meets the

y-axis at the point D.

(d) Find the area of 4BAD. (3)

2010-6 Paper(2) Q.6

22 y = tanx where tanx =sinx

cosx

(a) Show, by differentiation, thatdy

dx=

1

cos2 x(3)

(b) Hence differentiate e3x tanx (3)

(c) Solve, to 3 significant figures, the equation 3 sin θ = 6 cos θ, 0 6 θ 6 π2 (2)

2011-1 Paper(1) Q.1

23 Differentiate x2 cos 3x (3)


2011-1 Paper(1) Q.5


x+ 2,

(a) show thatdy

dx=y(kx+ 2k − 1)

x+ 2(5)

Given also thatdy

dx=

5

4when x = 0,




2011-6 Paper(1) Q.3


(a) finddy

dx(3)

(b) show thatd2y

dx2= 2

dy

dx− 9y + 6e2x cos 3x (4)

2012-1 Paper(1) Q.5


(a) y = x2ex (2)

(b) y = (x3 + 2x2 + 3)5 (3)


2013-1 Paper(2) Q.4


(a) 3x sin 5x (3)

(b)e2x

4− 3x2(3)

2014-1 Paper(1) Q.3


(a) e3x(5x− 7)2 (3)

(b)cos 2x

x+ 9(3)

2015-6 Paper(1) Q.2


(a) finddy

dx(3)


dx= 2y(1 + x) (2)

2016-6 Paper(2) Q.4


e2x cos 3x

(3)


2018-1 Paper(2) Q.5


show thatd2y

dx2− 2

dy

dx+ y = 12ex (7)


26 Quadratic Inequalities

1988-6 Paper(1) Q.3

1 Find the set of values of x for which

2x(x− 2) < x+ 3.

(5)

1996-6 Paper(1) Q.5


(2x+ 1)2 < 4(2x+ 1).

(5)

2007-6 Paper(2) Q.4

3 (a) Find the coordinates of the points where the line with equation y = 3x + 3 intersects the

curve with equation y = x2 + x− 12. (5)

(b) Find the set of values for which x2 + x− 12 > 3x+ 3. (2)

2012-1 Paper(1) Q.3

4 Solve the inequality 6x2 − 19x− 7 < 0 (4)

2012-6 Paper(1) Q.1

5 Find the set of values of x for which (2x+ 1)(4− x) > (x− 4)(2x− 3) (4)


2013-6 Paper(1) Q.2


3(x+ 1)2 < 9− x

(4)

2014-6 Paper(2) Q.4

7 (a) Find the coordinates of the points where the line with equation y = 4x− 4 meets the curve

with equation y = x2 − 3x+ 6 (5)

(b) Hence, or otherwise, find the set of value of x for which x2 − 3x+ 6 > 4x− 4 (2)

2016-1 Paper(1) Q.2

8 Find the set of values of x for which (2x− 3)2 > 7x− 3 (5)

2017-1 Paper(1) Q.3

9 Use algebra to find the set of values of x for which (3x− 1)(x− 1) < 2(3x− 1) (5)


27 Quotient Rule

1988-1 Paper(2) Q.1


x

sinx.

(3)

1988-1 Paper(2) Q.11


f: x 7→ x

(x− 1)(x+ 3)where x ∈ R and x 6= −3, x 6= 1.


(x− 1)2(x+ 3)2. (5)



x =x

(x− 1)(x+ 3).

(4)




1989-1 Paper(2) Q.6


(a) ex(x2 + 1), (3)

(b)sin 3x

x2. (3)


1991-6 Paper(2) Q.14

4 f(x) =(x− 3)2

(1− 2x), x 6= 1

2 .

(a) Show that f ′(x) =2(x+ 2)(3− x)

(1− 2x)2. (3)








1993-6 Paper(2) Q.2


(a) e2x cos 3x, (2)

(b)2x

e3x. (2)

1994-1 Paper(2) Q.2


(a) e−x sinx, (2)

(b)1

cos 2x. (2)


1995-1 Paper(2) Q.6


(a)2x

ex, (3)

(b) 3x sin 2x. (3)

1995-6 Paper(2) Q.2


(a) x3 cos 2x, (2)

(b)ex

1 + x. (2)


1996-1 Paper(2) Q.12

9 f(x) =k + x









in x3. (3)




1996-1 Paper(2) Q.13





N =4

r2 − 4r, r > 4.

(b) Show that

dN

dr=

8(2− r)(r2 − 4r)2

.

(3)


dt= 0.4.

(c) FinddN

dtwhen r = 5. (4)

1996-6 Paper(2) Q.6


(a) x3 cos 2x, (3)

(b)x3

sinx, 0 < x < π (2)

2007-6 Paper(2) Q.1


x2 + 3. (3)


2008-1 Paper(2) Q.4

13 Use tan θ =sin θ

cos θ

(a) to show thatd

dθ(tan θ) =

1

cos2 θ, (3)

(b) to solve, in degrees to one decimal place, the equation 5 sin θ = 7 cos θ, 0 6 θ < 360◦. (3)

2010-6 Paper(2) Q.6

14 y = tanx where tanx =sinx

cosx

(a) Show, by differentiation, thatdy

dx=

1

cos2 x(3)

(b) Hence differentiate e3x tanx (3)

(c) Solve, to 3 significant figures, the equation 3 sin θ = 6 cos θ, 0 6 θ 6 π2 (2)

2012-1 Paper(1) Q.5


(a) y = x2ex (2)

(b) y = (x3 + 2x2 + 3)5 (3)


2012-6 Paper(2) Q.4


(a)1

x2(2)

(b)1

(2x+ 1)2(2)

(c)1

1− cos2 x(3)

2013-1 Paper(2) Q.4


(a) 3x sin 5x (3)

(b)e2x

4− 3x2(3)

2014-1 Paper(1) Q.3


(a) e3x(5x− 7)2 (3)

(b)cos 2x

x+ 9(3)


2014-6 Paper(1) Q.3

19 Given that 2xy − 3y = e2x

(a) show thatdy

dx=

4e2x(x− 2)

(2x− 3)2(5)

(b) find the value ofdy

dxwhen x = 0 (1)

(c) find an equation, with integer coefficients, of the tangent to the curve with equation 2xy−3y =

e2x at the point on the curve where x = 0 (3)

2016-1 Paper(2) Q.7

20

O x

y

Figure 1

Figure 1 shows the curve with equation y =x2 − 2

2x− 3where x 6= 3

2


(b) Finddy

dx(3)

(c) Find the coordinates of the stationary points on the curve. (5)


2016-6 Paper(2) Q.8


y =3x2 − 1

3x+ 2where x 6= −2

3







O x

y

(3)




2017-1 Paper(2) Q.5

22 Given that y = 3x√

2x− 1 x >1

2

(a) show thatdy

dx=

3(3x− 1)√2x− 1

(5)

The straight line l is the normal to the curve with equation y = 3x√

2x− 1 at the point on the

curve where x = 1

(b) Find an equation, with integer coefficients, for l. (6)


28 Rational Function

1988-1 Paper(2) Q.6

1 State the equations of the two asymptotes of the curve

y = 2 +1

x, where x ∈ R and x 6= 0.

Sketch the curve showing clearly where it crosses a coordinate axis and how it approaches the

asymptotes. (6)

1989-1 Paper(2) Q.11

2 (a) Show thatx+ 1

x− 2= 1 +

3

x− 2, x 6= 2. (1)

The function f is defined by

f(x) =x+ 1

x− 2, x 6= 2.

(b) Write down the equations of the two asymptotes to the curve y = f(x). (2)

(c) Find the coordinates of the points at which the curve y = f(x) crosses the coordinate axes.

(2)

(d) Sketch the curve y = f(x) showing clearly how the curve approaches its asymptotes. (4)

(e) Find the coordinates of the two points at which the line y = 3x + 3 cuts the curve. Show

that this line is the normal to the curve at one of these points. (6)


1989-6 Paper(2) Q.14

3 f(x) =2x2 + 6

x− 1, x ∈ R, x 6= 1.

(a) Find the set of values of x for which f(x) > 0. (2)

(b) Write down the equation of that asymptote of the curve with equation y = f(x) which is

parallel to the y-axis. (1)

(c) Find the coordinates of the maximum and minimum points of the curve with equation y =

f(x), distingusing between these points. (7)

(d) Hence sketch the curve with equation y = f(x), showing the results you have in (a), (b), and

(c). (5)

1991-1 Paper(2) Q.8

4 (a) Sketch the graph of y =2x+ 5

x, x 6= 0. (2)

(b) State the equations of the asymptotes for this curve. (2)

(c) On the same set of axes sketch the graph of y = x− 2. (1)

(d) Calculate the values of the x-coordinates of the points of intersection of the two graphs. (3)


1991-6 Paper(2) Q.10

5 The curve C has equation y =x+ 1

x− 2, x 6= 2.

(a) State the equations of the asymptotes of C. (2)

(b) Show that, for all points on C,dy

dxis negative. (2)

(c) Sketch C, giving the coordinates of the points where C crosses the axes. (3)

The line with equation 3y = x+ 9 cuts the curve at the points A and B.

(d) Find the coordinates of A and B. (4)

(e) Show that the line with equation 3y = x+ 9 is a normal to C at one of these points, but not

at the other point. (4)

1993-6 Paper(2) Q.7

6 For the curve with equation y = f(x) where

f(x) =3x+ 1

x− 2

(a) find an equation of each asymptote, (2)

(b) find the coordinates of the points at which the curve intersects the coordinate axes. (2)

(c) Sketch the curve, showing clearly the asymptotes and the points where the curve cuts the

coordinate axes. (2)


1994-1 Paper(2) Q.8

7 Given that the point with coordinates (2,−1) lies on the curve with equation

y =2x+ 1

x+ k,

where k is a constant,

(a) find k. (2)

Hence

(b) write down an equation of each asymptote to the curve, (2)

(c) sketch the curve, showing how the curve approaches the asymptotes and the coordinates of

the points where the curve intersects the axes. (3)

1995-6 Paper(2) Q.8

8 For the curve with equation y = f(x) where

f(x) =3(x− 4)

(x− 2), x 6= 2,

(a) find the equations of the horizontal and vertical asymptotes. (2)

(b) find the coordinates of the points at which the curve intersects the coordinate axes. (2)

(c) Show that the curve with equation y = f(x) has no turning points. (2)

(d) Sketch the curve with equation y = f(x) showing clearly the asymptotes and the points where



1996-1 Paper(2) Q.12

9 f(x) =k + x









in x3. (3)



1996-6 Paper(2) Q.4

10 For the curve with equation y =2x+ 7

x− 3, x 6= 3, write down

(a) the equation of the asymptote which is parallel to the y-axis. (1)

(b) the equation of the asymptote which is parallel to the x-axis. (1)

(c) Sketch the curve showing clearly the asymptotes and the coordinates of the points where the

curve crosses the coordinate axes. (3)


1997-6 Paper(2) Q.8

11 f : x 7→ 2 +1

x− 3, x ∈ R, x 6= 3.

(a) Solve the equation f(x) = 0. (2)

The curve, C, has equation y = f(x).

(b) Write down the equations of the asymptotes of C. (2)

(c) Sketch a graph of C, showing how the curve approaches the asymptotes and the coordinates

of the points where C intersects the axes. (3)

2007-1 Paper(1) Q.10

12 A curve has equation y =5x2 + 10

2x− 1, x 6= 1

2 .



(c) Sketch the curve, showing the asymptote parallel to the y-axis and the coordinates of the

stationary points. (3)


(d) Find an equation for the tangent to the curve at A. (3)

(e) Find an equation for the normal to the curve at A. (2)

(f) Find the area enclosed by the tangent at A, the normal at A and the x-axis. (3)


2007-6 Paper(1) Q.3

13 A curve has equation y =x− 4

5− x, x 6= 5.


(i) the x-axis,


(b) Sketch the curve, showing clearly the asymptotes and the coordinates of the points at which

the curve crosses the coordinate axis. (5)

2008-1 Paper(2) Q.10


4x+ 2, x 6= −1

2 .

(a) Write down an equation for the asymptote to the curve which is parallel to

(i) the x-axis,


(b) Find the coordinates of the points where the curve crosses the coordinate axes. (2)



The curve intersects the y-axis at the point P .

(d) Find an equation for the normal to the curve at P . (5)

The normal at P meets the curve again at Q.

(e) Find the coordinates of Q. (5)


2008-6 Paper(2) Q.6

15 A curve has equation y = 3− 2

x+ 1, x 6= −1.

(a) Find an equation of the asymptote to the curve which is parallel to


(b) Find the coordinates of the point where the curve crosses




2009-6 Paper(2) Q.4

16 A curve has equationx− 2

x+ 3, x 6= −3


(i) the x-axis,


(b) Calculate the coordinates of the point where the curve crosses

(i) the x-axis,





2010-6 Paper(2) Q.10

17 A curve C has equation y = 4x+1

x− 1, x 6= 1


(b) Prove that C has a minimum point when x = 32 and a maximum point when x = 1

2 . (8)

(c) Find the y-coordinate of

(i) the minimum point,

(ii) the maximum point. (2)

(d) Sketch C, showing clearly the asymptote found in (a), the coordinates of the turning points,

and the coordinates of the point where the curve crosses the y-axis. (4)

2011-1 Paper(1) Q.4

18 A curve C has equation y =2

x− 3− 1, x 6= 3.

(a) Find an equation of the asymptote to C which is paralle to

(i) the x-axis,



(i) the x-axis,


(c) Sketch the graph of C, showing clearly the asymptotes and the coordinates of the points

where the graph crosses the coordinate axes. (3)


2011-6 Paper(1) Q.9


y =2x2 − 6

3x− 6x 6= 2




(c) Find an equation of the normal to the curve at A. (3)

The normal at A meets the curve again at B.

(d) Find the x-coordinate of B. (4)

2012-1 Paper(1) Q.7

20 The curve C with equation y =2x− 3

x− 3, x 6= 3, crosses the x-axis at the point A and the y-axis

at the point B.

(a) Find the coordinates of A and the coordinates of B. (2)

(b) Write down an equation of the asymptote to C which is

(i) parallel to the y-axis,

(ii) parallel to the x-axis. (2)

(c) Sketch C showing clearly the asymptotes and the coordinates of the points A and B. (3)

(d) Find an equation of the normal to C at the point B. (5)

The normal to C at the point B crosses the curve again at the point D.

(e) Find the x-coordinate of D. (4)


2012-6 Paper(2) Q.7

21 The curve G has equation y = 3− 1

x− 1, x 6= 1

(a) Find an equation of the asymptote to G which is parallel to

(i) the x-axis,


(b) Find the coordinates of the point where G crosses

(i) the x-axis,


(c) Sketch G, showing clearly the asymptotes and the coordinates of the points where the curve

crosses the coordinates axes. (3)

A straight line l intersects G at the points P and Q. The x-coordinate of P and the x-coordinate

of Q are roots of the equation 2x− 3 =1

x− 1

(d) Find an equation of l. (2)


2013-6 Paper(1) Q.3

22

O 1 3

d

b

x

y

Figure 1

Figure 1 shows a sketch of the curve with equation y = 1 +c

x+ a, where a and c are integers.

The equations of the asymptotes to the curve are x = 3 and y = b.


The curve crosses the x-axis at (1, 0) and the y-axis at (0, d).

(b) Find the value of c and the value of d. (4)


2014-1 Paper(2) Q.5

23 A curve C has equation y =2x− 5

x+ 3, x 6= −3

(a) Find an equation of the asymptote to C which is parallel to




(c) Sketch the graph of C, showing clearly its asymptotes and the coordinates of the points where

the graph crosses the coordinate axes. (3)

(d) Find the gradient of C at the point on C where x = −1 (3)

2014-6 Paper(2) Q.8


4x+ 5, x 6= −5

4


(i) the x-axis (ii) y-axis. (2)

(b) Find the coordinates of the points where the curve crosses

(i) the x-axis (ii) the y-axis. (2)



(d) Find an equation of the normal to the curve at the point where x = −1

Give your answer in the form ax+ by + c = 0 where a, b and c are integers. (7)


2015-6 Paper(1) Q.9

25 A curve C has equation y =3x+ 1

2x+ 3x 6= −3

2

(a) Write down an equation of the asymptote of C which is parallel to

(i) the x-axis,


(b) Find the coordinates of the points where C crosses

(i) the x-axis,


(c) Using the axes opposite, sketch the curve C, showing clearly the asymptotes and the coordi-

nates of the points where C crosses the axes. (3)

The curve C intersects the x-axis at the point A.

The line l is the normal to C at A.


The line l meets C again at the point B.

(e) Find the x-coordinate of B. (5)

O x

y


2016-6 Paper(2) Q.8


y =3x2 − 1

3x+ 2where x 6= −2

3







O x

y

(3)




2017-1 Paper(1) Q.6

27

O(s, 0)

(0, 3.5)

x

y

y = 3

x = −2

Figure 3

Figure 3 shows a sketch of the curve with equation

y =bx+ c

x+ ax 6= −a,

where a, b and c are integers.

