FP1 All Question Papers CONDENSED

N34694A This publication may only be reproduced in accordance with Edexcel Limited copyright policy.2009 Edexcel Limited.

Paper Reference(s)

6667/01

Edexcel GCEFurther Pure Mathematics FP1

Advanced LevelFriday 30 January 2009 AfternoonTime: 1 hour 30 minutesMaterials required for examination Items included with question papersMathematical Formulae (Orange) Nil

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebramanipulation, differentiation or integration, or have retrievable mathematical formulae stored in them.

Instructions to Candidates

In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP1), the paper reference (6667), your surname, initials and signature.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for Candidates

A booklet Mathematical Formulae and Statistical Tables is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 10 questions on this paper. The total mark for this paper is 75.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.

N34694A 2

1. f(x) = 2x3 8x2 + 7x 3.

Given that x = 3 is a solution of the equation f(x) = 0, solve f(x) = 0 completely.(5)

2. (a) Show, using the formulae for r and 2r , that

n

rrr

1

2 )146( = n(n + 2)(2n + 1).

(5)

(b) Hence, or otherwise, find the value of

20

11

2 )146(r

rr .

(2)

3. The rectangular hyperbola, H, has parametric equations x = 5t, y =t5 , t

(a) Write the cartesian equation of H in the form xy = c2.(1)

Points A and B on the hyperbola have parameters t = 1 and t = 5 respectively.

(b) Find the coordinates of the mid-point of AB.(3)

4. Prove by induction that, for n +,

n

r rr1 )1(1 =

1nn .

(5)

5. f(x) = 3x +x

18 20.

(a) Show that the equation f(x) = 0 has a root in the interval [1.1, 1.2].(2)

(b) Find f x).(3)

(c) Using x0 = 1.1 as a first approximation to , apply the Newton-Raphson procedure once to f(x) to find a second approximation to , giving your answer to 3 significant figures.

(4)

N34694A 3

6. A series of positive integers u1, u2, u3, ... is defined by

u1 = 6 and un + 1 = 6un 5, for n t 1.

Prove by induction that un = 5 6n 1 + 1, for n t 1.(5)

7. Given that X =

112 a

, where a is a constant, and a

(a) find X1 in terms of a.(3)

Given that X + X1 = I, where I is the 2 2 identity matrix,

(b) find the value of a.(3)

8. A parabola has equation y2 = 4ax, a > 0. The point Q (aq2, 2aq) lies on the parabola.

(a) Show that an equation of the tangent to the parabola at Q is

yq = x + aq2.(4)

This tangent meets the y-axis at the point R.

(b) Find an equation of the line l which passes through R and is perpendicular to the tangent at Q.

(3)

(c) Show that l passes through the focus of the parabola.(1)

(d) Find the coordinates of the point where l meets the directrix of the parabola.(2)

N34694A 4

9. Given that 1z = 3 + 2i and 2z =1

i512z ,

(a) find 2z in the form a + ib, where a and b are real.(2)

(b) Show, on an Argand diagram, the point P representing 1z and the point Q representing 2z .(2)

(c) Given that O is the origin, show that POQ =2S .

(2)

The circle passing through the points O, P and Q has centre C. Find

(d) the complex number represented by C,(2)

(e) the exact value of the radius of the circle.(2)

10. A =

230023 , B =

0110

, C =

21

21

21

21

.

(a) Describe fully the transformations described by each of the matrices A, B and C.(4)

It is given that the matrix D = CA, and that the matrix E = DB.

(b) Find D.(2)

(c) Show that E =

3333

.

(1)

The triangle ORS has vertices at the points with coordinates (0, 0), (15, 15) and (4, 21). This triangle is transformed onto the triangle 256by the transformation described by E.

(d) Find the coordinates of the vertices of triangle 256.(4)

(e) Find the area of triangle 256and deduce the area of triangle ORS.(3)

TOTAL FOR PAPER: 75 MARKSEND

M35146A This publication may only be reproduced in accordance with Edexcel Limited copyright policy.2009 Edexcel Limited.

Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1

Advanced SubsidiaryWednesday 17 June 2009 MorningTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Orange) Nil

Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical formulas stored in them.


Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP1), the paper reference (6667), your surname, initials and signature.


A booklet Mathematical Formulae and Statistical Tables is provided.Full marks may be obtained for answers to ALL questions.There are 8 questions in this question paper. The total mark for this paper is 75.



M35146A 2

1. The complex numbers 1z and 2z are given by

1z = 2 i and 2z = 8 + 9i

(a) Show 1z and 2z on a single Argand diagram.(1)

Find, showing your working,

(b) the value of 1z ,(2)

(c) the value of arg 1z , giving your answer in radians to 2 decimal places,(2)

(d)1

2

zz in the form a + bi, where a and b are real.