The equations of the asymptotes to the curve are x = −2 and y = 3

The curve crosses the y-axis at (0, 3.5)

(a) Write down the value of a and the value of b. (2)

(b) Find the value of c. (2)

Given that the curve crosses the x-axis at (s, 0)

(c) find the value of s. (2)


2017-6 Paper(1) Q.10

28 A curve C has equation y = 8x+1

2x− 1x 6= 1

2


(b) Show that C has a minimum point at x =3

4and a maximum point at x =

1

4(9)

(c) Find the y coordinate of

(i) the minimum point,

(ii) the maximum point,

(iii) the point where C crosses the y-axis. (3)

(d) Sketch the curve C, showing clearly the asymptote found in part (a), the coordinates of the

turning points and the coordinates of the point where C crosses the y-axis. (3)


29 Related Rates

1988-1 Paper(2) Q.8

1 A balloon, in the shape of a cylinder, is being inflated. Throughout the period of inflation, the

height of the cylinder is equal to the radius of a circular end and the rate of increase of this radius

is 2 cm/s.

At the instant when the radius of the circular end is 4 cm, calculate, in terms of π,

(a) the rate, in cm3/s, at which the volume of the balloon is increasing, (4)

(b) the rate, in cm2/s, at which the total surface area of the balloon is increasing. (3)


1988-6 Paper(2) Q.12

2

1 mh m

Fig. 1

Figure 1 shows a spherical tank of radius 1 m containing fuel oil to a height of h metres, measured

from the centre of the sphere. The volume, V m3, of fuel in the tank is given by

V = 13π(2 + 3h− h3).

(a) FinddV

dh. (1)

On a particular evening, the level of the fuel in the tank is decreasing at a constant rate of 0.002

m/min.

(b) Calculate the rate of decrease of the volume of fuel oil in the tank, in m3/min, to 2 significant

figures, at the instant when h = 0.3. (4)

(c) Calculate, to 2 significant figures, the two values of V when the rate of decrease of the volume

of the fuel oil in the tank is 0.00128π m3/min. (5)

The area of the surface of the fuel in contact with the air is A m2.

(d) Write down an expression for A in terms of h. (1)

(e) Find the rate of change of the surface area, in m2/min, to 2 significant figures, at the instant

when h = 0.5. (4)


1989-1 Paper(2) Q.7

3

A

PQ

B

C

Fig. 1

Figure 1 shows a trough whose length AB is 2 m. Its cross-section is an isosceles right angled

triangle and its upper edges lie in a horizontal plane. At time t seconds the height AC of the

water level above AB is h metres.

(a) Show that the area of 4APQ is h2 metres2. (1)

Given that water is flowing into the trough at a rate of 0.02 m3/s,

(b) find the rate, in m/s, at which the water level is rising when h = 0.4. (5)

1989-6 Paper(2) Q.5

4 The volume, V cm3, of water in a container is given by the expression

V = 12h2,

where h cm is the depth of the water. Water is pouring into the container at a steady rate of 90

cm3/s. Find the rate, in cm/s, at which the depth of water is increasing when h = 3. (5)


1990-1 Paper(2) Q.3

5 A hollow right circular cone is such that its height and the radius of its base are equal in length.

The cone is inverted and water poured in at a rate of 20 cm3/s. At time t seconds the height of

water is x cm and the volume of water is V cm3.

(a) Show that V = 13πx

3. (1)

(b) Calculate, in terms of π, the value ofdx

dtwhen x = 5. (4)

1991-1 Paper(2) Q.4

6 The edges of a cube are of length x cm. Given that the volume of the cube is being increased at

a rate of p cm3/s, where p is a constant, calculate, in terms of p, in cm2/s, the rate at which the

surface area of the cube is increasing when x = 5. (5)


1991-6 Paper(2) Q.12

7 The line with equation y = x cuts the curve C with equation 8y = x2 at the origin O and at the

point P .

(a) Find the coordinates of P . (1)

(b) Calculate the area of the finite region bounded by C and the line. (4)

The region bounded by C and the line y = h is rotated through 180◦ about the y-axis.

(c) Show that the volume of the solid generated is 4πh2. (2)

A set of coordinate axes are graduated with each unit being 1 cm and the point P defined as

above. A bowl is made with an inner surface in the shape formed by rotating the arc OP of the

curve with equation 8y = x2 through 360◦ about the y-axis. The bowl is held with the vertical

through O being an axis of symmetry and water is poured in at a rate of 24 cm3/s.

(d) At the instant when the depth of water is 2 cm calculate, leaving your answer in terms of π,

the rate, in cm/s, at which the depth of water is increasing. (5)

(e) Calculate, to the nearest second, the time taken to fill the bowl. (3)

1993-6 Paper(2) Q.4

8 The volume V cm3 of a cylinder of fixed height 4 cm is increasing at the rate of 4.8 π cm3/s.

Find, in cm/s, the rate at which the radius of the cylinder, r cm, is increasing when r = 3. (4)

1994-1 Paper(2) Q.6

9 The radius of a sphere is r cm. Given that the surface area of the sphere is increasing at a rate of

4 cm2/s, calculate the rate, in cm3/s, at which the volume of the sphere is increasing when r = 3.

(5)


1995-1 Paper(2) Q.7

10 V = kr3.

Given that V = 15 when r = 2,

(a) find the value of the constant k. (2)

Given also thatdr

dt= 0.2 when r = 4,

(b) find the value ofdr

dt= 0.2 when r = 4. (4)


1995-1 Paper(2) Q.12

11 (a) Starting with the results ford

dθ(sin θ) and

d

dθ(cos θ) show that

d

dθ(tan θ) =

1

cos2 θ. (5)

A S

P

B

θ

π3

Fig. 1

Figure 1 shows a point S which is 3 m from the nearest point A of a straight line. Point B is on

this line and is such that ∠ASB =π

3. A point P moves on the line in such as way that ∠ASP = θ,

and the rate of change of θ with respect to time is 2π radians per second.

(b) Show thatd(AP )

dθ=

3

cos2 θ. (3)

Find in terms of π the velocity, in m/s, of P when

(c) P is at the point A, (4)

(d) P is at the point B. (3)

1995-6 Paper(2) Q.5

12 The area of a circle of radius r cm is increasing at a rate of 3 cm2/s. Calculate, in cm/s, the rate

at which the radius is increasing when r =21

π. (5)


1996-1 Paper(2) Q.5

13 The radius of the base of a right circular cone is r cm and the vertical height of the cone is 2r

cm. Given that the radius of the base is increasing at a rate of 5 cm/s, find the rate, in cm3/s to

3 significant figures, at which the volume of the cone is increasing when r = 6. (5)

1996-1 Paper(2) Q.13





N =4

r2 − 4r, r > 4.

(b) Show that

dN

dr=

8(2− r)(r2 − 4r)2

.

(3)


dt= 0.4.

(c) FinddN

dtwhen r = 5. (4)


1996-6 Paper(2) Q.13

15

C

BA

4 m

50 cm

25 cm

Fig. 1

Figure 1 shows an empty water container 4 m in length and 25 cm wide at the top. The top is

horizontal. The cross-section of the container is a vertical isosceles triangle, ABC, with AB = 25

cm and the perpendicular from C to AB of length 50 cm. Water is poured into the container at

a rate of 1.25 m3 per second. At time t seconds after starting to fill the container the depth of

water is h metres and the volume of water is V cubic metres.

(a) Show that V = h2 (4)

(b) Find the rate, in m/s to 3 decimal places, at which the water is rising in the container when

t = 3. (4)

(c) A right circular cone is such that its height is twice the diameter of its base. The volume of

the cone is increased by x%, where x is small. The final shape of the cone is similar to its

original shape. An estimate for the corresponding increase in its base radius is kx%. Find,

using calculus methods, the value of the constant k. (7)


1997-6 Paper(1) Q.11

16 A can is made of thin sheet metal and it in the shape of a right circular cylinder closed at both

ends. The radius of the circular cross-section of the can is r cm, the height of the can is h cm and

its total surface area is A cm2. Given that the capacity of the can is 31.25π cm3,

(a) show that r2h = 31.25, (1)

(b) find an expression of A in terms of π and r only, (3)

(c) calculate, to one decimal place, the value ofdA

drwhen r = 3, (4)

(d) show that A has a stationary value when r = 2.5, (3)

(e) find, giving your answer as a multiple of π, the stationary value of A, establishing whether

this is a maximum or a minimum value. (4)

1997-6 Paper(2) Q.6

17 An expanding spherical balloon, of radius r m, has volume V m3 and surface area A m2.

(a) Show thatdA

dV=

2

r. (4)

Given that the volume of the balloon is increasing at a constant rate of 0.5 m3/s.

(b) find the rate, in m2/s, at which the surface area of the balloon is expanding when r = 1.2.

(2)

2007-1 Paper(1) Q.4

18 Oil drips from a leaking pipe and forms a circular pool on horizontal ground. The area of the pool

is increasing at a constant rate of 30 cm2/s. Find, to 2 significant figures, the rate of increase of

the radius when the area of the pool is 40 cm2. (6)


2007-6 Paper(2) Q.2

19 Oil is dripping from a pipe at a constant rate and forms a circular pool. The area of the pond is

increasing at 15 cm2/s. Find, to 3 significant figures, the rate of increase of the radius of the pool

when the area is 50 cm2. (6)

2008-1 Paper(2) Q.3

20 The volume of a right cylinder cone is increasing at the rate of 45 cm3/s. The height of the cone

is always three times of the base of the cone. Find the rate of increase of the radius of the base,

in cm/s to 3 significant figures, when the radius of the cone is 4 cm. (6)

2009-6 Paper(1) Q.4

21 Oil is dripping from a pipe onto horizontal ground, forming a circular pool.

Find an estimate for the percentage increase in the area of the pool when its radius increases by

1%. (5)

2010-1 Paper(1) Q.3

22 The volume of a sphere is increasing at a rate of 25 cm3/s.

Find the rate of increase of the surface area of the sphere when the radius is 2.5 cm. (6)

2010-6 Paper(2) Q.5

23 Oil is leaking from a hole in a pipe. The oil forms a circular pool of radius r cm. The area, A

cm2, of the surface of the pool is increasing at a constant rate of 0.3 cm2/s. Find, to 3 significant

figures, the rate of increase of the radius of the pool when the area is 40 cm2. (5)


2012-1 Paper(1) Q.11

24

O

h

x

y

Figure 3

The centre of the circle C, with equation x2 + y2 − 10y = 0, has coordinates (0, 5). The circlepasses through the origin O. The region bounded by the circle, the positive y-axis and the liney = h, where h < 5, is shown shaded in Figure 3. The shaded region is rotated through 2π radiansabout the y-axis.

(a) Show that the volume of the solid formed is1

3π h2(15− h). (5)

O x

y

A

Figure 4

The point A with coordinates (5, 5) lies on C. A bowl is formed by rotating the arc OA through2π radians about the y-axis, as shown in Figure 4. Water is poured into the bowl at a constantrate of 6 cm3/s. The volume of water in the bowl is V cm3 when the depth of water above O ish cm.

(b) Use the formula given in part (a) to find an expression fordV

dhin terms of h. (1)

(c) Find, to 3 significant figures, the rate at which h is changing when the water above O is 1.5cm deep. (4)

The area of the surface of the water is W cm2 when the depth of water above O is h cm.

(d) Show that, for 0 < h < 5, the rate of change of the depth of water above O isk

W, stating

the value of k. (3)


2012-1 Paper(2) Q.6

25

10 cm

h cm60◦

Figure 3

A container in the shape of a right circular cone of height 10 cm is fixed with its axis of symmetry

vertical. The vertical angle of the container is 60◦, as shown in Figure 3. Water is dripping out of

the container at a constant rate of 2 cm3/s. At time t = 0 the container is full of water. At time

t seconds the depth of water remaining is h cm.

(a) Show that h =

[1000− 18t

π

] 13

(6)

(b) Find, in cm2/s, to 3 significant figures, the rate of change of the area of the surface of the

water when t = 15 (6)


2012-6 Paper(1) Q.6

26

O

P

Q

r cmθ rad

Figure 1

The points P and Q lie on the circumference of a circle with center O and radius r cm. Angle

POQ = θ radians. The segment shaded in Figure 1 has area A cm2.

(a) Show that A =1

2r2(θ − sin θ) (3)

When angle POQ is increased to (θ + δθ) radians, where δθ is small, the area of the shaded

segment is increased to (A+ δA) cm2, where δA is small.

(b) Show that δA =1

2r2(1− cos θ)δθ (3)

For a circle of radius 4 cm, the area of the shaded segment is increased by 0.05 cm2 when angle

POQ increases by 0.02 radians.

(c) Find, to 1 decimal place, an estimate for θ (4)

2012-6 Paper(2) Q.2

27 Given that x = t3 + 4 and y = 1− t+ 5t2

(a) Find

(i)dx

dt

(ii)dy

dt(2)

(b) Finddy

dxin terms of t. (2)


2013-6 Paper(1) Q.5

28 The volume of liquid in a container is V cm3 when the depth of the liquid is h cm. Liquid is added

to the container at a rate of 36 cm3/s. Given that V = 4h3, find the rate at which the depth of

the liquid is increasing when V = 500 (7)

2014-1 Paper(2) Q.2

29 The volume of a right circular cone is increasing at a constant rate of 12 cm3/s. The radius of the

base of the cone is always half the height of the cone. Find, in cm/s, the exact value of the rate

of increase of the height of the cone when the height is 4 cm. (5)

2014-6 Paper(1) Q.5

30 The volume of a right circular cone is increasing at the rate of 72 cm3/s. The height of the cone

is always four times the radius of the base of the cone. Find the rate of increase of the radius of

the base, in cm/s to 3 significant figures, when the height of the cone is 12 cm. (6)

2015-1 Paper(1) Q.1

31 An equilateral triangle has sides of length x cm.

(a) Show that the area of the triangle is

√3

4x2 cm2 (2)

The length of each side of the equilateral triangle is increasing at a rate of 0.1 cm/s.

(b) Find the length of each side of the triangle when the area of the triangle is increasing at a

rate of

√3

10cm2/s. (4)


2015-6 Paper(2) Q.6

32

O

D

A

C

B

θ rad

Figure 1

Figure 1 shows a sector OAB of the circle with centre O and radius 10 cm.

The points C and D lie on OB and OA respectively and CD is an arc of the circle with centre O

and radius 6 cm. The size of angle AOB is θ radians. The shaded region is bounded by the arcs

AB and CD and the lines AD and BC.

The area of the shaded region is S cm2.

(a) Show that S = 32θ. (3)

The size of angle AOB is increasing at a constant rate of 0.2 rad/s.

(b) Find the rate of increase of S. (2)

When the area of the shaded region is 20 cm2

(c) calculate the perimeter of the shaded region. (5)

2016-1 Paper(1) Q.3

33 The volume, V cm3, of a sphere of radius r cm is increasing at the rate of 60 cm3/s.

Find the rate of increase of the radius, in cm/s correct to 2 significant figures, when the volume

is 36000π cm3. (7)


2016-6 Paper(2) Q.10

34

h cm

60◦

Figure 1

A conical container is fixed with its axis of symmetry vertical. Oil is dripping into the container

at a constant rate of 0.4 cm3/s. At time t seconds after the oil starts to drip into the container,

the depth of the oil is h cm. The vertical angle of the container is 60◦, as shown in Figure 1

When t = 0 the container is empty.

(a) Show that h3 =18t

5π(4)

Given that the area of the top surface of the oil is A cm2

(b) show thatdA

dt=

4

5h(6)

(c) Find, in cm2 to 3 significant figures, the rate of change of the area of the top surface of the

oil when t = 10 (2)

2017-1 Paper(2) Q.3

35 The radius of a circular pool of oil is increasing at a constant rate of 0.5 cm/s.

Find in cm2/s to 3 significant figures, the rate at which the area of the pool is increasing when

the radius of the pool is 200 cm. (5)


2017-6 Paper(1) Q.2

36 Sand is poured onto horizontal ground at a rate of 50 cm3/s. The sand forms a right circular

cone with its base on the ground. The volume of the cone increases in such a way that the radius

of the base is always three times the height of the cone. Find the rate of change, in cm/s to 3

significant figures, of the radius of the cone when the radius is 10 cm. (5)

2018-1 Paper(1) Q.3

37 The volume of a right circular cone is increasing at a constant rate of 27 cm3/s. The radius of the

base of the cone is always 1.5 times the height of the cone.

(6)


30 Simultaneous Equations

1988-1 Paper(2) Q.9


32x − 3(3x)− 4 = 0.



y = log3 x.

(3)

Given that

y = log3 x and y = 12 [1 + log3 9x],


tions. (6)

1989-1 Paper(1) Q.3

2 Given that

x+ y = 4

and x2 − y = 8,

calculate the two possible values of x. (5)

1990-1 Paper(1) Q.5

3 Solve for x and y the simultaneous equations

x− 2y = 5

3x+ 2y2 = 11

(5)


1991-1 Paper(1) Q.5

4 Find the coordinates of the points at which the straight line with equation y = x + 1 intersects

the curve with equation 4x2 + y2 = 1. (5)

1992-1 Paper(1) Q.6

5 Solve the simultaneous equations

y = x+ 1

2x2 + y2 = 9.

(5)

2007-1 Paper(1) Q.3

6 Find the coordinates of the points of intersection of the curve with equation y = 3x2− 4x+ 2 and

the line with equation 7x+ y = 8. (5)

2009-6 Paper(2) Q.2

7 Find the coordinates of the points where the line with equation y = 2x− 5 meets the curve with

equation xy = 12. (5)

2010-1 Paper(1) Q.4


xy = 6.

xy + x+ y = 11.

(6)


2010-6 Paper(1) Q.3


x− 2y = 3

2y2 + 2xy + x2 = 1

(6)

2010-6 Paper(1) Q.8

10 (a) Solve the equation log5 625 = x (2)

(b) Solve the equation log3 (5y + 3) = 5 (2)

(c) (i) Factorise 5x lnx+ 3 lnx− 10x− 6

(ii) Hence find the exact solution of the equation

5x lnx+ 3 lnx− 10x− 6 = 0 (5)

(d) Given that p 6= q, solve the simultaneous equations

logp q + 3 logq p = 4

pq = 81

(5)

2011-1 Paper(1) Q.3

11 Find the coordinates of the points where the line with equation y = x+ 8 crosses the curve with

equation y = 2x2 + 3x− 4 (5)


2011-6 Paper(1) Q.1

12 Solve equations

y = x2 − 3x+ 2

y − x = 7

(5)

2012-1 Paper(2) Q.3

13 Find the coordinates of the points of intersection of the curve with equation y = 3 + 6x− x2 and

the line with equation y − x = 7 (5)

2012-6 Paper(2) Q.3


2x2 + xy − y2 = 36

x+ 2y = 1

(6)

2014-1 Paper(2) Q.3


x2 + xy − 3x = 2

5y + 6x = 22

(6)


2016-1 Paper(1) Q.10

16 Given that 2 logy x+ 2 logx y = 5

(a) show that logy x =1

2or logy x = 2 (5)

(b) Hence, or otherwise, solve the equations

xy = 27

2 logy x+ 2 logx y = 5

(6)

2016-1 Paper(2) Q.3


3y = 12− 4x

(x+ 1)2 + (y − 2)2 = 4

(7)

2017-6 Paper(1) Q.7

18 (a) Solve loga 1024 = 5 (1)

(b) Solve log3 (6c+ 9) = 4 (2)

(c) Solve 2(logb 25 + logb 125) = 5 (4)

(d) Solve the equations, giving the values of x and y to 3 significant figures,

3 log2 x+ 4 log3 y = 10

log2 x− 2 log3 y = 1

(6)


2017-6 Paper(2) Q.2


y = x2 − 6x+ 5

y + x = 11

(5)


31 Sine Rule

1988-1 Paper(1) Q.4

1 In 4ABC, AB = 8 cm, BC = 9 cm and angle ABC = 42◦.

Calculate

(a) the length of AC, in cm, to 3 significant figures, (3)

(b) the magnitude of ∠ACB, to the nearest one tenth of a degree. (2)

1988-6 Paper(1) Q.8

2 In 4ABC, AB = 2x cm, AC = x cm, BC = 14 cm and ∠BAC = 120◦.

(a) Calculate, to 3 significant figures, the value of x. (3)

Given also that ∠ABC = y◦,

(b) without evaluating y, show that 2 sin y◦ = sin (60− y)◦. (3)

1990-1 Paper(1) Q.1

3 In 4ABC, AB = 6 cm, ∠CAB = 105◦ and ∠ABC = 40◦.

Calculate

(a) the length, in cm to 3 significant figures, of BC, (2)

(b) the area, in cm2 to 2 significant figures, of 4ABC. (2)


1990-1 Paper(2) Q.12




cos 2x = 1− 2 sin2 x.

(3)

(c) Show that

∫2 sin2 x dx = x− 1


O x

y

P

Q

R

π

Fig. 2









1990-6 Paper(1) Q.8

5

A

B C

D4 cm

3 cm

2 cm

x◦x◦

Fig. 2

Figure 2 shows a quadrilateral ABCD in which AB = 4 cm, BC = 3 cm, CD = 2 cm and

∠ABC = 90◦. The diagonal AC bisects ∠BCD and ∠ACB = ∠ACD = x◦.

(a) State the values of cosx◦ and sinx◦. (1)

(b) Calculate, in cm2, the exact value of AD2. (2)

(c) Calculate, in cm2, the exact value of the area of the quadrilateral ABCD. (4)

1991-6 Paper(1) Q.1

6 In 4ABC, AB = 12 cm, BC = 15 cm and CA = 10 cm. Calculate ∠ABC, in degrees to 1

decimal place. (3)


1991-6 Paper(2) Q.11






(3)



(ii) sin

(x+

π

3

)+ sin

(x− π

3

)= 1. (7)

(c) Find

∫ √(1 + cos 2x) dx. (2)



1992-1 Paper(1) Q.7

8 In 4ABC, AB = 6 cm, BC = 8√

3 cm and ∠ABC is obtuse. The area of the triangle is 36 cm2.

(a) Show that ∠ABC = 120◦.

(b) Calculate, in cm to 3 significant figures, the length of the side AC.

(c) Calculate, in cm to 2 significant figures, the distance of B from the side AC.


1996-1 Paper(1) Q.7

9

A

C

B

x◦ 6 cm

y cm

Fig. 1

Figure 1 shows 4ABC in which AB = y cm, ∠BAC = x◦ and AC = 6 cm. Given that the area

of the triangle is 15 cm2,

(a) find, in terms of y, the value of sinx◦. (3)

Given that y2 cos2 x◦ = y2 − k2,

(b) find the value of the positive constant k. (3)

Given that x = 30,

(c) find the value of y. (1)

1997-6 Paper(1) Q.1

10 In 4ABC,AB = 4 cm, AC = 6 cm, ∠ACB = 40◦ and ∠ABC is obtuse.

Calculate, to the nearest 0.1◦, the size of ∠ABC. (4)

2007-1 Paper(2) Q.1

11 A triangle has sides of length 4.6 cm, 5.3 cm and 6.5 cm. Find, to the nearest degree, the size of

the largest angle of the triangle. (3)


2007-6 Paper(2) Q.3

12 In 4LMN,LM = 5.6 cm, LN = 8.2 cm and ∠MLN = 57◦. Find, to 3 significant figures,

(a) the length of MN , (3)

(b) the size of ∠LNM . (3)

2008-1 Paper(1) Q.1

13 Triangle LMN has LM = 5 cm, LN = 8.2 cm and MN = 6.4 cm. Calculate, in degrees to the

nearest 0.1◦, the size of ∠LMN . (3)

2009-6 Paper(1) Q.9

14

B C

D

A

P

12 cm

9 cm

8 cm

Figure 3


The point D is on BA produced and the bisector of ∠DAC meets BC produced at P .

(a) Find, to the nearest 0.1◦, the size of each of the three angles of 4ABC. (6)

(b) Find, to the nearest cm, the length of BP . (5)


2010-1 Paper(1) Q.1

15 In 4ABC,AB = 5.7 cm, BC = 8.4 cm and ∠ACB is 42◦.

Find, to the nearest 0.1◦, the two possible sizes of ∠BAC. (4)

2010-6 Paper(1) Q.4

16 The lengths of the sides of a triangle are 5 cm, 6 cm and 8 cm.

(a) Find, in degrees to one decimal place, the size of the smallest of the triangle. (4)

(b) Find, to the nearest cm2, the area of the triangle. (3)

2011-1 Paper(1) Q.2

17 A triangle ABC has AB = 4.6 cm, AC = 5.7 cm and ∠C = 52◦

Angle B is acute.

Calculate, to the nearest 0.1◦, the size of ∠B. (3)

2011-6 Paper(2) Q.3

18 In triangle ABC, AB = 5 cm, AC = 3 cm, angle B = 25◦ and angle C is obtuse.

(a) Find, to the nearest degree, the size of angle C. (3)

The point D lies on BC produced and AD = 3 cm.

(b) Find, to 3 significant figures, the length of CD. (3)


2012-6 Paper(1) Q.2

19 In triangle ABC, AB = 8 cm, BC = 5 cm and CA = 7 cm.

(a) Find, to the nearest 0.1◦, the size of angle BAC. (3)

(b) Find, to 3 significant figures, the area of triangle ABC. (2)

2013-1 Paper(1) Q.6

20

A CD

B

6 cm10 cm

6 cm

28◦

Figure 1

Figure 1 shows triangle ABC with AB = 10 cm, BC = 6 cm and ∠BAC = 28◦. The point D lies

on AC such that BD = 6 cm.

(a) Find, to the nearest 0.1◦, the size of ∠DBC. (4)

(b) Find, to 3 significant figures, the length of AD. (3)

(c) Find, to 3 significant figures, the area of the triangle ABC. (3)


2013-6 Paper(2) Q.1

21

A C

B

10 cm

16 cm

35◦

Figure 1

In triangle ABC, AB = 10 cm, AC = 16 cm and ∠BAC = 35◦, as shown in Figure 1.

(a) Find, to 3 significant figures, the area of the triangle ABC. (2)

(b) Find, in degrees to the nearest 0.1◦, the size of the angle ABC. (5)

2014-1 Paper(1) Q.6

22 In triangle ABC, AB = x cm, BC = 7 cm, AC = (5x− 6) cm and ∠BAC = 60◦

(a) Find, to 3 significant figures, the value of x. (5)

Using your value of x

(b) find, in degrees to 1 decimal place, the size of ∠ACB. (3)


2014-6 Paper(2) Q.11

23 In triangle ABC, ∠BAC = 60◦, AB = (3x− 1) cm, AC = (3x+ 1) cm and BC = 2√

7x cm.

(a) Show that (9x− 1)(x− 3) = 0 (3)

(b) Hence find the value of x, justifying your answer. (2)

(c) Find, to the nearest 0.1◦, the size of angle ABC. (3)

(d) Find the exact value, in cm2, of the area of triangle ABC. (2)

2015-1 Paper(2) Q.1

24

A

B

C

x cm

2x cm

100◦

Figure 1

In triangle ABC, AB = x cm, AC = 2x cm and ∠ABC = 100◦, as shown in Figure 1.

(a) Find, in degrees to the nearest 0.1◦, the size of ∠BAC. (4)

Given that the area of triangle ABC is 16 cm2,

(b) find, to 3 significant figures, the value of x. (3)


2015-6 Paper(1) Q.6

25

A C

B

22 cm 20 cm

14 cm

Figure 1


(a) Find, to 3 decimal places, the size of each of the three angles of 4ABC. (5)

The bisector of angle BAC meets BC at P .

(b) Find, in cm to 3 significant figures, the length of AP . (3)

(c) Find, to the nearest cm2, the area of 4ABC. (2)


2016-1 Paper(1) Q.7

26

B C

A

D4 cm

5 cm

4 cm

30◦

Figure 1

Figure 1 shows the triangle ABC with AB = 4 cm, BC = 5 cm and angle BCA = 30◦

The point D lies on AC such that BD = 4 cm and angle BDC is obtuse.

Find

(a) the size of angle BDC, giving your answer in degrees correct to 1 decimal place, (3)

(b) the length, in cm, of AD, giving your answer correct to 3 significant figures, (3)

(c) the area, in cm2, of triangle ABD, giving your answer correct to 3 significant figures. (2)

2016-6 Paper(2) Q.1

27 A triangle has sides of length 10 cm, 8 cm and 9 cm.

(a) Calculate, in degrees to the nearest 0.1◦, the size of the largest angle of this triangle. (3)

(b) Find, to 3 significant figures, the area of this triangle. (2)


2017-1 Paper(1) Q.5

28

B

A

C

D

8 cm

12 cm

120◦

35◦

Figure 2

Figure 2 shows the quadrilateral ABCD in which AB = BC.

DC = 8 cm AC = 12 cm ∠ABC = 120◦ ∠CAD = 35◦

Find

(a) the exact length, in cm, of AB. (2)

Given that angle ADC is obtuse, find

(b) the size, in degrees to 1 decimal place, of angle ADC, (3)

(c) the area, in cm2 to 3 significant figures, of the quadrilateral ABCD. (6)


2017-6 Paper(1) Q.5

29 In triangle ABC, AB = 10 cm, BC = 7 cm and angle BAC = 40◦

(a) Find, in degrees to the nearest 0.1◦, the two possible sizes of angle ACB. (4)

(b) Find, in cm to 3 significant figures, the difference between the two possible lengths of AC.

(4)

2017-6 Paper(2) Q.5

30 In triangle ABC, AB = x cm, BC = (4x− 5) cm, AC = (2x+ 3) cm and angle ABC = 60◦.

Find, to 3 significant figures,

(a) the value of x, (5)

(b) the area of triangle ABC. (3)


2018-1 Paper(1) Q.6

31

B

A

C

θ◦

x cm (x+ 4) cm

(2x− 2) cm

Figure 1

Figure 1 shows the triangle ABC with AB = x cm, BC = (2x − 2) cm, AC = (x + 4) cm and

∠BAC = θ◦

Given that tan θ◦ =√

255 and without finding the value of θ,

(a) show that cos θ◦ =1

16(2)

Hence find


(c) the size, in degrees to 1 decimal place, of ∠ABC, (2)

(d) the area, in cm2 to 3 significant figures, of triangle ABC. (2)


32 Solve by Graphing

1988-1 Paper(2) Q.5

1 Using the same scale and axes, sketch the graphs of y = x− 1 and y = x3.

Hence find the number of solutions of the equation

x3 = x− 1.

(5)

1988-6 Paper(2) Q.4

2 Using the same scales and axes, sketch the graphs of y = e−x and y = x3. Hence find the number

of roots of the equation

e−x = x3.

(4)


1991-1 Paper(1) Q.14

3 (a) Copy and complete the following table for which

f(x) = cos2 x◦ − sinx◦, 0 6 x 6 180,

giving your answers to 2 decimal places where appropriate.

x 0 30 60 90 120 150 180

f(x) 1 0.25

(2)

(b) On graph paper and using a scale of 2 cm to represent 30◦ on the x-axis and 4 cm to represent

1 unit on the y-axis, draw the graph of y = f(x). (3)

(c) From your graph estimate the solutions of the equation

cos2 x◦ − sinx◦ = 0, 0 6 x 6 180.

(3)

(d) Solve the equation

cos2 x◦ − sinx◦ = 0, 0 6 x 6 180,

giving your answers to the nearest degree. (4)

(e) From your graph find the solutions of the equation

2 cos2 x◦ − 2 sinx◦ + 1 = 0, 0 6 x 6 180.

(3)


1991-1 Paper(2) Q.14

4 (a) Copy and complete the following table for y = ln (1 + 3x) and y = 14(x − 2)2, giving your

values to 2 decimal places where appropriate.

x 0 1 2 3 4 5 5.5 6

y = ln (1 + 3x) 0 1.95 2.56 2.86

y = 14(x− 2)2 0.25 1 3.06 4

(3)

(b) Using scales of 2 cm to represent 1 unit on the x-axis and 4 cm to represent 1 unit on the

y-axis draw, with the same axes, the graphs of

y = ln (1 + 3x) and y = 14(x− 2)2 for 0 6 x 6 6.

(3)

(c) State the number of roots of the equation

4 ln (1 + 3x) = (x− 2)2 in the range 0 6 x 6 6

and use your graph to estimate these roots to 2 significant figures. (6)

(d) By drawing a suitable straight line on the graph, estimate, to 2 decimal places, the root of

the equation

2 ln (1 + 3x) = 4− x.

(3)


1991-6 Paper(2) Q.13

5 (a) Copy and complete the table which shows the values of y, to 2 decimal places, for y = ex− 3

and y = log2 x, for values of x in the interval 0.1 6 x 6 1.6.

x 0.1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

y = ex − 3 −1.89 −1.78 −1.18 0.32 1.95

y = log2 x −3.32 −2.32 −1.32 −0.32 0.26 0.49

(4)

(b) On graph paper draw, in the interval 0.1 6 x 6 1.6, the graphs of the curves with equations

y = ex − 3 and y = log2 x, using a scale of 2 cm to represent 0.2 units on the x-axis and 2

cm to represent 0.5 units on the y-axis. (3)

By drawing appropriate lines on your graph estimate the solutions,

in the interval 0.1 6 x 6 1.6, of

(c) log2 x = −0.5, (1)

(d) ex = 4, (2)

(e) 2x = 0.9, (2)

(f) ex − log2 x = 3. (3)


1994-1 Paper(2) Q.14

6 (a) Copy and complete the following table for

y = log3 x and y =3

x+ 1, giving your values to 2 decimal places where appropriate.

x 1 1.5 2 3 4 5 6 7 8

y = log3 x 0 0.37 0.63 1.46 1.77 1.89

y =3

x+ 11.2 1 0.75 0.6 0.38 0.33

(3)

(b) Using scales of 2 cm to represent one unit on the x-axis and 10 cm to represent one unit on

the y-axis draw, with the same axes, the graphs of

y = log3 x and y =3

x+ 1

for 1 ≤ x ≤ 8. (4)

(c) Use your graph to estimate to 2 decimal places the solution of

(x+ 1) log3 x = 3.