(3)

2. (a) Using the formulae for

n

rr

1

,

n

rr

1

2 and

n

rr

1

3 , show that

n

rrrr

1)3)(1( =

121 n(n + 1)(n + 2)(3n + k),

where k is a constant to be found.(7)

(b) Hence evaluate

40

21)3)(1(

rrrr .

(2)

3. f(x) = (x2 + 4)(x2 + 8x + 25)

(a) Find the four roots of f(x) = 0.(5)

(b) Find the sum of these four roots.(2)

M35146A 3

4. *LYHQWKDWLVWKHRQO\UHDOURRWRIWKHHTXDWLRQ

x3 x2 6 = 0,

(a) VKRZWKDW(2)

(b) TakinJ DV D ILUVW DSSUR[LPDWLRQ WR DSSO\ WKH 1HZWRQ-Raphson procedure once to f(x) = x3 x2 WRREWDLQ D VHFRQGDSSUR[LPDWLRQ WR JLYLQJ\RXU DQVZHU WR 3 decimal places.

(5)

(c) Use linear interpolation once on the interval [2.2, 2.3] to find another approximation WRgiving your answer to 3 decimal places.

(3)

5. R =

baa 2

, where a and b are constants and a > 0.

(a) Find R2 in terms of a and b.(3)

Given that R2 represents an enlargement with centre (0, 0) and scale factor 15,

(b) find the value of a and the value of b.(5)

6. The parabola C has equation y2 = 16x.

(a) Verify that the point P(4t2, 8t) is a general point on C.(1)

(b) Write down the coordinates of the focus S of C.(1)

(c) Show that the normal to C at P has equation

y + tx = 8t + 4t3 .(5)

The normal to C at P meets the x-axis at the point N.

(d) Find the area of triangle PSN in terms of t, giving your answer in its simplest form.(4)

M35146A 4

7. A =

412a

, where a is a constant.

(a) Find the value of a for which the matrix A is singular.(2)

B =

4123

(b) Find B1.(3)

The transformation represented by B maps the point P onto the point Q.

Given that Q has coordinates (k 6, 3k + 12), where k is a constant,

(c) show that P lies on the line with equation y = x + 3.(3)

8. Prove by induction that, for n + ,

(a) f(n) = 5n + 8n + 3 is divisible by 4,(7)

(b)n

1223

=

nn

nn

212

212

.

(7)

TOTAL FOR PAPER: 75 MARKS

END


Paper Reference(s)

6667/01


Advanced/Advanced SubsidiaryMonday 1 February 2010 AfternoonTime: 1 hour 30 minutesMaterials required for examination Items included with question papersMathematical Formulae (Orange) Nil








N35143A 2

1. The complex numbers 1z and 2z are given by

1z = 2 + 8i and 2z = 1 i


(a)2

1

zz in the form a + bi, where a and b are real,

(3)

(b) the value of 2

1

zz ,

(2)

(c) the value of arg2

1

zz , giving your answer in radians to 2 decimal places.

(2)

2. f(x) = 3x2 211x

.

(a) Write down, to 3 decimal places, the value of f(1.3) and the value of f(1.4).(1)

The equation f(x) = 0 has a root between 1.3 and 1.4

(b) Starting with the interval [1.3, 1.4], use interval bisection to find an interval of width 0.025 which contains .

(3)

(c) Taking 1.4 as a first approximation to , apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to , giving your answer to 3 decimal places.

(5)

3. A sequence of numbers is defined by

1u = 2,

1nu = 5 nu 4, n t 1.

Prove by induction that, for n , nu = 5n 1 + 1.(4)

N35143A 3

4.

Figure 1

Figure 1 shows a sketch of part of the parabola with equation y2 = 12x .

The point P on the parabola has x-coordinate 31 .

The point S is the focus of the parabola.

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

The points A and B lie on the directrix of the parabola.The point A is on the x-axis and the y-coordinate of B is positive.

Given that ABPS is a trapezium,

(b) calculate the perimeter of ABPS.(5)

N35143A 4

5. A =

425

aa

, where a is real.

(a) Find det A in terms of a.(2)

(b) Show that the matrix A is non-singular for all values of a.(3)

Given that a = 0,

(c) find A1.(3)

6. Given that 2 and 5 + 2i are roots of the equation

x3 12x2 + cx + d = 0, c, d,

(a) write down the other complex root of the equation.(1)

(b) Find the value of c and the value of d.(5)

(c) Show the three roots of this equation on a single Argand diagram.(2)

7. The rectangular hyperbola H has equation xy = c2, where c is a constant.

The point P

tcct, is a general point on H.

(a) Show that the tangent to H at P has equation

t2y + x = 2ct.(4)

The tangents to H at the points A and B meet at the point (15c, c).

(b) Find, in terms of c, the coordinates of A and B.(5)

N35143A 5

8. (a) Prove by induction that, for any positive integer n,

n

rr

1

3 =41 n2(n + 1)2.