(3)

(d) By drawing a suitable straight line on the graph, estimate to 2 decimal places the solutions

of the equation

x4 = 3x.

(5)


1995-1 Paper(2) Q.11

7 y = cos 2θ, 0.5 6 θ 6 0.6, where θ is measured in radians.

(a) Copy and complete the following tables of values

θ 0.5 0.52 0.54 0.56 0.58 0.6

y 0.540 0.436

(4)

(b) Plot the graph of y = cos 2θ for 0.5 6 θ 6 0.6. Use a scale of 2 cm to represent 0.1 units on

the y-axis and 10 cm to represent 0.1 radians on the θ-axis. (3)

(c) By drawing a suitable line on your graph estimate the equation cos 2θ = θ. (3)

(d) Using the identity cos 2θ = 2 cos2 θ − 1, and plotting another suitable straight line on your

graph, estimate, to 3 decimals places, a value of θ which satisfies the equation 2 cos2 θ =

θ + 0.9. (5)


1995-6 Paper(2) Q.9

8 y = 2x+1

(a) Copy and complete the table below, giving values to 2 decimals places where appropriate.

x −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

y 1 2 8

(2)

(b) On graph paper with a scale of 2 cm to represent 1 unit on the y-axis and 4 cm to represent

1 unit on the x-axis, draw the graph of y = 2x+1. (3)

(c) By drawing suitable lines on your graph obtain estimates, to 2 decimal places, for the value

of

(i) 2−0.4, (ii) log2 6. (4)

(d) By drawing suitable lines on your graph obtain estimates, to 2 decimal places, for the value

of

(i) 2x = 2− x, (ii) x = log2 (3− 2x)− 1. (4)


1996-1 Paper(2) Q.14

9 (a) Using the same scales and the same axes, sketch, for −π 6 x 6 2π, the graphs of y = 2−x

and y = sin 2x, where x is in radians, stating the coordinates of the points where each curve

cuts the axes. (3)

(b) Deduce, from your sketch, the number of solutions, in the range −π 6 x 6 2π, of the equation

2−x = sin 2x. (1)

The following is a table for y = 2−x, where p and q are constants.

x 0 0.2 0.4 0.6 0.8 1.0 1.3 1.6

y 1 0.87 0.76 p 0.57 0.5 0.41 q

(c) Find, to 2 decimal places, the values of p and q needed to complete the table. (2)

(d) Using all the values in the completed table and scales of 10 cm to represent one unit on

the x-axis and 20 cm to represent one unit on the y-axis, draw the graph of y = 2−x for

0 6 x 6 1.6. (2)

The following is a table for y = sin 2x, where x is in radians and r and s are constants.

x 0 0.2 0.4 0.5 0.6 0.8 1.0 1.2 1.3 1.4 1.5

y 0 0.39 0.72 0.84 r 1.00 0.91 0.68 0.52 s 0.14

(e) Find, to 2 decimal places, the values of r and s needed to complete the table. (2)

(f) On the same axes used in (d) and using all the values in the completed table, draw a graph

of the curve with equation y = sin 2x for 0 6 x 6 1.5. (2)

(g) Use your graph to estimate to 2 decimal places, the solutions, for 0 6 x 6 1.5 of the equation

2−x = sin 2x.

(3)


1996-6 Paper(2) Q.14

10 f(x) = 6 ln (1 + 5x) and g(x) = ex2 + 2.

(a) Copy and complete the given table, giving values to 1 decimal place where appropriate

x 0 1 2 3 4 5 6

f(x) 0 10.8 16.6 18.3 20.6

g(x) 3 4.7 6.5 9.4 14.2

(4)

(b) On graph paper, using 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit

on the y-axis, draw the graphs of y = f(x) and y = g(x) for 0 6 x 6 6. By drawing suitable

straight lines on your graph obtain estimates, to 2 decimal places, for the roots, in the interval

0 6 x 6 6, of the equations (3)

(c) 9 ln (1 + 5x) = 30− 5x, (4)

(d) 3ex2 = 30− x. (4)


1997-6 Paper(2) Q.13

11 The following is a table for y = 2x.

x 0 0.5 1 1.5 2 2.5 2.75 3

y 1 1.41 2 p 4 q 6.73 8

(a) Find, to 2 decimal places, the values of p and q. (2)

(b) Using all values in the completed table and scales of 4 cm to represent one unit on the x-axis

and 2 cm to represent one unit on the y-axis, draw a graph of the curve with equation y = 2x

for 0 6 x 6 3. (2)

(c) Using the same axes and by drawing suitable straight lines estimate, to 2 decimal places, the

solutions of the equations

(i) 2x+1 = 7,

(ii) x = log2 (5− x). (4)

The following is a table for y = 6 sinx, where x is in radians.

x 0 0.25 0.5 1 1.5 2 2.5 2.75 3

y 0 1.48 r 5.05 5.98 5.46 3.59 s 0.85

(d) Find, to 2 decimal places, the values or r and s. (2)

(e) With the same axes as those used in (b) and using all values in the completed table, draw a

graph of y = 6 sinx for 0 6 x 6 3. (2)

(f) Use your graphs to estimate, to 2 decimal places, the solutions of the equation 6 sinx = 2x

for 0 6 x 6 3. (3)


2007-1 Paper(2) Q.8

12 (a) Complete the table below of values for y = e− 1

2x

+ 1, giving your values of y to 2 decimal

places.

x −1 0 1 2 3 4 5

y 2 1.61 1.22 1.14

(2)

(b) Using a scale of 2 cm to 1 unit on the x-axis and 4 cm to 1 unit on the y-axis, draw the graph

of y = e− 1

2x

+ 1 for −1 6 x 6 5. (2)

(c) Use your graph to estimate, to 2 significant figures, the solution of the equation

e− 1

2x

= 0.8

showing your method clearly. (2)

(d) By drawing a straight line on your graph, estimate, to 2 significant figures, the solution of

the equation x = −2 ln (2x− 7). (4)


2007-6 Paper(1) Q.5

13 (a) Complete the table for y = e12x− 1

2x, giving your values of y to 3 significant figures, where

appropriate.

x 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

y 1.03 1.37 1.72 2.98 5.39

(2)

(b) Using a scale of 4 cm to 1 unit on both axes, draw the graph of y = e12x− 1

2x for 0 6 x 6 4.

(2)

(c) Use your graph to solve, to 2 significant figures,

(i) 2e12x

= x+ 3,

(ii) x = 2 ln (2 + 12x). (5)


2008-1 Paper(2) Q.6

14 (a) Complete the table for y = 2x − 1

x2, giving the values of y to 3 significant figures where

appropriate.

x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

y −3 2.56 4.84 7.94

(2)

(b) Using a scale of 4 cm to 1 unit on the x-axis and 2 cm to 1 unit on the y-axis, draw the graph

of y = 2x− 1

x2for 0.5 6 x 6 4.0. (2)

(c) Express 2x− 1

x2as a single fraction, and hence your graph to estimate, to 2 significant figures,

the value of 3√

0.5. (3)

(d) By drawing a suitable straight line on your graph, find an estimate, to 2 significant figures,

of the root of the equation 3x− 6− 1

x2= 0, in the interval 0.5 6 x 6 4.0. (3)


2009-6 Paper(2) Q.3

15 The diagram on shows a sketch of the line with equation y = 16− 4x.

The line crosses the x-axis at the point A and the y-axis at the point B.

(a) Write down the coordinates of

(i) the point A,

(ii) the point B. (2)

On the diagram

(b) sketch the line with equation x = 3 and the line with equation y = 3x+ 8, (2)

(c) show, by shading, the region for which y > 16− 4x, x 6 3 and y 6 3x+ 8.

O

B

A x

y

y = 16− 4x

(1)


2009-6 Paper(2) Q.5

16 (a) Complete the table for y = 2x− 3 +1

x2, giving your values of y to 2 decimal places.

x 0.3 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

y 2 0 1.25 3.11 5.06

(2)

The grid on the facing page shows the graph of y = 2 +1

x, 0.3 6 x 6 4.0.

(b) On the same grid, draw the graph of y = 2x− 3 +1

x2for 0.3 6 x 6 4.0. (2)

(c) Use algebra to show that the x-coordinates of the points of intersection of the curve with

equation y = 2x−3+1

x2and the curve with equation y = 2+

1

xare the roots of the equation

2x3 − 5x2 − x+ 1 = 0. (2)

(d) Hence use your graph to obtain estimates, in the interval 0.3 6 x 6 4.0, to one decimal place,

of the roots of the equation 2x3 − 5x2 − x+ 1 = 0. (2)


y

x

0 0.5 1 1.5 2 2.5 3 3.5 4

1

2

3

4

5

6

7

8

9

10


2010-6 Paper(1) Q.1

17 The grid opposite shows the graph of y = 3x− 4 +6

x2for 0.74 6 x 6 4

The line with equation y = 5x− 4 intersects the curve with equation y = 3x− 4 +6

x2at the point

P .

(a) Using algebra, show that the x-coordinate of P satisfies x2 = 3 (3)

(b) By drawing a suitable straight line on the grid, obtain an estimate, to 1 decimal place, for

the value of 3√

3 (2)


y

x0 1 2 3 4

1

2

3

4

5

6

7

8

9

10


2012-1 Paper(1) Q.6

18 For x radians,

y = 3 cosx

2

(a) Complete the table, giving the three missing values correct to 2 decimal places.

x 0 0.5 1 1.5 2 2.5 3 3.5

y 3 2.63 2.20 0.95 0.21

(2)

On the axes opposite,

(b) draw the graph of y = 3 cosx

2for 0 6 x 6 3.5 (2)

(c) Using your graph, find an estimate, to 1 decimal place, for the root of the equation

2x = 1 + 2 cosx

2

(4)


y

xO 1 2 3 4 5

−2

−1

1

2

3

4

5


2013-6 Paper(2) Q.7

19 (a) Complete the table of values for y = 5 log10 (x+ 2) − x, giving your answers to 2 decimal

places.

x −1 0 1 2 3 4 5

y 1 1.51 1.39 −0.77

(2)

(b) On the grid opposite, draw the graph y = 5 log10 (x+ 2)− x for −1 6 x 6 5 (2)

(c) Use your graph to obtain an estimate, to 1 decimal place, of the root of the equation

10 log10 (x+ 2)− 2x = 11

2in the interval −1 6 x 6 5 (2)

(d) By drawing an appropriate straight line on your graph, obtain an estimate, to 1 decimal

place, of the root of the equation x = 1012x− 2 in the interval −1 6 x 6 5 (4)

y

xO−1 1 2 3 4 5

−1

−0.5

0.5

1

1.5

2


2014-1 Paper(1) Q.7

20 (a) Complete the table of values for y = 2x − 4 +5

x2, giving your answers to 2 decimal places

where appropriate.

x 0.8 1 1.5 1.7 2 2.5 3 4

y 5.41 1.22 1.8 4.31

(2)

(b) On the grid opposite, draw the graph y = 2x− 4 +5

x2for 0.8 6 x 6 4 (2)

(c) Use your graph to obtain estimates, to 1 decimal place, of the roots of the equation 2x+5

x2= 6

in the interval 0.8 6 x 6 4 (2)

(d) By drawing a straight line on your graph obtain an estimate, to 1 decimal place, of the root

of the equation 4x+5

x2= 12 in the interval 0.8 6 x 6 4. (4)


y

xO1 2 3 4

1

2

3

4

5


2015-1 Paper(2) Q.5

21 The grid opposite shows the graph of y = 3x sinx for −1 6 x 6 3, where x is measured in radians.

(a) Use the graph to estimate, to 1 decimal place, the roots of the equation

x sinx = 1 in the interval −1 6 x 6 3

(3)

(b) By drawing a suitable straight line on the grid, obtain estimates, to 1 decimal place, of the

roots of the equation

2x sinx− x = 1 in the interval −1 6 x 6 3

y

xO−1 1 2 3

1

2

3

4

5

6

(5)


2015-6 Paper(2) Q.2

22 (a) Complete the table of values for y = x+6

x2

Give your answers to 2 decimal places where necessary.

x 1.0 1.25 1.5 1.75 2.0 2.25 2.5 2.75 3.0

y 4.17 3.71 3.44 3.54 3.67

(2)

(b) On the grid opposite, draw the graph of y = x+6

x2for 1 6 x 6 3 (2)

(c) By drawing a suitable straight line on the grid, obtain estimates, to 1 decimal place, for the

solutions of the equation x3 − 3x2 + 3 = 0 in the interval 1 6 x 6 3 (4)


y

xO 1 2 3 4

1

2

3

4

5

6

7

8


2016-1 Paper(2) Q.11

23 (a) Complete the table of values for y = e(x−1) + 2

Give your answers to 2 decimal places where appropriate.

x −2 −1 0 1 2 3

y 2.05 4.72 9.39

(2)

(b) On the grid opposite, draw the graph of y = e(x−1) + 2 for −2 6 x 6 3 (2)

(c) Using your graph to obtain an estimate, to 1 decimal place, of the root of the equation

4 = e(x−1) in the interval −2 6 x 6 3 (2)

(d) By drawing a straight line on the grid, obtain an estimate, to 1 decimal place, of the root of

the equation ln (4x− 4) = x− 1 in the interval −2 6 x 6 3 (5)

y

xO−3 −2 −1 1 2 3 4

−4

−3

−2

−1

1

2

3

4

5

6

7

8

9

10


2016-6 Paper(1) Q.7

24 (a) Complete the table of values for y = 2x − 4, giving your answers to 2 decimal places.

x 0 0.5 1 1.5 2 2.5 2.75 3

y −3 −2 0 2.73 4

(2)

(b) On the grid opposite, draw the graph y = 2x − 4 for 0 6 x 6 3 (2)

(c) Use your graph to obtain an estimate, to one decimal place, of the value of log2 7

Show clearly how you used the graph. (3)

(d) By drawing a straight line on your graph, obtain an estimate to one decimal place of the root

of the equation 2x + 3x = 7 in the interval 0 6 x 6 3 (4)


y

xO0.5 1 1.5 2 2.5 3

−4

−3

−2

−1

1

2

3

4

5


2017-1 Paper(1) Q.7

25 (a) Complete the table of values for y = ln (5x+ 1) + 2 giving your answers to 2 decimal places.

x 0 1 2 3 4 5 6 7

y 2 4.40 4.77 5.04 5.43

(2)

(b) On the grid opposite draw the graph of y = ln (5x+ 1) + 2 for 0 6 x 6 7 (2)

(c) By drawing an appropriate straight line on the grid, obtain an estimate, to 1 decimal place,

of the positive root of the equation ln (5x+ 1)− x = 0 in the interval 0 6 x 6 7 (3)

(d) By drawing an appropriate straight line on the grid, obtain an estimate, to 1 decimal place,

of the root of the equation e(3x−1) = 5x+ 1 in the interval 0 6 x 6 7 (4)

y

xO 1 2 3 4 5 6 7

1

2

3

4

5

6


2018-1 Paper(1) Q.5

26 (a) Complete the table of values for y =x3 + 2

x+ 1giving your answers to 2 decimal places where

appropriate.

x 0 0.5 1 1.5 2 3 4

y 1.42 2.15 7.25

(2)

(b) On the grid opposite draw the graph of y =x3 + 2

x+ 1for 0 6 x 6 4 (2)

(c) By drawing a suitable straight line on your graph obtain an estimate, to 1 decimal place, of

the root of the equation x3 + x2 − 3x− 2 = 0 in the interval 0 6 x 6 4 (5)


33 Trigonometric Equations

1988-1 Paper(2) Q.7

1 By using the identity

tan (A+B) =tanA+ tanB

1− tanA tanB,

find the solutions, in the range 0 6 x < 180, of the equation

3 tanx◦ = tan 2x◦.

(6)

1989-6 Paper(2) Q.8

2 By using the identity tan 2x◦ =2 tanx◦

1− tan2 x◦, or otherwise, find the elements of the set

{x : 0 6 x 6 180}

which satisfy the equation

tan 2x◦ = 3 tanx◦.

(6)


1989-6 Paper(2) Q.12

3 Using the basic formulae for the expansion of sin (A+B) and cos (A+B), prove that

(a) sin 2x = 2 sinx cosx, (2)

(b) cos 2x = 1− 2 sin2 x, (3)

(c) sin 3x = 3 sinx− 4 sin3 x. (5)

(d) Use the result in (c) to evaluate ∫0

π2 sin3 x dx.

(5)

1990-1 Paper(1) Q.4

4 Given that 0 < θ < 180, find θ when

(a) tan θ◦ = −1, (2)

(b) sin 2θ = −12 . (3)


1990-6 Paper(1) Q.12

5 Solve, for 0 6 x < 360, giving your answers to 1 decimal place, where appropriate, the equations

(a) tan 2x◦ = −1, (3)

(b) 2 cos2 x◦ = sinx◦ cosx◦, (4)

(c) 3 sinx◦ + 3 = cos2 x◦. (3)

A cuboid has a square horizontal base of side 2 cm and a height of 3 cm. Calculate, in degrees to

3 significant figures, the angle that a diagonal of the cuboid makes with

(d) the base (2)

(e) a vertical face of the cuboid. (3)

1991-1 Paper(2) Q.9


tanx =1 + 2 tanα

2 + tanα.

(4)


Hence show that

∫0

π12 sin2 3x dx =

1

24(π − 2). (5)


(c) 3 sin 2x = cosx, (3)



1991-6 Paper(2) Q.11






(3)



(ii) sin

(x+

π

3

)+ sin

(x− π

3

)= 1. (7)

(c) Find

∫ √(1 + cos 2x) dx. (2)



1992-1 Paper(1) Q.10

8 Solve, for 0 6 θ < 360, the equations

(a) tan (θ◦ − 30◦) = 1,

(b) (2 cos θ◦ − 1)(sin 2θ◦ + 1) = 0,

(c) cos2 θ◦ − 2 sin2 θ◦ = 1.

(d) Prove that cos4 θ + 2 sin2 θ − sin4 θ = 1.

(e) Given that sin θ = x − 1 and cos θ = 2y for all values of θ, find an equation, not containing

θ, relating x and y.


1994-1 Paper(1) Q.6

9 Given the 0 ≤ x ≤ 180, find the values of x such that

(a) 2 cosx◦ = −1 (2)

(b) sin 2x◦ = 0.7, giving your answers to the nearest whole number. (3)

1994-1 Paper(2) Q.13

10 Using the formula


(a) show that


. (3)

(b) Hence show that


(1)




(d) Evaluate ∫0

π4 sin 3x sinx dx.

(4)


1995-6 Paper(2) Q.10

11 (a) Using the formula sin(A+B) = sinA cosB + cosA sinB show that

sin (A+B) + sin (A−B) = 2 sinA cosB.

(1)

Hence show that

(b) sinP + sinQ = 2 sin(P +Q)

2cos

(P −Q)

2, (2)

(c) sin 3x+ sin 2x+ sinx = sin 2x(2 cosx+ 1). (2)

(d) Sketch, on separate axes, for the range 0 6 x 6 4π, the curves with equations

(i) y = cosx, (ii) y = sin 2x. (4)

(e) Solve, for 0 6 x 6 2π, the equation

sin 3x+ sin 2x+ sinx = 2 cosx+ 1,

giving your answers in terms of π. (6)


1996-1 Paper(2) Q.11



to show that

cos2 θ = 12(1 + cos 2θ).

(3)


f(θ) = 3 sin2 θ + 4 cos2 θ.



∫0

π4 f(θ) dθ.


1996-6 Paper(2) Q.5

13 Using the substitution y = sinx◦, solve, to the nearest integer, in the interval 0 6 x < 360,

cos2 x◦ + 3 sinx◦ = 2

(5)


2007-1 Paper(1) Q.8


f(θ) = 5 cos θ − 12 sin θ.

Given that f(θ) = p cos (θ + β), p > 0, 0 < α < π2 ,

(a) (i) Show that p = 13,

(ii) find in radians to 3 significant figures, the value of α. (5)

(b) Hence solve, to 2 significant figures, for 0 6 θ < 2π, 5 cos θ − 12 sin θ = 9. (4)

(c) Evaluate∫0

π3 f(θ) dθ, giving your answer in the form c + d

√3, where c and d are rational

numbers. (5)

2008-6 Paper(2) Q.8

15 Solve, to 3 significant figures, for 0 6 θ 6 π,

(a) (4 sin θ − 1)(2 sin θ + 5) = 0, (3)

(b) tan (2θ − 1

3π) = 2.4, (4)

(c) 9 sin2 θ − 9 cos θ = 11. (5)


2011-1 Paper(1) Q.8


(a) Show that

(i) sin2 θ = 12(1− cos 2θ)

(ii) cos2 θ = 12(cos 2θ + 1) (3)

f(θ) = 8 sin4 θ + 4 sin2 θ − 5




Given that 4

∫ π4

π8

f(θ) dθ = m+ n√

2


2011-1 Paper(2) Q.7


tanA =sinA

cosA

(a) Show that the equation

2 sin (x+ α) = 5 sin (x− α)

can be written in the form

3 tanx = 7 tanα

(5)

(b) Hence solve, to one decimal place,

2 sin (2y + 50◦) = 5 sin (2y − 50◦) for 0 6 y 6 180◦

(5)


2011-6 Paper(1) Q.4

18 (a) sin (A+B) = sinA cosB + cosA sinB


Write down an expression for sin 2A in terms of sinA and cosA (1)

(b) Find an expression for cos 2A in terms of sinA (2)

(c) Show that sin 3A+ sinA = 4 sinA− 4 sin3A (4)

2012-1 Paper(2) Q.8



tanA =sinA

cosA

(a) Show that tan (A+B) =tanA+ tanB

1− tanA tanB(3)

(b) Hence write down an expression for tan 2θ in terms of tan θ (1)

(c) Show that tan 3θ =3 tan θ − tan3 θ

1− 3 tan2 θ(4)

Given that tan 3θ = −1 and tan θ 6= ±√

3

3

(d) without finding the value of θ, show that tan3 θ + 3 tan2 θ − 3 tan θ − 1 = 0 (1)

Given also that tan θ 6= 1

(e) find the exact values of tan θ, giving your answers in the form a ±√b where a and b are

integers. (4)


2013-1 Paper(1) Q.8

20 Solve, for 0 6 θ 6 π, giving each solution to 3 significant figures,

(a) 5 sin θ − 1 = 0 (3)

(b) tan

(2θ +

π

3

)= 0.4 (4)

(c) 4 sin2 θ − 7 cos θ = 2 (4)

2014-1 Paper(1) Q.8


tanA =sinA

cosA


4 sin (x+ α) = 7 sin (x− α)


3 tanx = 11 tanα

(5)

(b) Hence solve, to 1 decimal place,

4 sin (3y + 45)◦ = 7 sin (3y − 45)◦ for 0 6 y 6 180

(6)


2014-6 Paper(1) Q.2

22 Solve, in degrees to 1 decimal place, for 0 6 θ < 180◦

(a) tan 2θ = 1.5 (3)

(b) (3 cos θ + 1)(2 cos θ + 3) = −2 (4)

2014-6 Paper(2) Q.10








(d) Find

∫(24 sin3 θ + 6 cos θ) dθ (2)

(e) Hence evaluate

∫ π3


√c, where a,



2015-1 Paper(1) Q.8

24 tan θ =sin θ

cos θ

(a) Using the above identity, show that 1 + tan2 θ =1

cos2 θ(3)

(b) Show that1 + sin θ cos θ + sin2 θ

cos2 θ= 1 + tan θ + 2 tan2 θ (3)

(c) Solve the equation 1 + sin θ cos θ + sin2 θ = 4 cos2 θ for 0 6 θ 6 180◦.

Give your answer to 1 decimal place, where appropriate. (6)

2015-1 Paper(2) Q.4



(a) Write down the exact value of sin 45◦ (1)

Given that sin θ =

√5

2√

2and cos θ =

√3

2√

2

(b) show that sin (45◦ + θ) =

√3 +√

5

4(2)

(c) Find the exact value of cos (45◦ + θ) (2)

(d) Show that sin (45◦ + θ)× cos (45◦ + θ) = −1

8(2)


2015-6 Paper(1) Q.8








(4)

(d) (i) Find

∫sin3 θ dθ

(ii) Evaluate

∫ π4


a− b√

2


integers. (5)

2016-1 Paper(1) Q.6

27 Giving your solutions to 3 decimal places, solve the equation

(a) cosx = 0.4 −π < x < π (2)

(b) tan

(2θ +

π

4

)= 1.5 0 < θ < π (4)


2016-6 Paper(1) Q.5

28 Using the identities


tanA =sinA

cosA

(a) show that the equation

3 sin (x+ α) = 5 sin (x− α)

can be written in the form tanx = 4 tanα (5)

(b) Hence solve, to the nearest integer, the equation

3 sin (2y + 30)◦ = 5 sin (2y − 30)◦ for 90 6 y < 180

(4)


2016-6 Paper(2) Q.9








6 cos θ − 8 cos3 θ + 1 = 0

(4)

(e) find

(i)



∫ π3

0(8 cos3 θ + 4 sin θ) dθ (4)

2017-1 Paper(2) Q.2

30 (a) Show that the equation 6 cos2 α− sinα = 5 can be written as

6 sin2 α+ sinα− 1 = 0

(2)

(b) Solve, to 1 decimal place where appropriate, for 0 6 θ 6 90

6 cos2 (2θ + 40)◦ − sin (2θ + 40)◦ = 5

(5)


2017-1 Paper(2) Q.4

31 tan (A+B) =tanA+ tanB

1− tanA tanB

(a) (i) Write down an expression for tan (2x) in terms of tanx

(ii) Hence show that tan (3x) =3 tanx− tan3 x

1− 3 tan2 x(6)

Given that a is the acute angle such that cosα =1

3

(b) find the exact value of tanα (2)

(c) Hence use the identity in part (a) to find the exact value of tan (3α)

Give your answer in the forma√

2

bwhere a and b are integers. (2)

2017-6 Paper(1) Q.4

32 Solve, for 0 6 θ 6 π, to 4 significant figures,

(a) (tan θ − 3)(tan θ + 2) = 0 (3)

(b) 6 cos2 θ − sin θ = 5 (4)


2017-6 Paper(1) Q.9

33 Using



2(cos 2θ + 1) (2)

f(θ) = 8 cos4 θ + 4 cos2 θ − 5


Hence


8 cos4 x+ 4 cos2 x− 6 cos 2x = 4.5

(4)

(d) find

(i)∫

f(θ) dθ


30 f(θ) dθ (5)


2018-1 Paper(1) Q.10



2(cos 2θ + 1) (3)




2, the equation

16 cos4(θ − π

6

)+ 16 sin2

(θ − π

6

)− 15 = 0

(4)


0(8 cos4 θ + 8 sin2 θ + 2 sin 2θ) dθ

(4)


34 Trigonometric Function

1988-1 Paper(2) Q.13


π

4.

(a) Find f ′(x). (3)



6 x 6π



x. (3)


π


1

x.


−π46 x 6

π

4. (8)

1988-6 Paper(1) Q.2

2 Given that 0 6 x 6 180, find the values of x such that

(a) 2 cosx◦ = −1, (2)

(b) tan 2x◦ = 1. (3)


1993-6 Paper(1) Q.7

3 The height above sea-level, x meters, of high tide at a sea port in January 1993 was given by

x = 10 + 3 cos 12t◦,

where t is the day of the month. For example, January 16th corresponds to t = 16.

(a) Find the value of the x on January 20th. (2)

(b) Find the values of t for which x = 1112 . (4)

1993-6 Paper(2) Q.8

4 (a) Using the formula for sin (A−B) find the values of R and θ for which R sin (3x− θ) =

2 sin 3x− 2√

3 cos 3x. (4)

Deduce

(b) the greatest value of y, where y = 2 sin 3x− 2√

3 cos 3x, (1)

(c) the values of x, in terms of π, in the interval −π < x 6 π, at which y has this greatest value.

(4)

1996-6 Paper(2) Q.3

5 Using the formula for the expansion of sin (A−B)

(a) show that 2 sin

(x− π

3

)= sinx−

√3 cosx. (2)

Hence write down

(b) the greatest value of sinx−√

3 cosx, (1)

(c) the smallest positive value of x, in the form kπ, where k is a constant, for which this greatest

value occurs. (1)


1996-6 Paper(2) Q.11

6 f(x) = cos 4x+ 12

(a) Solve, for 0 6 x 6 π4 , the equation f(x) = 0. (3)

(b) Sketch, for 0 6 x 6 π4 , the curve with equation y = f(x). (2)

(c) Shade on your sketch the region for which 0 6 x 6 π4 and 0 6 y 6 f(x) (1)

(d) Calculate, to 3 significant figures, the area of the shaded region. (3)

(e) Show that [f(x)]2 = 12 cos 8x+ cos 4x+ 3

4 . (3)

(f) Calculate, to 3 significant figures, the volume generated when the shaded region is rotated

through 2π radians about the x-axis. (3)

2012-6 Paper(1) Q.7


(a) Express cos (2x+ 45◦) in the form M cos 2x+N sin 2x, where M and N are constants, giving

exact value of M and the exact value of N . (2)

(b) Solve, for 0◦ 6 x 6 180◦, the equation cos 2x− sin 2x = 1 (5)

The maximum value of cos 2x− sin 2x is k.

(c) Find the exact value of k. (2)

(d) Find the smallest positive value of x for which a maximum occurs. (3)


35 Trigonometric Identities

1988-1 Paper(2) Q.7

1 By using the identity

tan (A+B) =tanA+ tanB

1− tanA tanB,

find the solutions, in the range 0 6 x < 180, of the equation

3 tanx◦ = tan 2x◦.

(6)


1988-1 Paper(2) Q.14

2

x

y

y = sin 2xy = sin (2x+π

3)

O A

B

The figure shows the graphs of

y = sin 2x and y = sin (2x+π

3)

in the interval −π66 x 6

π

2.

(a) Determine the coordinates of the point A. (3)

(b) By using the basic addition formula

sin (P +Q) = sinP cosQ+ cosP sinQ,

show that the x-coordinate of the point B isπ

6. (7)



1989-1 Paper(2) Q.13

3 (a) By expanding each side of the equation

cos (x◦ + 30◦) = 2 sin (x◦ + 60◦),

solve the equation for 0 < x < 180. (6)

(b) Show that

cos

(x+

π

4

)+ cos

(x− π

4

)= 2 cosx cos

π

4.

(3)

(c) Hence, or otherwise, find the maximum value of

cos

(x+

π

4

)+ cos

(x− π

4

).

(2)

(d) Find

∫0

π2

[cos

(x+

π

4

)+ cos

(x− π

4

)]dx. (4)

1989-6 Paper(2) Q.8

4 By using the identity tan 2x◦ =2 tanx◦

1− tan2 x◦, or otherwise, find the elements of the set

{x : 0 6 x 6 180}

which satisfy the equation

tan 2x◦ = 3 tanx◦.

(6)


1991-1 Paper(2) Q.9


tanx =1 + 2 tanα

2 + tanα.

(4)


Hence show that

∫0

π12 sin2 3x dx =

1

24(π − 2). (5)


(c) 3 sin 2x = cosx, (3)


1992-1 Paper(1) Q.10

6 Solve, for 0 6 θ < 360, the equations

(a) tan (θ◦ − 30◦) = 1,

(b) (2 cos θ◦ − 1)(sin 2θ◦ + 1) = 0,

(c) cos2 θ◦ − 2 sin2 θ◦ = 1.

(d) Prove that cos4 θ + 2 sin2 θ − sin4 θ = 1.

(e) Given that sin θ = x − 1 and cos θ = 2y for all values of θ, find an equation, not containing

θ, relating x and y.


1993-6 Paper(2) Q.12

7 (a) Starting with the formulae for the expansions of sin (A+B) and sin (A−B) show that


(3)

(b) Hence show that

sin 3x+ sin 5x = 2 sin 4x cosx.

(1)

(c) Solve, for 0 6 x 6 π, the equation

sin 3x+ sin 4x+ sin 5x = 0.

(7)

(d) Using the result of (b), evaluate

∫0

π4 2 sin 4x cosx dx, giving your answer to 2 decimal places.

(4)


1994-1 Paper(2) Q.13

8 Using the formula


(a) show that


. (3)

(b) Hence show that


(1)




(d) Evaluate ∫0

π4 sin 3x sinx dx.

(4)


1995-1 Paper(2) Q.14

9 Using the identity cos (A+B) = cosA cosB − sinA sinB show that

(a) cos 2x = 2 cos2 x− 1, (3)

(b) cos 6θ = 1− 2 sin2 3θ. (2)

Hence show that

(c) cos 2x+ cos 6θ = 2(cosx+ sin 3θ)(cosx− sin 3θ). (2)

I =

∫0

π12 (cosx+ sin 3θ)(cosx− sin 3θ) dx.

Given that θ is a constant

(d) show that I =a+ π cos 6θ

b, where a and b are constants to be found.

Given 0 6 θ 6π

3find the values of θ, in radians to 3 significant fgures, for which I = 0. (8)

1995-6 Paper(2) Q.4

10 Using the formula sin(A+B) = sinA cosB + cosA sinB,

(a) show that 2 sin (x+π

3) = sinx+

√3 cosx. (2)

Given that y = sinx+√

3 cosx, find,

(b) the least value of y, (1)

(c) the value of x, in terms of π, in the interval 0 < x < 2π, for which y has this least value. (2)


1995-6 Paper(2) Q.10

11 (a) Using the formula sin(A+B) = sinA cosB + cosA sinB show that


(1)

Hence show that

(b) sinP + sinQ = 2 sin(P +Q)

2cos

(P −Q)

2, (2)

(c) sin 3x+ sin 2x+ sinx = sin 2x(2 cosx+ 1). (2)

(d) Sketch, on separate axes, for the range 0 6 x 6 4π, the curves with equations

(i) y = cosx, (ii) y = sin 2x. (4)

(e) Solve, for 0 6 x 6 2π, the equation

sin 3x+ sin 2x+ sinx = 2 cosx+ 1,

giving your answers in terms of π. (6)

1995-6 Paper(2) Q.13


(3)

(b) Hence show that

∫4 sin2 3x dx =

1


(c) Evalute

∫ π4





1996-1 Paper(2) Q.11



to show that

cos2 θ = 12(1 + cos 2θ).