(5)

(b) Using the formulae for

n

rr

1

and

n

rr

1

3 , show that

n

rrr

1

3 )23( =41 n(n + 2)(n2 + 7).

(5)

(c) Hence evaluate

25

15

3 )23(r

rr .

(2)

9. M =

21

21

21

21

.

(a) Describe fully the geometrical transformation represented by the matrix M.(2)

The transformation represented by M maps the point A with coordinates (p, q) onto thepoint B with coordinates (32, 42).

(b) Find the value of p and the value of q.(4)

(c) Find, in its simplest surd form, the length OA, where O is the origin.(2)

(d) Find M2.(2)

The point B is mapped onto the point C by the transformation represented by M2.

(e) Find the coordinates of C.(2)


H35387A This publication may only be reproduced in accordance with Edexcel Limited copyright policy.2010 Edexcel Limited.

Paper Reference(s)


Advanced SubsidiaryTuesday 22 June 2010 AfternoonTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Pink) Nil

Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical formulas stored in them.







H35387A 2

1. z = 2 3i

(a) Show that z2 = L(2)


(b) the value of ~z2~,(2)

(c) the value of arg (z2), giving your answer in radians to 2 decimal places.(2)

(d) Show z and z2 on a single Argand diagram.(1)

2. M =

aa6

32, where a is a real constant.

(a) Given that a = 2, find M1.(3)

(b) Find the values of a for which M is singular.(2)

3. f(x) = x3 x7 + 2, x > 0.

(a) Show that f (x) = 0 has a root D between 1.4 and 1.5.(2)

(b) Starting with the interval [1.4, 1.5], use interval bisection twice to find an interval of width 0.025 that contains D .

(3)

(c) Taking 1.45 as a first approximation to D, apply the Newton-Raphson procedure onceto f(x) = x3

x7 + 2, x > 0 to obtain a second approximation to D, giving your answer to

3 decimal places.(5)

H35387A 3

4. f(x) = x3 + x2 + 44x + 150.

Given that f (x) = (x + 3)(x2 + ax + b), where a and b are real constants,

(a) find the value of a and the value of b.(2)

(b) Find the three roots of f(x) = 0.(4)

(c) Find the sum of the three roots of f (x) = 0.(1)


(a) Verify that the point P(5t2 ,10t) is a general point on C.(1)

The point A on C has parameter t = 4.The line l passes through A and also passes through the focus of C.

(b) Find the gradient of l.(4)

6. Write down the 2 2 matrix that represents

(a) an enlargement with centre (0, 0) and scale factor 8,(1)

(b) a reflection in the x-axis.(1)

Hence, or otherwise,

(c) find the matrix T that represents an enlargement with centre (0, 0) and scale factor 8,followed by a reflection in the x-axis.

(2)

A =

2416

and B =

61

ck

, where k and c are constants.

(d) Find AB.(3)

Given that AB represents the same transformation as T,

(e) find the value of k and the value of c.(2)

H35387A 4

7. f(n) = 2n + 6n.

(a) Show that f(k +1) = 6f(kk ).(3)

(b) Hence, or otherwise, prove by induction that, for n +, f(n) is divisible by 8.(4)

8. The rectangular hyperbola H has equation xy = c2, where c is a positive constant.

The point A on H has x-coordinate 3c.

(a) Write down the y-coordinate of A.(1)

(b) Show that an equation of the normal to H at A is

3y = 27x c.(5)

The normal to H at A meets H again at the point B.

(c) Find, in terms of c, the coordinates of B.(5)

9. (a) Prove by induction that

n

rr

1

2 =61 n(n + 1)(2n + 1).

(6)

Using the standard results for

n

rr

1

and

n

rr

1

2 ,

(b) show that

n

rrr

1)3)(2( =

31 n(n2 + an + b),

where a and b are integers to be found.(5)

(c) Hence show that

n

nrrr

2

1)3)(2( =

31 n(7n2 + 27n + 26).

(3)


END


Paper Reference(s)

6667/01


Advanced/Advanced SubsidiaryMonday 31 January 2011 AfternoonTime: 1 hour 30 minutesMaterials required for examination Items included with question papersMathematical Formulae (Pink) Nil








N35406A 2

1. z = 5 3i, w = 2 + 2i

Express in the form a + bi, where a and b are real constants,

(a) z2,(2)

(b)wz .

(3)

2. A =

3502

, B =

2513

(a) Find AB.(3)

Given that

C =

1001

(b) describe fully the geometrical transformation represented by C,(2)

(c) write down C100.(1)

3. f(x) = 5x2 23

4x 6, x t 0.

The root D of the equation f (x) = 0 lies in the interval [1.6,1.8].

(a) Use linear interpolation once on the interval [1.6, 1.8] to find an approximation to D.Give your answer to 3 decimal places.