(3)


f(θ) = 3 sin2 θ + 4 cos2 θ.



∫0

π4 f(θ) dθ.


1996-6 Paper(2) Q.7

14 Using the formulae for the expansions of cos (A+B) and cos (A−B)

(a) show that cos (A−B)− cos (A+B) = 2 sinA sinB. (2)

(b) Hence solve for x, in the interval 0 6 x 6 π, the equation

cosx− cos 5x = sin 3x.

(4)


2007-1 Paper(2) Q.5


(a) Find an expression for cos 2θ in terms of cos2 θ. (2)

The region enclosed by the curve with equation y = 3 cos 2x, the y-axis, the x-axis and the line

x = π8 is rotated through 360◦ about the x-axis.

(b) Find, in terms of π, the volume of the solid formed. (6)

2007-6 Paper(2) Q.10


sin (A+B) = sinA cosB + cosA sinB.

(a) Write down an expression for sin 2θ in terms of sin θ and cos θ. (1)

Show that

(b) sin2 θ = 12(1− cos 2θ). (2)

(c) sin2 (A+B)− sin2 (A−B) = sin 2A sin 2B. (5)

(d) Hence show that

(i) sin2 3θ − sin2 θ = sin 4θ sin 2θ,

(ii) sin2 3θ − sin2 θ = 12(cos 2θ − cos 6θ). (4)

(e) find the exact value of∫ π

30 (6 sin 4θ sin 2θ + 2) dθ. (5)


2008-1 Paper(1) Q.9


cos (A+B) = cosA cosB − sinA sinB.

(a) Obtain an expression for cos 2θ in terms of cos2 θ. (2)

(b) Write down an expression for sin 2θ in terms of sin θ and cos θ. (1)

(c) Show that cos 3θ = 4 cos3 θ − 3 cos θ. (4)

(d) Solve, for 0 6 θ 6 π, the equation 9 cos θ− 12 cos3 θ = 2, giving your answers to 3 significant

figures. (4)

(e) Find∫ π

20 (3 cos3 θ + 2 sin θ) dθ. (5)

2009-6 Paper(2) Q.10

18 cos (A+B) = cosA cosB − sinA sinB,

cos (A−B) = cosA cosB + sinA sinB.

(a) Prove that cos 2A = 2 cos2A− 1 (2)

f(θ) = cos 5θ + cos 3θ + 2 cos θ

(b) Show that

(i) cos 5θ + cos 3θ = 2 cos 4θ cos θ,

(ii) f(θ) = 16 cos5 θ − 16 cos3 θ + 4 cos θ. (6)

(c) Hence or otherwise solve, for −π 6 θ 6 π, giving the values of θ in terms of π, the equation

cos 5θ + cos 3θ − 2 cos θ = 0 (5)

(d) Find, to 3 significant figures, the value of∫ π

30 (cos5 θ − cos3 θ) dθ. (5)


2010-6 Paper(2) Q.9


Show that

(a) cos (A+B) + cos (A−B) = 2 cosA cosB, (1)

(b) cos 2A = 2 cos2A− 1, (2)

(c) cosP + cosQ = 2 cos

(P +Q

2

)cos

(P −Q

2

). (2)

(d) Hence show that cos 8x+ 2 cos 6x+ cos 4x = 4 cos 6x cos2 x. (4)

(e) Find the exact value of∫ π

40 cos 6x cos2 x dx. (6)

2011-1 Paper(1) Q.8


(a) Show that

(i) sin2 θ = 12(1− cos 2θ)

(ii) cos2 θ = 12(cos 2θ + 1) (3)

f(θ) = 8 sin4 θ + 4 sin2 θ − 5




Given that 4

∫ π4

π8

f(θ) dθ = m+ n√

2



2011-1 Paper(2) Q.7


tanA =sinA

cosA


2 sin (x+ α) = 5 sin (x− α)


3 tanx = 7 tanα

(5)

(b) Hence solve, to one decimal place,

2 sin (2y + 50◦) = 5 sin (2y − 50◦) for 0 6 y 6 180◦

(5)

2011-6 Paper(1) Q.4

22 (a) sin (A+B) = sinA cosB + cosA sinB


Write down an expression for sin 2A in terms of sinA and cosA (1)

(b) Find an expression for cos 2A in terms of sinA (2)

(c) Show that sin 3A+ sinA = 4 sinA− 4 sin3A (4)


2012-1 Paper(2) Q.8



tanA =sinA

cosA

(a) Show that tan (A+B) =tanA+ tanB

1− tanA tanB(3)

(b) Hence write down an expression for tan 2θ in terms of tan θ (1)

(c) Show that tan 3θ =3 tan θ − tan3 θ

1− 3 tan2 θ(4)

Given that tan 3θ = −1 and tan θ 6= ±√

3

3

(d) without finding the value of θ, show that tan3 θ + 3 tan2 θ − 3 tan θ − 1 = 0 (1)

Given also that tan θ 6= 1

(e) find the exact values of tan θ, giving your answers in the form a ±√b where a and b are

integers. (4)

2012-6 Paper(1) Q.7


(a) Express cos (2x+ 45◦) in the form M cos 2x+N sin 2x, where M and N are constants, giving

exact value of M and the exact value of N . (2)

(b) Solve, for 0◦ 6 x 6 180◦, the equation cos 2x− sin 2x = 1 (5)

The maximum value of cos 2x− sin 2x is k.

(c) Find the exact value of k. (2)

(d) Find the smallest positive value of x for which a maximum occurs. (3)


2013-1 Paper(2) Q.2

25 Using the identities sin (A+B) = sinA cosB + cosA sinB


tanA =sinA

cosA

(a) show that tan (A+B) =tanA+ tanB

1− tanA tanB(3)

(b) Hence show that

(i) tan 105◦ =1 +√

3

1−√

3(ii) tan 15◦ =

√3− 1

1 +√

3(4)

2013-1 Paper(2) Q.5


(a) Use the above identity to show that 2 sin2A = 1− cos 2A (3)

(b) Hence find the value of k such that sin2 2A = k(1− cos 4A) (1)

O x

y

R

Figure 2

Figure 2 shows part of the curve with equation y = 3 sin 2x. The region R, bounded by the curve,

the positive x-axis and the line x =π

6, is rotated through 360◦ about the x-axis.

(c) Use calculus to find, to 3 significant figures, the volume of the solid generated. (6)


2013-6 Paper(1) Q.4

27 Solve, for −90 < x 6 90, the equation

6 sin2 x◦ − cosx◦ − 4 = 0

(6)

2013-6 Paper(2) Q.10

28 tan θ =sin θ

cos θ


A particle P is moving along a straight line. At time t seconds (t > 0) the displacement, s metres,

of P from a fixed point O on the line is given s =√

3 sin1

2t+ cos

1

2t

(a) Find the value exact of s when t =π

3(2)

(b) Find the exact value of t when P first passes through O. (4)

The velocity of P at time t seconds is v m/s.

(c) Find an expression for v in terms of t. (2)

(d) Show that v = cos

(π

6+

1

2t

)(2)

(e) Find the exact value of t for which v =1

2when

(i) 0 6 t < 2π

(ii) 2π 6 t < 4π (4)


2014-1 Paper(1) Q.8


tanA =sinA

cosA


4 sin (x+ α) = 7 sin (x− α)


3 tanx = 11 tanα

(5)

(b) Hence solve, to 1 decimal place,

4 sin (3y + 45)◦ = 7 sin (3y − 45)◦ for 0 6 y 6 180

(6)

2014-6 Paper(2) Q.10








(d) Find

∫(24 sin3 θ + 6 cos θ) dθ (2)

(e) Hence evaluate

∫ π3


√c, where a,



2015-1 Paper(1) Q.8

31 tan θ =sin θ

cos θ

(a) Using the above identity, show that 1 + tan2 θ =1

cos2 θ(3)

(b) Show that1 + sin θ cos θ + sin2 θ

cos2 θ= 1 + tan θ + 2 tan2 θ (3)

(c) Solve the equation 1 + sin θ cos θ + sin2 θ = 4 cos2 θ for 0 6 θ 6 180◦.

Give your answer to 1 decimal place, where appropriate. (6)

2015-1 Paper(2) Q.4



(a) Write down the exact value of sin 45◦ (1)

Given that sin θ =

√5

2√

2and cos θ =

√3

2√

2

(b) show that sin (45◦ + θ) =

√3 +√

5

4(2)

(c) Find the exact value of cos (45◦ + θ) (2)

(d) Show that sin (45◦ + θ)× cos (45◦ + θ) = −1

8(2)


2015-6 Paper(1) Q.8








(4)

(d) (i) Find

∫sin3 θ dθ

(ii) Evaluate

∫ π4


a− b√

2


integers. (5)

2016-1 Paper(2) Q.6



sinA

cosA= tanA

Using the above formulae, show that

(a) sin 2x = 2 sinx cosx (1)

(b) cos 2x = cos2 x− sin2 x (1)

(c)sin 2x

1 + cos 2x= tanx (4)


2016-6 Paper(1) Q.5

35 Using the identities


tanA =sinA

cosA

(a) show that the equation

3 sin (x+ α) = 5 sin (x− α)

can be written in the form tanx = 4 tanα (5)

(b) Hence solve, to the nearest integer, the equation

3 sin (2y + 30)◦ = 5 sin (2y − 30)◦ for 90 6 y < 180

(4)


2016-6 Paper(2) Q.9








6 cos θ − 8 cos3 θ + 1 = 0

(4)

(e) find

(i)



∫ π3

0(8 cos3 θ + 4 sin θ) dθ (4)

2017-1 Paper(2) Q.2

37 (a) Show that the equation 6 cos2 α− sinα = 5 can be written as

6 sin2 α+ sinα− 1 = 0

(2)

(b) Solve, to 1 decimal place where appropriate, for 0 6 θ 6 90

6 cos2 (2θ + 40)◦ − sin (2θ + 40)◦ = 5

(5)


2017-1 Paper(2) Q.4

38 tan (A+B) =tanA+ tanB

1− tanA tanB

(a) (i) Write down an expression for tan (2x) in terms of tanx

(ii) Hence show that tan (3x) =3 tanx− tan3 x

1− 3 tan2 x(6)

Given that a is the acute angle such that cosα =1

3

(b) find the exact value of tanα (2)

(c) Hence use the identity in part (a) to find the exact value of tan (3α)

Give your answer in the forma√

2

bwhere a and b are integers. (2)

2017-6 Paper(1) Q.9

39 Using



2(cos 2θ + 1) (2)

f(θ) = 8 cos4 θ + 4 cos2 θ − 5


Hence


8 cos4 x+ 4 cos2 x− 6 cos 2x = 4.5

(4)

(d) find

(i)∫

f(θ) dθ


30 f(θ) dθ (5)


2018-1 Paper(1) Q.10



2(cos 2θ + 1) (3)




2, the equation

16 cos4(θ − π

6

)+ 16 sin2

(θ − π

6

)− 15 = 0

(4)


0(8 cos4 θ + 8 sin2 θ + 2 sin 2θ) dθ

(4)


36 Trigonometric Ratios

1988-6 Paper(2) Q.7

1 Given that cosx◦ =15

17, where 0 < x < 90, find, as fractions, the exact values of

(a) sin 2x◦, (4)

(b) tan 2x◦. (2)

1989-1 Paper(2) Q.8

2 Given that A and B are acute angles and that sinA =5

13, cosB =

4

5, find the exact values of

(a) sin (A+B), (3)

(b) tan (A+B). (3)

1994-1 Paper(1) Q.1

3 Given that sinx =5

13calculate

(a) the exact value of cos2 x, (2)

(b) the possible values of tanx. (2)


2013-1 Paper(2) Q.2

4 Using the identities sin (A+B) = sinA cosB + cosA sinB


tanA =sinA

cosA

(a) show that tan (A+B) =tanA+ tanB

1− tanA tanB(3)

(b) Hence show that

(i) tan 105◦ =1 +√

3

1−√

3(ii) tan 15◦ =

√3− 1

1 +√

3(4)


37 Unit Vectors

1988-1 Paper(1) Q.5

1 The unit vectors i and j are due East and due North respectively. The point A has position vector

(i + j) m and the point B position vector (4i− 4j) m.

Calculate

(a) the distance AB, to the nearest 0.1 m, (3)

(b) the bearing of B from A, to the nearest degree. (2)

1988-6 Paper(1) Q.7

2 With respect to an origin O, the points M , N and P have position vectors

(2i + 3j) m, (5i− 12j) m and (11i + aj) m respectively.

(a) Calculate, to 3 significant figures, the value of | # »

OM |. (1)

(b) Find a unit vector parallel to# »

ON . (2)

(c) Given that M , N and P are collinear, find the value of a. (2)


1989-1 Paper(1) Q.13

3 With respect to an origin, O, the points A, B and C have position vectors (2i+ 5j) m, (−i+ 2j)m

and (7i + 15j) m respectively.

(a) Express# »

BA, in terms of i and j. (1)

Given that P is the point on BA, between B and A, such thatBP

PA=

1

2,

(b) write down# »

BP , in terms of i and j. (1)

(c) Hence, or otherwise, find# »

OP , in terms of i and j. (2)

Given that Q is the point on AC, between A and C, such thatAQ

QC=

2

3.

(d) find# »

OQ, in terms of i and j. (4)

Given that R is the point, on CB produced, such thatCR

BR=

3

1,

(e) find# »

OR, in terms of i and j. (4)

(f) Show that# »

RQ = λ# »

RP , where λ is a constant. (1)

(g) State the value of λ. (2)


1989-6 Paper(1) Q.8

4 Referred to a fixed origin O, the position vectors of the points A, B and C are

(i + 8j), (3i + 14j) and (2pi + 7pj)

respectively, where p is a constant.

(a) Write down, in terms of i and j, an expression for# »

AB. (1)

(b) Write down, in terms of i, j and p, an expression for# »

BC. (2)

Given that# »

BC = λ# »

AB is a constant,

(c) find the value of p. (4)

1990-1 Paper(1) Q.3

5 The position vectors of points A and B relative to a fixed origin O are (3i + j) and (−i + 2j)

respectively.

Find the position vector of

(a) the mid-point M of AB, (2)

(b) the point C where OACB is a parallelogram. (2)

1990-6 Paper(1) Q.3

6 Referred to a fixed origin O, the position vectors of the points A and B are (6i−9j) m and (i+3j)

m respectively. Find, in terms of i and j, the unit vector in the direction# »

AB. (4)


1991-6 Paper(1) Q.14

7

O

B

A

M

X

Y

Fig. 2

In Fig. 2, the points A and B have position vectors a and b respectively relative to the point O

as origin. The point M is the mid-point of OA. The point X is on OB such that X divides OB

in the ratio 3 : 1 and the point Y is on AX such that Y divides AX in the ratio 4 : 1.

(a) Write down in terms of a, b, or a and b, expressions for# »

OM,# »

OX and# »

OY . (4)

(b) Show that# »

BY = 15(a− 2b). (2)

(c) Deduce that B, Y and M are collinear. (2)

(d) Calculate the ratio BY : YM . (2)

The point E lies on XM produced and is such that E divides XM externally in the ratio 3 : 2.

Given that a = 3i + 2j and b = 2i− j,

(e) find, in terms of i and j, the position vector of E. (5)


1994-1 Paper(1) Q.2

8 Referred to a fixed origin O, the position vectors of the points A and B are (3i+ 4j) and (7i+ 7j)

respectively.

(a) Write down, in terms of i and j, an expression for# »

AB. (2)

(b) Hence show that 4AOB is isosceles. (2)

1996-1 Paper(1) Q.3

9 Referred to a fixed origin O, the position vectors of the points A and B are (2i−3j) and (7i−15j)

respectively. Find, in terms of i and j,

(a)# »

AB, (2)

(b) the unit vector in the direction of# »

AB. (2)

1996-6 Paper(1) Q.3

10 Referred to a fixed origin O, the position vectors of points A and B are (−i + 4j) and (i − 2j)

respectively. C is the point on AB produced such that AC = 3AB. Find, in terms of i and j.

(a)# »

AB, (2)

(b)# »

OC. (2)


1997-6 Paper(1) Q.2

11 Referred to a fixed origin O, the position vector of the points P and Q are (−2i+ 3j) and (ki+ lj)

respectively, where k and l are constants. The points M is the mid-point of PQ. Given that the

vector position of M relative to O is (2i + 4j),

(a) find the values of k and l. (2)

The point R is on PQ produced such that PQ : QR = 1 : 2.

(b) Find, in terms of i and j, an expression for# »

OR. (2)

2007-1 Paper(2) Q.4

12 The position vectors of the points R and S are (2i + 6j) and (6i + 14j) respectively, referred to a

fixed origin O. The point T divides RS internally in the ratio 3 : 1. Find, in terms of i and j.

(a)# »

OT . (3)

(b) the unit vectors in the direction of# »

OT . (2)

2008-1 Paper(2) Q.2

13 Referred to a fixed origin O, the position vectors of the points P and Q are (6i−5j) and (10i+3j)

respectively. The midpoint of PQ is R.