(4)

(b) Differentiate f(x) to find f x).(2)

(c) Taking 1.7 as a first approximation to D, apply the Newton-Raphson process once to f(x)to obtain a second approximation to D. Give your answer to 3 decimal places.

(4)

N35406A 3

4. Given that 2 4i is a root of the equation

z2 + pz + q = 0,

where p and q are real constants,

(a) write down the other root of the equation,(1)

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

5. (a) Use the results for

n

rr

1

,

n

rr

1

2 and

n

rr

1

3 , to prove that

n

rrrr

1)5)(1( =

41 n(n + 1)(n + 2)(n + 7)

for all positive integers n.(5)

(b) Hence, or otherwise, find the value of

50

20)5)(1(

rrrr .

(2)

N35406A 4

6.

Figure 1

Figure 1 shows a sketch of the parabola C with equation y2 = 36x.The point S is the focus of C.

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

(b) Write down the equation of the directrix of C.(1)

Figure 1 shows the point P which lies on C, where y > 0, and the point Q which lies on the directrix of C. The line segment QP is parallel to the x-axis.

Given that the distance PS is 25,

(c) write down the distance QP,(1)

(d) find the coordinates of P,(3)

(e) find the area of the trapezium OSPQ.(2)

N35406A 5

7. z = 24 7i

(a) Show z on an Argand diagram.(1)

(b) Calculate arg z, giving your answer in radians to 2 decimal places.(2)

It is given that

w = a + bi, a , b .

Given also that w = 4 and arg w =6

5S ,

(c) find the values of a and b,(3)

(d) find the value of zw.(3)

8. A =

3122

(a) Find det A.(1)

(b) Find A1.(2)

The triangle R is transformed to the triangle S by the matrix A.Given that the area of triangle S is 72 square units,

(c) find the area of triangle R.(2)

The triangle S has vertices at the points (0, 4), (8, 16) and (12, 4).

(d) Find the coordinates of the vertices of R.(4)

N35406A 6

9. A sequence of numbers u1, u2, u3, u4, . . ., is defined by

un + 1 = 4un + 2, u1 = 2.

Prove by induction that, for n +,

un =32 (4n 1).

(5)

10. The point P

tt 6,6 , t 0, lies on the rectangular hyperbola H with equation xy = 36.

(a) Show that an equation for the tangent to H at P is

y = xt 21 +

t12 .

(5)

The tangent to H at the point A and the tangent to H at the point B meet at the point 12).

(b) Find the coordinates of A and B.(7)


P38168A This publication may only be reproduced in accordance with Edexcel Limited copyright policy.2011 Edexcel Limited.

Paper Reference(s)


Advanced/ Advanced SubsidiaryWednesday 22 June 2011 MorningTime: 1 hour 30 minutes


Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical formulae stored in them.


Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics FP1), the paper reference (6667), your surname, initials and signature.





P38168A 2

1. f (x) = 3x + 3x 7

(a) Show that the equation f (x) = 0 has a root D between x =1 and x = 2.(2)

(b) Starting with the interval [1, 2], use interval bisection twice to find an interval of width 0.25 which contains D.

(3)

2. 1z = 2 + i

(a) Find the modulus of 1z .(1)

(b) Find, in radians, the argument of 1z , giving your answer to 2 decimal places.(2)

The solutions to the quadratic equation

z2 10z + 28 = 0

are 2z and 3z .

(c) Find 2z and 3z , giving your answers in the form p r iq, where p and q are integers.(3)

(d) Show, on an Argand diagram, the points representing your complex numbers 1z , 2z and 3z .(2))

P38168A 3

3. (a) Given that

A =

1221 ,

(i) find A2,

(ii) describe fully the geometrical transformation represented by A2.(4)

(b) Given that

B =

0110

,

describe fully the geometrical transformation represented by B.(2)

(c) Given that

C =

9121

kk

,

where k is a constant, find the value of k for which the matrix C is singular.(3)

4. f(x) = x2 +x2

5 3x 1, x z 0.

(a) Use differentiation to find f x).(2)

The root D of the equation f(x) = 0 lies in the interval [0.7, 0.9].

(b) Taking 0.8 as a first approximation to D, apply the Newton-Raphson process once to f(x) to obtain a second approximation to D. Give your answer to 3 decimal places.

(4)

P38168A 4

5. A =

24b

a, where a and b are constants.

Given that the matrix A maps the point with coordinates (4, 6) onto the point with coordinates

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

A quadrilateral R has area 30 square units.It is transformed into another quadrilateral S by the matrix A.Using your values of a and b,

(b) find the area of quadrilateral S.(4)

6. Given that z = x + iy, find the value of x and the value of y such that

z + 3iz* + 13i

where z* is the complex conjugate of z.(7)

7. (a) Use the results for

n

rr

1

and

n

rr

1

2 to show that

n

rr

1

2)12( =31 n(2n + 1)(2n 1)


(b) Hence show that

n

nrr

3

1

2)12( =32 n(an2 + b)

where a and b are integers to be found.(4)

P38168A 5


The point P(12t 2, 24t) is a general point on C.

(a) Find the equation of the directrix of C.(2)

(b) Show that the equation of the tangent to C at P(12t 2, 24t) is

x ty + 12t 2 = 0.(4)

The tangent to C at the point (3, 12) meets the directrix of C at the point X.

(c) Find the coordinates of X.(4)

9. Prove by induction, that for n +,

(a)n

1603

=

1)13(303

n

n

(6)

(b) f(n) = 72n 1 + 5 is divisible by 12.(6)


END


Paper Reference(s)

6667/01


Advanced SubsidiaryMonday 30 January 2012 AfternoonTime: 1 hour 30 minutesMaterials required for examination Items included with question papersMathematical Formulae (Pink) Nil








P40086A 2

1. Given that 1z =1 i,

(a) find arg ( 1z ).(2)

Given also that 2z = 3 + 4i, find, in the form a + ib, a, b ,

(b) 1z 2z ,(2)

(c)1

2

zz .

(3)

In part (b) and part (c) you must show all your working clearly.

2. (a) Show that f(x) = x4 + x 1 has a real root D in the interval [0.5, 1.0].(2)

(b) Starting with the interval [0.5, 1.0], use interval bisection twice to find an interval ofwidth 0.125 which contains D.

(3)

(c) Taking 0.75 as a first approximation, apply the Newton Raphson process twice to f(x) to obtain an approximate value of D. Give your answer to 3 decimal places.

(5)

3. A parabola C has cartesian equation y2 = 16x. The point P(4t2, 8t) is a general point on C.

(a) Write down the coordinates of the focus F and the equation of the directrix of C.(3)

(b) Show that the equation of the normal to C at P is y + tx = 8t + 4t3.(5)

P40086A 3

4. A right angled triangle T has vertices A(1, 1), B(2, 1) and C(2, 4). When T is transformed by

the matrix P =

0110

, the image is T .

(a) Find the coordinates of the vertices of T (2)

(b) Describe fully the transformation represented by P.(2)

The matrices Q =

1324

and R =

4321

represent two transformations. When T is

transformed by the matrix QR, the image is T s.

(c) Find QR.(2)

(d) Find the determinant of QR.(2)

(e) Using your answer to part (d), find the area of Ts.(3)

5. The roots of the equation

z3 8z2 + 22z 20 = 0

are 1z , 2z and 3z .

(a) Given that 1z = 3 + i, find 2z and 3z .(4)

(b) Show, on a single Argand diagram, the points representing 1z , 2z and 3z .(2)

6. (a) Prove by induction

n

rr

1

3 =41 n2(n + 1)2.

(5)(b) Using the result in part (a), show that

n

rr

1

3 )2( =41 n(n3 + 2n2 + n 8).

(3)

(c) Calculate the exact value of

50

20

3 )2(r

r .

(3)

P40086A 4

7. A sequence can be described by the recurrence formula

1nu = 2 nu + 1, n t 1, 1u = 1.

(a) Find 2u and 3u .(2)

(b) Prove by induction that nu = 2n 1.

(5)

8. A =

3210

.

(a) Show that A is non-singular.(2)

(b) Find B such that BA2 = A.(4)

9. The rectangular hyperbola H has cartesian equation xy = 9.

The points P

pp 3,3 and Q

qq 3,3 lie on H, where p q.

(a) Show that the equation of the tangent at P is x + p2y = 6p.(4)

(b) Write down the equation of the tangent at Q.(1)

The tangent at the point P and the tangent at the point Q intersect at R.

(c) Find, as single fractions in their simplest form, the coordinates of R in terms of p and q.(4)



Paper Reference(s)


Advanced SubsidiaryFriday 1 June 2012 MorningTime: 1 hour 30 minutes


Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical formulae stored in them.


Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics FP1), the paper reference (6667), your surname, initials and signature.





P40688A 2

1. f(x) = 2x3 6x2 7x 4.

(a) Show that f(4) = 0.(1)

(b) Use algebra to solve f(x) = 0 completely.(4)

2. (a) Given that

A =

554313

and B =

102111

,

find AB.(2)

(b) Given that

C =

6823

and D =

kk

425

, where k is a constant

and

E = C + D,

find the value of k for which E has no inverse.(4)

3. f(x) = x2 +x4

3 3x 7, x > 0.

A root D of the equation f(x) = 0 lies in the interval [3, 5].

Taking 4 as a first approximation to D, apply the Newton-Raphson process once to f(x) to obtain a second approximation to D . Give your answer to 2 decimal places.