(a) Find the position vector of R. (2)

The midpoint of OP is S.

(b) Prove that SR is parallel to OQ. (3)


2010-1 Paper(1) Q.5

14 Relative to a fixed origin O, the position vector of the point A is 3i + 8j and the position vector

of the point B is 12i + qj.

The point C divides AB internally in the ratio 1 : 2 and# »

OC = pi + 4j.

(a) Find the value of p and the value of q. (5)

The point D lies on OB and the line DC is parallel to OA. The mid-point of DC is M .

(b) Find, in terms of i and j, the position vector of M . (3)

2010-6 Paper(2) Q.4

15 Referred to a fixed origin O, the position vectors of the points A and B are ai + 8j and 14i + bj

respectively and the mid-point, M , of AB has position vector 9i + 40j.


(b) Find the unit vector in the direction of# »

OM . (2)

2012-1 Paper(2) Q.1

16 Referred to a fixed origin O, the position vectors of the points P and Q are (10i−3j) and (4i+6j)

respectively. The point R divides PQ internally in the ratio 2:1

(a) Find the position vector of R (2)

The point S divides OQ internally in the ratio 5 : 4 and area 4OPQ = λ area 4SRQ.

(b) Find the exact value of λ. (4)


2015-6 Paper(2) Q.4

17 Referred to a fixed origin O, the position vectors of the points P and Q are (3i+ 6j) and (4i− 2j)

respectively.

(a) Find, as a simplified expression in terms of i and j.# »

PQ. (2)

(b) Find a unit vector which is parallel to# »

PQ. (2)

(c) Show that# »

OP is perpendicular to# »

OQ. (4)

2016-6 Paper(2) Q.2

18 Relative to a fixed origin O, the point A has position vector 6i + 5j and the point B has position

vector 3i + 9j

(a) Find# »

AB as a simplified vector in terms of i and j (2)

The line PQ is parallel to AB. Given that# »

PQ = 12i + λj

(b) find the value of λ (2)

(c) Find a unit vector parallel to AB. (2)


2017-1 Paper(2) Q.8

19 [In this question, p and q are non-zero and non-parallel vectors.]

O, A, B and C are fixed points such that# »

OA = 5p− 3q# »

OB = 11p# »

OC = 13p + q

(a) (i) Show that the points A, B and C are collinear.

(ii) Write down the ratio AB : BC. (4)

The midpoint of OA is M and the midpoint of OB is N .

(b) Show that the ratio of the area of the quadrilateral ABNM to the area of the triangle OAC

is 9 : 16 (7)


38 Vector Diagram

1988-1 Paper(1) Q.13

1

OP

A

R

Q

B

S

Fig. 2

In Fig. 2,# »

OA = a and# »

OB = b. The points P and Q lie on OA and OB respectively, so that

OP : PA = 2 : 1 and OQ : QB = 1 : 2.

(a) Find, in terms of a, b or a and b, the vectors

(i)# »

OP , (ii)# »

OQ, (iii)# »

PQ.

(4)

The point R is such that OR : AR = 2 : 1.

(b) Find, in terms of a and b, vector# »

RB. (2)

(c) Show that RB is parallel to PQ and find the ratio RB : PQ. (3)

The line segment QP is produced to a point S so that QP = PS.

(d) Find, in terms of a and b, the vectors# »

PS and# »

AS. (3)

(e) Show that the points B, A and S are collinear. (3)


1988-6 Paper(1) Q.13

2

A

C

B

E

D

X

Fig. 2

In Fig. 2,# »

AB = p and# »

AC = 2q. D is the point on BC such that BD : DC = 1 : 3 and E is the

mid-point of AC.

(a) Write down, in terms of p and q, expressions for

(i)# »

BE, (ii)# »

BD. (3)

(b) Show that# »

AD =3

4p +

1

2q. (1)

# »

AX = λ# »

AD and# »

BX = µ# »

BE, where λ and µ are scalar constants.

By considering the triangle BXA

(c) find a relationship between λ, µ, p and q. (2)

(d) Deduce the values of λ and µ. (5)

(e) Write down the ratios

(i) area4BXA : area4BXD,

(ii) area4BXA : area4EXA,

(iii) area4BXD : area4EXA.

(4)


1989-6 Paper(1) Q.12

3

O G B C

A

D

F

Fig. 2

In Fig. 2,# »

OA = 5a,# »

AB = 3b,# »

OC = 32

# »

OB and# »

OD = 35

# »

OA.

The line DC meets AB at F .

(a) Write down, in terms of a and b, expressions for# »

OB,# »

OC and# »

DC. (5)

Given that# »

DF = λ(a + b) and# »

AF = µb

(b) use the vector triangle ADF to form an equation relating a, b, λ and µ. (2)

(c) Use your equation from part (b) to find the values of λ and µ. (3)

(d) Deduce the ratios

(i) AF : FB,

(ii) DF : FC. (3)

A line is drawn through F parallel to AO to meet OB at G.

(e) Write down an expression, in terms of a and b, for# »

OG. (2)


1990-6 Paper(1) Q.14

4

A

BC

O

D

E

Fig. 4

In Fig. 4,# »

OA = a,# »

OB = b and C divides AB in the ratio 5 : 1.

(a) Write down, in terms of a and b, expressions for# »

AB,# »

AC and# »

OC. (4)

Given that# »

OE = λb, where λ is a scalar,

(b) write down, in terms of a, b and λ, an expression for# »

CE. (3)

Given that# »

OD = µ(b− a), where µ is a scalar,

(c) write down, in terms of a, b, λ and µ, an expression for# »

ED. (2)

Given also the E is the mid-point of CD,

(d) deduce the values of λ and µ, (5)

(e) state what kind of quadrilateral OCBD is. (1)


1991-1 Paper(1) Q.11

5 In 4OAB, P is the mid-point of AB and Q is the point on OP such that OQ = 34OP . Given

that# »

OA = a and# »

OB = b, find, in terms of a and b,

(a)# »

AB, (1)

(b)# »

OP , (1)

(c)# »

OQ, (2)

(d)# »

AQ. (3)

The point R on OB is such that OR = kOB, where 0 < k < 1.

(e) Find, in terms of a, b and k, the vector# »

AR. (1)

Given that AQR is a straight line

(f) find the ratio in which Q divides AR and the value of k. (7)


1991-6 Paper(1) Q.14

6

O

B

A

M

X

Y

Fig. 2

In Fig. 2, the points A and B have position vectors a and b respectively relative to the point O

as origin. The point M is the mid-point of OA. The point X is on OB such that X divides OB

in the ratio 3 : 1 and the point Y is on AX such that Y divides AX in the ratio 4 : 1.

(a) Write down in terms of a, b, or a and b, expressions for# »

OM,# »

OX and# »

OY . (4)

(b) Show that# »

BY = 15(a− 2b). (2)

(c) Deduce that B, Y and M are collinear. (2)

(d) Calculate the ratio BY : YM . (2)

The point E lies on XM produced and is such that E divides XM externally in the ratio 3 : 2.

Given that a = 3i + 2j and b = 2i− j,

(e) find, in terms of i and j, the position vector of E. (5)


1992-1 Paper(1) Q.14

7

O

A

C

B

D

E

Fig. 1

In Fig. 1 the points A and B have position vectors a and b respectively, relative to the point O

as origin. The point C is on OB produced such that OB : BC = 3 : 4 The point D is on AB such

that AD : DB = 2 : 3, and the point E is on OD such that# »

BE = λ# »

CA, where λ is a constant.

(a) Write down, in terms of a and b or a and b, expressions for# »

OC and# »

CA and hence show

that

# »

BE =λ

3(3a− 7b).

(b) Write down, in terms of a and b, an expression for# »

AB and show that

# »

OD =1

5(3a + 2b).

Given that# »

OE = µ# »

OD, where µ is a constant, and using the fact that# »

OE =# »

OB +# »

BE.

(c) form an equation relating a, b, λ and µ and hence show that λ = 13 and µ = 5

9 .

(d) Write down, in terms of a and b, the position vector of E.


1994-1 Paper(1) Q.12

8

O

D

A

B

C

E

F

Fig. 3

In Fig. 3# »

OA = a,# »

OB = b, 3# »

OC = 2# »

OA and 4# »

OD = 7# »

OB.

The line DC meets the line AB at E.

(a) Write down, in terms of a and b, expressions for (i)# »

AB, and (ii)# »

DC. (3)

Given that# »

DE = λ# »

DC and# »

EB = µ# »

AB where λ and µ are constants

(b) use 4EBD to form an equation relating a, b, λ and µ. (2)

Hence

(c) show that λ =9

13, (2)

(d) find the exact value of µ, (1)

(e) express# »

OE in terms of a and b. (3)

The line OE produced meets the line AD at F .

Given that# »

OF = k# »

OE where k is a constant and that# »

AF =1

10(7b− 4a)

(f) find the value of k. (4)


1995-1 Paper(1) Q.6

9

O D

C

A

M

B

Fig. 3

In Fig. 1, the points A and B have position vectors a and b, respectively, relative to the point O

as origin.# »

OC = 2# »

OA and# »

OD = 3# »

OB and M is the point on CD such that CM :MD = 3:5.

(a) Find, in terms of a and b, an expression for# »

CD. (1)

(b) Hence, or otherwise, show that# »

OM = 18(10a + 9b) (4)


1995-1 Paper(1) Q.14

10

O D

C

A

B

E

F

Fig. 3

In Fig. 3,# »

OA = a,# »

OB = b,# »

OC = 4# »

OA,# »

OD = 3# »

OB, 5# »

CF = 3# »

CD,# »

AE = λ# »

AD and# »

BE = µ# »

BC,

where λ and µ are constants.

(a) Find, in terms of a and b, expressions for# »

AD and# »

BC. (2)

(b) Show that# »

OE = (1− λ)a + 3λb. (2)

(c) Find in terms of a, b and µ, another expression for# »

OE. (2)

(d) Use your answers to (b) and (c) to find λ and µ. (3)

(e) Write down, in terms of a and b only, an expression for# »

OE. (1)

(f) Find, in terms of a and b, an expression for# »

OF and, hence, show that AD, BC and OF are

concurrent. (5)


1996-1 Paper(1) Q.13

11

P O

Q

R

S

T

Fig. 3

In Fig. 3,# »

OR = a,# »

OP = 2# »

OR,# »

OS = 2b and 2# »

OQ = 3# »

OS.

Find, in terms of a and b,

(a) (i)# »

RQ and (ii)# »

PS. (3)

Given that# »

RT = λ# »

RQ, where λ is a constant,

(b) use 4PRT to express# »

PT in terms of λ, a and b. (3)

Given also that# »

PT = µ# »

PS, where µ is a constant,

(c) express# »

PT in terms of µ, a and b. (2)

(d) Hence find the values of λ and µ. (5)

(e) Calculate the ratio

(area 4OPQ) : (area quadrilateral PQSR).

(2)


1996-6 Paper(1) Q.14

12

O B

A

C

E

D

3a

3b

2a + 2b

Fig. 3

In Fig. 3,# »

OA = 3a,# »

OB = 3b, and# »

OC = 2a + 2b. The point D is on AC such that AD : DC =

2 : 1 and E is the point on BC produced such that# »

BE = k# »

BC, where k is constant.

Express, in terms of a and b,

(a) (i)# »

BC, (ii)# »

AC, (iii)# »

OD. (4)

(b) Show that# »

OE = 2ka + (3− k)b. (3)

Given also that ODE is a straight line, find

(c) the value of k, (4)

(d)# »

OE, in terms of a and b only, (2)

(e) the ratio OD : DE. (2)


1997-6 Paper(1) Q.13

13

O B

A

C

D

EF

Fig. 4

In Fig. 4,# »

OA = 3a,# »

OB = 2b,# »

OB = 2# »

OD,# »

OA = 3# »

OC,# »

DE = p# »

DA,# »

CE = q# »

CB and# »

BF = r# »

BA,

where p, q, and r are positive constants.

(a) Find, in terms of a and b, an expression for

(i)# »

DA, (ii)# »

CB. (2)

(b) Show that# »

OE = 3pa + (1− p)b. (2)

(c) Find, in terms of a, b, and q, another expression for# »

OE. (2)

(d) Using your answers to (b) and (c), find the values of p and q. (3)

(e) Write down, in terms of a and b only, an expression for# »

OE. (2)

(f) Find the value of r. (4)


2007-6 Paper(2) Q.7

14

O B

A

D

P Q

Figure 1

Figure 1 shows 4OAB where# »

OA = a and# »

OB = b. The point P divides OA in the ratio 2 : 3

and the point Q divides AB in the ratio 5 : 2. The side OB is produced to the point D, where

OD = 52OB.

(a) Find, in terms of a and b,

(i)# »

PB, (ii)# »

AD, (iii)# »

AB, (iv)# »

PQ, (v)# »

PD. (8)

(b) Show that

(i) PB is parallel to AD,

(ii) P , Q and D are collinear. (4)


2011-1 Paper(2) Q.4

15

O B

A

C

D

Figure 1

In 4OAB , # »

OA = a and# »

OB = b.

The point C divides AB in the ratio 2 : 3 and D is the mid-point of OB as shown in Figure 1.

(a) Find, in terms of a and b,# »

OC. (2)

OC and AD meet at K.

K divides AD in the ratio λ : 3.

(b) Find the value of λ. (4)


2012-1 Paper(1) Q.10

16

O

A

C

B

P

Q

T

Figure 2

Figure 2 shows a trapezium OABC in which AB is parallel to OC and AB =1

2OC. The point

P divides OA in the ratio 1:3 and the point Q divides BC in the ratio 1:2

The line AC intersects the line PQ at the point T .# »

OA = a and# »

OC = c

(a) Find, as simplified expressions in terms of a and c

(i)# »

BC

(ii)# »

PQ (5)

(b) (i) Given that# »

PT = λ# »

PQ, find an expression for# »

AT in terms of λ, a and c

(ii) Given also that# »

AT = µ# »

AC, find an expression for# »

AT in terms of µ, a and c (2)

(c) Use your answers from part (b) to find the value of λ and hence write down the ratio PT : TQ

(6)


2013-1 Paper(2) Q.8

17

O B

A

X

P

M

Figure 3

In Figure 3,# »

OA = a,# »

OB = b and M is the mid-point of AB.

The point P is on OA such that OP : PA = 3 : 2

The point X lies on OB produced.

(a) Find, as simplified expressions in terms of a and b,

(i)# »

AB (ii)# »

OM (iii)# »

PM (6)

Given that P , M , and X are collinear

(b) find, in terms of b,# »

OX (4)

(c) Find the ratio (area4OAM) : (area4OAX). (3)


2013-6 Paper(1) Q.11

18 O, A, B and C are fixed points such that# »

OA = p + q# »

OB = 3p− q# »

OC = 6p− 4q

(a) Find# »

AB in terms of p and q. (1)

(b) Show that the points A, B and C are collinear. (2)

(c) Find the ratio AB : BC (1)

The point D lies on AC produced such that AC = 2CD

(d) Find# »

OD in terms of p and q, simplifying your answer. (4)


2014-1 Paper(2) Q.8

19

O

A

E

B

D

C

120◦

a

e

Figure 4

Figure 4 shows a hexagon OABCDE. Each internal angle of the hexagon is 120◦.

OA = OE, AB = ED = 2×OA and OC = 3×OA# »

OA = a and# »

OE = e.

Find, as simplified expressions in terms of a and e

(a)# »

AB, (2)

(b)# »

BE. (2)

The point P divides AB internally in the ratio 2 : 3

(c) Find# »

PC as a simplified expression in terms of a and e. (3)

The point Q lies on ED produced so that the points P , C and Q are collinear.

(d) Find# »

OQ in the form λa + µe, stating the value of λ and the value of µ. (6)


2015-1 Paper(1) Q.3

20

O

A

C

B

ab

c

Figure 1

Figure 1 shows the quadrilateral OABC.

# »

OA = a,# »

OB = b and# »

OC = c

(a) Find, in terms of a and b,# »

AB. (1)

The mid-point of OA is P and the mid-point of AB is Q.

(b) Show that# »

PQ = µb, where µ is a scalar, stating the value of µ. (2)

The point S lies on OC and the point R lies on BC such that# »

OS = λ# »

OC and# »

BR = λ# »

BC.

(c) Show that PQ is parallel to SR. (4)

Given that# »

PQ =3

2

# »

SR,

(d) find the value of λ. (2)


2016-1 Paper(2) Q.9

21

O C

A B

a b b− 2a

Figure 2

Figure 2 shows a quadrilateral OABC

# »

OA = a,# »

OB = b and# »

BC = b− 2a

(a) (i) Prove that# »

AB is parallel to# »

OC

(ii) Show that AB : OC = 1 : 2 (4)

The point D lies on OB such that OD : DB = 2 : 3

(b) Find the ratio of the area of 4ODC : the area of 4OAB. (6)


2016-6 Paper(1) Q.8

22

O B

A

D

E

Figure 1

In Figure 1,# »

OA = a,# »

OB = b and# »

OD =2

3b

The point E divides AD in the ratio 2 : 3

(a) Find as simplified expressions in terms of a and b

(i)# »

AD (ii)# »

OE (iii)# »

BE

(5)

The point F lies on OA such that# »

OF = λ# »

OA and F , E and B are collinear.