(6)

P40688A 3

4. (a) Use the standard results for

n

rr

1

3 and

n

rr

1

to show that

n

rrr

1

3 )36( =41 n2(n + 2n + 13)


(b) Hence find the exact value of

30

16

3 )36(r

rr .

(2)

P40688A 4

5.

Figure 1

Figure 1 shows a sketch of the parabola C with equation y2 = 8x.The point P lies on C, where y > 0, and the point Q lies on C, where y < 0.The line segment PQ is parallel to the y-axis.

Given that the distance PQ is 12,

(a) write down the y-coordinate of P,(1)

(b) find the x-coordinate of P.(2)

Figure 1 shows the point S which is the focus of C.

The line l passes through the point P and the point S.

(c) Find an equation for l in the form ax + by + c = 0, where a, b and c are integers.(4)

P40688A 5

6. f(x) = tan

2x + 3x 6, S < x < S.

(a) Show that the equation f(x) = 0 has a root D in the interval [1, 2].(2)

(b) Use linear interpolation once on the interval [1, 2] to find an approximation to D.Give your answer to 2 decimal places.

(3)

7. z L.

(a) Calculate arg z, giving your answer in radians to 2 decimal places.(2)

Use algebra to express

(b) z + z2 in the form a + bL3, where a and b are integers,(3)

(c)17

zz in the form c + dLZKHUHc and d are integers.

(4)

Given that

w = O 3i,

where O is a real constant, and arg (4 5i + 3w) = 2S ,

(d) find the value of O.(2)

P40688A 6

8. The rectangular hyperbola H has equation xy = c2, where c is a positive constant.

The point P

tcct, , t , is a general point on H.

(a) Show that an equation for the tangent to H at P is

x + t 2 y = 2ct.(4)

The tangent to H at the point P meets the x-axis at the point A and the y-axis at the point B.

Given that the area of the triangle OAB, where O is the origin, is 36,

(b) find the exact value of c, expressing your answer in the form k where k is an integer.(4)

P40688A 7

9. M =

5243

.

(a) Find det M.(1)

The transformation represented by M maps the point S(2a 7, a 1), where a is a constant, onto the point S c(25, 14).

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

The point R has coordinates (6, 0).

Given that O is the origin,

(c) find the area of triangle ORS.(2)

Triangle ORS is mapped onto triangle OR'S ' by the transformation represented by M.

(d) Find the area of triangle OR'S '.(2)

Given that

A =

0110

(e) describe fully the single geometrical transformation represented by A.(2)

The transformation represented by A followed by the transformation represented by B is equivalent to the transformation represented by M.

(f) Find B.(4)

10. Prove by induction that, for n +,

f(n) = 22n 1 + 32n 1

is divisible by 5.(6)


END


Paper Reference(s)


Advanced Subsidiary/Advanced LevelMonday 28 January 2013 MorningTime: 1 hour 30 minutes






A booklet Mathematical Formulae and Statistical Tables is provided.Full marks may be obtained for answers to ALL questions.The marks for the parts of questions are shown in round brackets, e.g. (2).There are 9 questions in this question paper. The total mark for this paper is 75.



P41485A 2

1. Show, using the formulae for

n

rr

1

and

n

rr

1

2 , that

n

rr

1

2)12(3 = n(2n + 1)(2n 1), for all positive integers n.

(5)

2. z =i43

50

.

Find, in the form a + ib where a, b ,

(a) z,(2)

(b) z2.(2)

Find

(c) ~z~,(2)

(d) arg z2, giving your answer in degrees to 1 decimal place.(2)

3. f(x) = 21

2x + 21

x 5, x > 0.

(a) Find f x).(2)

The equation f(x) = 0 has a root D in the interval [4.5, 5.5].

(b) Using x0 = 5 as a first approximation to D, apply the Newton-Raphson procedure once to f(x)to find a second approximation to D, giving your answer to 3 significant figures.

(4)

P41485A 3 Turn over

4. The transformation U, represented by the 2 u 2 matrix P, is a rotation through 90 anticlockwise about the origin.

(a) Write down the matrix P.(1)

The transformation V, represented by the 2 2 matrix Q, is a reflection in the line y x.

(b) Write down the matrix Q.(1)

Given that U followed by V is transformation T, which is represented by the matrix R,

(c) express R in terms of P and Q,(1)

(d) find the matrix R,(2)

(e) give a full geometrical description of T as a single transformation.(2)

5. f(x) = (4x2 + 9)(x2 x + 34).

(a) Find the four roots of f (x) = 0.

Give your answers in the form x = p + iq , where p and q are real.(5)

(b) Show these four roots on a single Argand diagram.(2)

P41485A 4

6. X =

231 a

, where a is a constant.

(a) Find the value of a for which the matrix X is singular.(2)

Y =

2311

.

(b) Find Y.(2)

The transformation represented by Y maps the point A onto the point B.

Given that B has coordinates (1 O , 7O 2), where O is a constant,

(c) find, in terms of O, the coordinates of point A.(4)

7. The rectangular hyperbola, H, has cartesian equation xy = 25.

The point P

pp 5,5 and the point Q

qq 5,5 , where p, q z 0, p z q, are points on the

rectangular hyperbola H.

(a) Show that the equation of the tangent at point P is

p2y + x = 10p.(4)

(b) Write down the equation of the tangent at point Q.(1)

The tangents at P and Q meet at the point N.

Given p + q z 0,

(c) show that point N has coordinates

qpqppq 10,10 .

(4)

The line joining N to the origin is perpendicular to the line PQ.

(d) Find the value of p2q2.(5)

P41486A 5

8. (a) Prove by induction that, for n +,

n

rrr

1)3( =

31 n(n + 1)(n + 5).

(6)

(b) A sequence of positive integers is defined by

u1 = 1,

un + 1 = un + n(3n + 1), n +.

Prove by induction thatun = n2(n 1) + 1, n +.

(5)

9.

Figure 1

Figure 1 shows a sketch of part of the parabola with equation y2 = 36x.

The point P (4, 12) lies on the parabola.

(a) Find an equation for the normal to the parabola at P.(5)

This normal meets the x-axis at the point N and S is the focus of the parabola, as shown inFigure 1.

(b) Find the area of triangle PSN.(4)


END

P42828AThis publication may only be reproduced in accordance with Edexcel Limited copyright policy.2013 Edexcel Limited.

Paper Reference(s)

6667/01REdexcel GCEFurther Pure Mathematics FP1 (R)

Advanced/Advanced SubsidiaryMonday 10 June 2013 MorningTime: 1 hour 30 minutes


Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation or symbolic differentiation/integration, or haveretrievable mathematical formulae stored in them.

This paper is strictly for students outside the UK.

Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature.Check that you have the correct question paper.Answer ALL the questions.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for CandidatesA booklet Mathematical Formulae and Statistical Tables is provided.Full marks may be obtained for answers to ALL questions.The marks for the parts of questions are shown in round brackets, e.g. (2).There are 10 questions in this question paper. The total mark for this paper is 75.There are 36 pages in this question paper. Any blank pages are indicated.

Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.

P42828A 2

1. The complex numbers z and w are given by

z = 8 + 3i, w = 2i

Express in the form a + bi, where a and b are real constants,

(a) z w,(1)

(b) zw.(2)

2. (i)2 1

3 5k k

A , where k is a constant

Given that

B = A + 3I

where I is the 2 2 identity matrix, find

(a) B in terms of k,(2)

(b) the value of k for which B is singular.(2)

(ii) Given that

234

C , D = (2 1 5)

and

E = CD

find E.(2)

P42828A 3

3. f(x) = 4 31 32

x x x

(a) Show that the equation f(x) = 0 has a root between x = 2 and x = 2.5.(2)

(b) Starting with the interval [2, 2.5] use interval bisection twice to find an interval of width 0.125 which contains .

(3)

The equation f(x) = 0 has a root in the interval [2, 1].

(c) Taking 1.5 as a first approximation to , apply the Newton-Raphson process once to f(x) to obtain a second approximation to .Give your answer to 2 decimal places.

(5)

4. f(x) = (4x2 +9)(x2 2x + 5)

(a) Find the four roots of f(x) = 0.(4)

(b) Show the four roots of f(x) = 0 on a single Argand diagram.(2)

P42828A 4

5.

Figure 1

Figure 1 shows a rectangular hyperbola H with parametric equations

x = 3t, y = 3t

, t

The line L with equation 6y = 4x 15 intersects H at the point P and at the point Q as shown in Figure 1.

(a) Show that L intersects H where 4t2 5t 6 = 0.(3)

(b) Hence, or otherwise, find the coordinates of points P and Q.(5)

P42828A 5

6.0 11 0

A ,

2 31 4

B

The transformation represented by B followed by the transformation represented by A is equivalent to the transformation represented by P.

(a) Find the matrix P.(2)

Triangle T is transformed to the triangle T by the transformation represented by P.

Given that the area of triangle T is 24 square units,

(b) find the area of triangle T.(3)

Triangle T is transformed to the original triangle T by the matrix represented by Q.

(c) Find the matrix Q.(2)

P42828A 6

7. The parabola C has equation y2 = 4ax, where a is a positive constant.

The point P(at2, 2at) is a general point on C.

(a) Show that the equation of the tangent to C at P(at2, 2at) is

ty = x + at2(4)

The tangent to C at P meets the y-axis at a point Q.