(b) Find the value of λ. (5)

The area of triangle OFB is 5 square units.

(c) Find the area of triangle OAD.

Give your answer in the formp

q, where p and q are integers. (3)


2017-1 Paper(2) Q.8

23 [In this question, p and q are non-zero and non-parallel vectors.]

O, A, B and C are fixed points such that# »

OA = 5p− 3q# »

OB = 11p# »

OC = 13p + q

(a) (i) Show that the points A, B and C are collinear.

(ii) Write down the ratio AB : BC. (4)

The midpoint of OA is M and the midpoint of OB is N .

(b) Show that the ratio of the area of the quadrilateral ABNM to the area of the triangle OAC

is 9 : 16 (7)


2017-6 Paper(1) Q.3

24

OB

A

E

C

D

Figure 1

In Figure 1,# »

OA = a and# »

OB = b

The point C is the midpoint of# »

OA and the point D divides OB in the ratio 2 : 1

(a) Find# »

CD in terms of a and b (2)

The point E lies on AB produced such that# »

OE = 2b− a

(b) Find# »

CE in terms of a and b (2)

(c) Hence show that C, D and E are collinear. (2)


2018-1 Paper(2) Q.6

25

O

A

B

a

b

Figure 3

Figure 3 shows the triangle OAB with# »

OA = a and# »

OB = b.

(a) Find# »

AB in terms of a and b. (1)

The point P is such that# »

OP =3

4

# »

OA, and the point Q is the midpoint of AB.

(b) Find# »

PQ as a simplified expression in terms of a and b. (2)

The point R is such that PQR and OBR are straight lines where# »

QR = µ# »

PQ and# »

BR = λ# »

OB

(c) Express# »

QR in terms of

(i) a, b and µ

(ii) a, b and λ (3)

(d) Hence find the value of

(i) µ

(ii) λ (4)


39 Volume by Revolution

1988-1 Paper(2) Q.11


f: x 7→ x

(x− 1)(x+ 3)where x ∈ R and x 6= −3, x 6= 1.


(x− 1)2(x+ 3)2. (5)



x =x

(x− 1)(x+ 3).

(4)





1989-1 Paper(1) Q.14

2

x

y

O (1, 0) Q

P (2, 1)

Fig. 4

Figure 4 shows a sketch of the graph of y = (x− 1)2 for x > 0. The line PQ is the normal to the

curve at the point P (2, 1).

(a) Find the equation of PQ. (5)

Given that PQ intersects the x-axis at Q,

(b) find the coordinates of the point Q. (1)

(c) Write down the binomial expansion of (x− 1)4 in descending powers of x. (2)

The finite region A is bounded by the curve y = (x− 1)2, the x-axis and the line PQ. The region

A is rotated through 2π about the x-axis. Using your answer to (c), or otherwise,

(d) find the volume of revolution so formed, leaving your answer in terms of π. (7)


1989-6 Paper(1) Q.11

3

x

y

PO

Q

y = 4x3 + x2 − 2x+ 1

y = 8x2 + 1

A

B

C

Fig. 1

Figure 1 shows, for x > 0, the graphs of y = 8x2 + 1 and y = 4x3 + x2 − 2x + 1. The curvesintersect at the points P and Q.

(a) State the coordinates of P and show that the x-coordinate of Q is 2. (3)

Region A is bounded by the curve with equation y = 4x3 + x2 − 2x+ 1, the x-axis and the linesx = 0 and x = 2.

(b) Show that the area of region A is 1623 . (3)

Region B lies in the first quadrant and is bounded by the two curves.

(c) Calculate the area of region B. (4)

Region C is bounded by the curve with equation y = 8x2 + 1, the y-axis and the line through Qparallel to the x-axis.

(d) Calculate, in terms of π, the volume generated when region C is rotated through 360◦ aboutthe y-axis. (5)


1989-6 Paper(2) Q.10




cosxshow that

d


(3)





terms of π. (5)


1990-1 Paper(2) Q.11

5

O A x

y

D

B

C

S

R

y = 2x(3− x)

Fig. 1

The curve with equation y = 2x(3− x) crosses the x-axis at O and A.

(a) State the coordinates of the point A. (1)

A straight line, which crosses the y-axis at the point B with coordinates (0, 5), meets the curveat the points C and D, as shown in Fig. 1. The coordinates of the point D are (k, k).

(b) Show that the value of k is 212 . (1)

(c) Find the equation of the straight line passing through B and D. (2)

(d) Show that the x-coordinate of the point C is 1. (2)

(e) Calculate the area of the shaded region R. (5)

(f) Calculate, in terms of π, the volume generated when the shaded region S, bounded by thecurve, the x-axis and the line x = 1, is rotated through 360◦ about the x-axis. (4)


1990-6 Paper(1) Q.9

6

O A C x

y

B

x+ y = 2

y2 = 3x− 2

Fig. 3

Figure 3 shows, for x > 0, y > 0, the curve with equation y2 = 3x− 2 and the line with equation

x + y = 2. The curve meets the x-axis at the point A. The line meets the curve at the point B,

and the x-axis at the point C.

(a) State the coordinates of A and C. (2)


(c) Calculate the area of the region bounded by the curve, the x-axis, the y-axis and the line

through B parallel to the x-axis. (4)

(d) Calculate, in terms of π, the volume of the solid generated when the shaded region bounded

by the curve, the line x+ y = 2 and the x-axis is rotated through 360◦ about the x-axis. (6)


1991-6 Paper(1) Q.11

7 f(x) = x3 − 12x+ 2

(a) Find the coordinates of the turning points of the curve with equation y = f(x), and determine

their nature. (8)

The region R is bounded by the x-axis, the curve with equation y = x2 + 1, and the lines with

equations x = 1 and x = 2.

(b) Draw a sketch to show the region R. (2)

The region R is rotated through 2π about the x-axis.

(c) Calculate, to 3 significant figures, the volume of the solid generated. (5)

1993-6 Paper(2) Q.14

8 (a) Using the formula for cos (A+B), show that

cos 2x = 2 cos2 x− 1.

(4)

(b) Evaluate

∫0

π4 2 cos2 2x dx, giving your answer in terms of π. (4)

(c) Sketch for −π < x 6 π the curve with equation y = cos 2x. (2)

The line l has equation y = 1− 4x

π.

(d) Draw the line l on your sketch. (1)

The finite region bounded by the curve with equation y = cos 2x, and the line l, for 0 6 x 6 14π,

is rotate through 360◦ about the x-axis.

(e) Calculate, in terms of π, the volume of the solid generated. (4)


1994-1 Paper(1) Q.11

9

OA C

B

x

y

Fig. 1

Figure 1 shows part of the curve with equation y = 3 + 2x− x2. The curve cuts the x-axis at Aand C and cuts the y-axis at B.

(a) calculate the coordinates of A, B and C. (4)

(b) Show that the line OB divides the area of the finite region bounded by the curve and thex-axis in the ratio of 5:27. (6)

0 5 10 x

y

Fig. 2

Figure 2 shows part of the curve with equation y = 10x− 3

2 .

(c) Calculate, in terms of π, the volume of the solid generated when the region bounded by thiscurve, the x-axis and the lines x = 5 and x = 10 is rotated through 360◦ about the x-axis.

(5)


1994-1 Paper(2) Q.9

10

O

A

P

x

y

Fig. 1

Figure 1 shows part of the curve with equation y = 2 sinx. P is the point with coordinates

(π3 ,√

3).

The normal to the curve at P cuts the x-axis at A.

(a) Show that an equation of the normal AP is

y + x =π

3+√

3.

(5)

(b) Show thatd

dx(2x− sin 2x) = 4 sin2 x. (4)

The shaded area is rotated through 360◦ about the x-axis.

(c) Using the result of (b), or otherwise, calculate, in terms of π, the volume of the solid generated.

(6)


1996-6 Paper(2) Q.11

11 f(x) = cos 4x+ 12

(a) Solve, for 0 6 x 6 π4 , the equation f(x) = 0. (3)

(b) Sketch, for 0 6 x 6 π4 , the curve with equation y = f(x). (2)

(c) Shade on your sketch the region for which 0 6 x 6 π4 and 0 6 y 6 f(x) (1)

(d) Calculate, to 3 significant figures, the area of the shaded region. (3)

(e) Show that [f(x)]2 = 12 cos 8x+ cos 4x+ 3

4 . (3)

(f) Calculate, to 3 significant figures, the volume generated when the shaded region is rotated

through 2π radians about the x-axis. (3)

1997-6 Paper(1) Q.7

12

O 2 4 x

y

Fig. 1

Figure 1 shows part of the curve with equation y =4

x. The shaded region is bounded by the

curve, the x-axis and the lines with equations x = 2 and x = 4. The shaded region is rotated

through 360◦ about the x-axis. Find, in terms of π, the volume of the solid generated. (6)


1997-6 Paper(2) Q.11

13 The curve C has equation y = ex + 2e−x.

(a) Show that C has a stationary point, P , when x = 12 ln 2. (5)

(b) Determine whether P is a maximum or a minimum point. (2)

(c) Find the y-coordinate of P , giving your answer in the form k√

2, stating the value of the

constant k. (3)

The finite region bounded by C, the y-axis, the x-axis and the line x = 1 is rotated through 2π

radians about the x-axis.

(d) Calculate, giving your answer in terms of e and π, the volume of the solid generated. (5)

2007-1 Paper(2) Q.5


(a) Find an expression for cos 2θ in terms of cos2 θ. (2)

The region enclosed by the curve with equation y = 3 cos 2x, the y-axis, the x-axis and the line

x = π8 is rotated through 360◦ about the x-axis.

(b) Find, in terms of π, the volume of the solid formed. (6)

2007-6 Paper(1) Q.1

15 The region enclosed by the curve with equation y2 = 16x, the x-axis and the lines x = 2 and x = 4

is rotated through 360◦ about the x-axis. Find, in terms of π, the volume of the solid generated.

(4)


2008-1 Paper(2) Q.1

16 The region enclosed by the curve with equation y = e2x + 4, the x-axis, the y-axis and the line

x = 2 is rotated through 360◦ about the x-axis. Find, in terms of e and π, the volume of the solid

generated. (5)

2008-6 Paper(2) Q.4

17 The finite region enclosed by the curve with equation y = 9−x2 and the x-axis is rotated through

360◦ about the x-axis. Find, to 3 significant figures, the volume of the solid generated. (6)

2009-6 Paper(2) Q.7

18 A curve has equation a32 y = x

52 , where x > 0 and a is a positive constant.

(a) Show that an equation of the normal to the curve at the point with coordinates(a, a) is

5y + 2x = 7a. (6)

(b) Find the coordinates of the point where this normal meets the x-axis. (1)

The finite region bounded by the curve, the normal to the curve at the point (a, a) and the x-axis

is rotated 360◦ about the x-axis.



2010-1 Paper(1) Q.6

19

O

C1

C2

x

y

A

B

Figure 1

Figure 1 shows the curve C1 with equation y2 = 8x+4 and the curve C2 with equation y2 = 8−4x.

The curve C1 and C2 intersect at the points A and B.

(a) Find the exact coordinates of A. (3)

The shaded region enclosed by C1, C2 and the x-axis is rotated through 360◦ about the x-axis.

(b) Find, in terms of π, the volume of the solid generated. (6)


2010-6 Paper(1) Q.6

20

O x

y

y = 5

Figure 1

Figure 1 shows the curve with equation y = 9− x2 and the line y = 5

The shaded region is rotated through 360◦ about the x-axis.

Find, to 3 significant figures, the volume of the solid generated. (11)

2011-1 Paper(2) Q.11

21 A curve has equation 5y = x2 + 4.

The x-coordinate of point P on the curve is 4.

(a) Find an equation, with integer coefficients, for the tangent to the curve at P . (6)

(b) Find an equation, with integer coefficients, for the normal to the curve at P . (2)

(c) Find the area of the triangle formed by the tangent at P , the normal at P and the x-axis.

(3)

The finite region bounded by the curve, the normal at P and the coordinate axes is rotated




2011-6 Paper(1) Q.5

22

O

A (a, a)

x

y

l

C

Figure 1

The curve C, with equation y2 = 5x an the line intersect at the point A with coordinates

(a, a), a 6= 0, as show in Figure 1.

(a) Find the value of a. (2)

The line l has gradient −5

7and intersects the x-axis at the point B.

(b) Find the x-coordinate of B. (3)

The shaded region is rotated through 360◦ about the x-axis.



2012-1 Paper(1) Q.11

23

O

h

x

y

Figure 3

The centre of the circle C, with equation x2 + y2 − 10y = 0, has coordinates (0, 5). The circlepasses through the origin O. The region bounded by the circle, the positive y-axis and the liney = h, where h < 5, is shown shaded in Figure 3. The shaded region is rotated through 2π radiansabout the y-axis.

(a) Show that the volume of the solid formed is1

3π h2(15− h). (5)

O x

y

A

Figure 4

The point A with coordinates (5, 5) lies on C. A bowl is formed by rotating the arc OA through2π radians about the y-axis, as shown in Figure 4. Water is poured into the bowl at a constantrate of 6 cm3/s. The volume of water in the bowl is V cm3 when the depth of water above O ish cm.

(b) Use the formula given in part (a) to find an expression fordV

dhin terms of h. (1)

(c) Find, to 3 significant figures, the rate at which h is changing when the water above O is 1.5cm deep. (4)

The area of the surface of the water is W cm2 when the depth of water above O is h cm.

(d) Show that, for 0 < h < 5, the rate of change of the depth of water above O isk

W, stating

the value of k. (3)


2013-1 Paper(2) Q.5


(a) Use the above identity to show that 2 sin2A = 1− cos 2A (3)

(b) Hence find the value of k such that sin2 2A = k(1− cos 4A) (1)

O x

y

R

Figure 2

Figure 2 shows part of the curve with equation y = 3 sin 2x. The region R, bounded by the curve,

the positive x-axis and the line x =π

6, is rotated through 360◦ about the x-axis.

(c) Use calculus to find, to 3 significant figures, the volume of the solid generated. (6)


2014-1 Paper(1) Q.11

25 The curve C has equation 5y = 4(x2 + 1). The coordinates of the point P on the curve are

(p, 8), p > 0

The line l with equation 5y − 24x+ q = 0 is the tangent to C at P .

(a) (i) Show that p = 3

(ii) Find the value of q (4)

(b) Find an equation, with integer coefficients, for the normal to C at P . (5)

(c) Find the exact value of the area of the triangle formed by the tangent to C at P , the normal

to C at P and the x-axis. (3)

The finite region bounded by C, the tangent to C at P , the x-axis and the y-axis is rotated




2014-6 Paper(2) Q.7

26

O

A

x

y l

Figure 2

Figure 2 shows the curve C with equation y2 = 8(x− 2) and the line l with equation y = x

The line l is the tangent to C at the point A.

(a) Find the coordinates of A. (4)

The region shown shaded in Figure 2 is rotated through 360◦ about the x-axis.

(b) Use algebraic integration to find the volume of the solid formed.

Give your answers in terms of π. (5)


2015-1 Paper(1) Q.7

27 The curve C has equation y = x2 + 3

The point A with coordinates (0, 3) and the point B with coordinates (4, 19) lie on C, as shown

below in Figure 3.

O

A

B

x

y

C

Figure 3

The finite area enclosed by the arc AB of curve C, the axes and the line with equation x = 4 is

rotated through 360◦ about the x-axis.

(a) Using algebraic integration, calculate, to 1 decimal place, the volume of the solid generated.

(6)

O

A

B

x

y

C

Figure 4

(b) Using algebraic integration, calculate the region between the chord AB and the arc AB of

C, shown shaded in Figure 4. (6)


2015-6 Paper(1) Q.1

28 The region enclosed by the curve with equation y = 4x2 − 9, the positive x-axis and the negative

y-axis is rotated through 360◦ about the x-axis.

Use algebraic integration to find, to 3 significant figures, the volume of the solid generated. (5)

2017-1 Paper(1) Q.11

29 The curve C has equation y = px+ qx2 where p and q are integers.

The curve C has a stationary point at (3, 9).

(a) (i) Show that p = 6 and find the value of q.

(ii) Determine the nature of the stationary point (3, 9). (7)

The straight line l with equation y + x− 10 = 0 intersects C at two points.

(b) Determine the x-coordinate of each of these two points of intersection. (3)

The finite region bounded by the curve c and the straight line l is rotated through 360◦ about the

x-axis.

(c) Use algebraic integration to find the volume of the solid formed. Give your answer in terms

of π. (5)


2018-1 Paper(2) Q.3

30

O

x2 + y2 = 5

1x

y

Figure 2

The region enclosed by the circle with equation x2 + y2 = 5 and the straight line with equation

x = 1, shown shaded in Figure 2, is rotated through 360◦ about the y-axis.

Use algebraic integration to find the exact volume of the solid generated. (5)


13D Trigonometry - Weebly...Figure 3 shows a tetrahedron ABCE. The horizontal base is the isosceles...

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