(b) Find the coordinates of Q.(1)

Given that the point S is the focus of C,

(c) show that PQ is perpendicular to SQ.(3)

8. (a) Prove by induction, that for n ,

1

1(2 1) ( 1)(4 1)6

n

rr r n n n

(6)

(b) Hence, show that

32

1

1(2 1) ( )3

n

r nr r n an bn c

where a, b and c are integers to be found.(4)

P42828A 7

9. The complex number w is given by

w = 10 5i

(a) Find w .(1)

(b) Find arg w, giving your answer in radians to 2 decimal places(2)

The complex numbers z and w satisfy the equation

(2 + i)(z + 3i) = w

(c) Use algebra to find z, giving your answer in the form a + bi,where a and b are real numbers.

(4)

Given that

arg( + 9i + w) = 4S

where is a real constant,

(d) find the value of .(2)

P42828A 8

10. (i) Use the standard results for 31

n

rr

and

1

n

rr

to evaluate

243

1( 4 )

rr r

(2)

(ii) Use the standard results for 21

n

rr

and

1

n

rr

to show that

2

0

1( 2 2 1) ( 1)( )( )6

n

rr r n n n a bn c

for all integers n ZKHUHa, b and c are constant integers to be found.(6)


END

P43138AThis publication may only be reproduced in accordance with Edexcel Limited copyright policy.2013 Edexcel Limited.

Paper Reference(s)


Advanced/Advanced SubsidiaryMonday 10 June 2013 MorningTime: 1 hour 30 minutes


Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation or symbolic differentiation/integration, or haveretrievable mathematical formulae stored in them.

Instructions to CandidatesIn the boxes above, write your centre number, candidate number, your surname, initials and signature.Check that you have the correct question paper.Answer ALL the questions.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for CandidatesA booklet Mathematical Formulae and Statistical Tables is provided.Full marks may be obtained for answers to ALL questions.The marks for the parts of questions are shown in round brackets, e.g. (2).There are 9 questions in this question paper. The total mark for this paper is 75.There are 32 pages in this question paper. Any blank pages are indicated.

Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.

P43138A 2

1.

M =2

3 6 4 11x x

x x

Given that the matrix M is singular, find the possible values of x.(4)

2. f(x) = cos(x2) x + 3, 0 < x <

(a) Show that the equation f(x) = 0 has a root in the interval [2.5, 3].(2)

(b) Use linear interpolation once on the interval [2.5, 3] to find an approximation for ,giving your answer to 2 decimal places.

(3)

3. Given that x = 12

is a root of the equation

2x3 9x2 + kx 13 = 0, k

find

(a) the value of k,(3)

(b) the other 2 roots of the equation.(4)

4. The rectangular hyperbola H has Cartesian equation xy = 4.

The point 22 ,P tt

lies on H, where t .

(a) Show that an equation of the normal to H at the point P is

ty t3x = 2 2t4(5)

The normal to H at the point where t = 12

meets H again at the point Q.

(b) Find the coordinates of the point Q.(4)

P43138A 3

5. (a) Use the standard results for1

n

rr

and 2

1

n

rr

to show that

2

1

1( 2)( 3) ( 9 26)3

n

rr r n n n

for all positive integers n.

(6)

(b) Hence show that

32

1

2( 2)( 3) ( )3

n

r nr r n an bn c

where a, b and c are integers to be found.(4)

6. A parabola C has equation y2 = 4ax, a > 0

The points P(ap2, 2ap) and Q(aq2, 2aq) lie on C, where p q p q.

(a) Show that an equation of the tangent to the parabola at P is

py x = ap2(4)

(b) Write down the equation of the tangent at Q.(1)

The tangent at P meets the tangent at Q at the point R.

(c) Find, in terms of p and q, the coordinates of R, giving your answers in their simplestform.

(4)

Given that R lies on the directrix of C,

(d) find the value of pq.(2)

P43138A 4

7. z1 = 2 + 3i, z2 = 3 + 2i, z3 = a + bi, a, b

(a) Find the exact value of |z1 + z2|.(2)

Given that w = 1 32

z zz

,

(b) find w in terms of a and b, giving your answer in the form x + iy, x, y .(4)

Given also that w = 17 7 i13 13

,

(c) find the value of a and the value of b,(3)

(d) find arg w, giving your answer in radians to 3 decimal places.(2)

8.

A =6 24 1

and I is the 2 2 identity matrix.

(a) Prove that

A2 = 7A + 2I(2)

(b) Hence show that

A1 = 12

(A 7I)

(2)

The transformation represented by A maps the point P onto the point Q.

Given that Q has coordinates (2k + 8, 2k 5), where k is a constant,

(c) find, in terms of k, the coordinates of P.(4)

P43138A 5

9. (a) A sequence of numbers is defined by

u1 = 8

un + 1 = 4un 9n,n 1

Prove by induction that, for n ,

un = 4n + 3n +1(5)

(b) Prove by induction that, for m ,

3 4 2 1 41 1 1 2

m m mm m

(5)


END

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