ational 4XDOLÛFDWLRQV 2017 - Scottish … 2017 Total marks — 60 Attempt ALL questions. You may...

*X7477611*©

NationalQualications2017 H

Total marks — 60

Attempt ALL questions.

You may NOT use a calculator.

Full credit will be given only to solutions which contain appropriate working.

State the units for your answer where appropriate.

Answers obtained by readings from scale drawings will not receive any credit.

Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space.

Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting.

Use blue or black ink.

Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper.

X747/76/11 MathematicsPaper 1

(Non-Calculator)FRIDAY, 5 MAY

9:00 AM – 10:10 AM

A/PB

FORMULAE LIST

Circle:

The equation 2 2 2 2 0x y gx fy c+ + + + = represents a circle centre (−g, −f ) and radius 2 2g f c+ − .

The equation ( ) ( )2 2 2x a y b r− + − = represents a circle centre (a, b) and radius r.

Scalar Product: cos , where is the angle between and θ θ=a.b a b a b

or 1 1

1 1 2 2 3 3 2 2

3 3

where and .a b

a b a b a b a ba b

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= + + = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

a.b a b

Trigonometric formulae:

sin (A B) sin A cos B cos A sin B± = ±

cos (A B) cos A cos B sin A sin B± = ∓

sin A sin A cos A2 2=

cos A cos A sin A2 22 = −

cos A22 1= −

sin A21 2= −

Table of standard derivatives:( )f x ( )f x′

sin ax

cosax

cosa ax

sina ax−

Table of standard integrals:( )f x ( )f x dx∫

sin ax

cosax

cos1 ax ca

− +

sin1 ax ca

+

MARKSATTEMPT ALL QUESTIONS

Total marks — 60

1. Functions f and g are defined on suitable domains by ( ) 5f x x= and ( ) cos2g x x= .

(a) Evaluate ( )( )0f g .

(b) Find an expression for ( )( )g f x .

2. The point P (−2, 1) lies on the circle 2 2 8 6 15 0x y x y+ − − − = .

Find the equation of the tangent to the circle at P.

3. Given ( )124 1y x= − , find dydx

.

4. Find the value of k for which the equation ( )2 4 5 0x x k+ + − = has equal roots.

5. Vectors u and v are

5

1

1

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠

and

3

8

6

⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟⎝ ⎠

respectively.

(a) Evaluate u .v.

(b)

u

w3π

Vector w makes an angle of 3π

with u and 3=w .

Calculate u .w.

1

2

4

2

3

1

3

[Turn over

MARKS

6. A function, h, is defined by ( ) 3 7h x x= + , where x∈ .

Determine an expression for ( )1h x− .

7. A (−3, 5), B (7, 9) and C (2, 11) are the vertices of a triangle.

Find the equation of the median through C.

8. Calculate the rate of change of ( ) 12

d tt

= , t ≠ 0, when t = 5.

9. A sequence is generated by the recurrence relation 1 6n nu mu+ = + where m is a constant.

(a) Given u1 = 28 and u2 = 13, find the value of m.

(b) (i) Explain why this sequence approaches a limit as n→ ∞ .

(ii) Calculate this limit.

3

3

3

2

1

2

MARKS 10. Two curves with equations y = x3 − 4x2 + 3x + 1 and y = x2 − 3x + 1 intersect as shown

in the diagram.

y

xO 2

y = x3 − 4x2 + 3x + 1

y = x2 − 3x + 1

(a) Calculate the shaded area.

The line passing through the points of intersection of the curves has equation y = 1 − x.

y = x3 − 4x2 + 3x + 1

y = x2 − 3x + 1

y = 1 − x

y

xO 2

(b) Determine the fraction of the shaded area which lies below the line y = 1 − x.

[Turn over

5

4

MARKS 11. A and B are the points (−7, 2) and (5, a).

AB is parallel to the line with equation 3y − 2x = 4.

Determine the value of a.

12. Given that log log 136 4

2a a− = , find the value of a.

13. Find ( )

12

1

5 4dx

x−

⌠⎮⌡

, 54

x < .

14. (a) Express sin cos3 ° °x x− in the form ( )sin °k x a− ,where k > 0 and 0 < a < 360.

(b) Hence, or otherwise, sketch the graph with equationsin cos3 ° °y x x= − , 0 ≤ x ≤ 360.

Use the diagram provided in the answer booklet.

3

3

4

4

3

MARKS

15. A quadratic function, f, is defined on , the set of real numbers.

Diagram 1 shows part of the graph with equation ( )y f x= .

The turning point is (2, 3).

Diagram 2 shows part of the graph with equation ( )y h x= .

The turning point is (7, 6).

Diagram 2

(7, 6)y y

x xO O3 6 81

(2, 3)

Diagram 1

( )y f x=

( )y h x=

(a) Given that ( ) ( )h x f x a b= + + .

Write down the values of a and b.

(b) It is known that ( )3

14f x dx =∫ .

Determine the value of ( )8

6h x dx∫ .

(c) Given ( )1 6f ′ = , state the value of ( )8h′ .

[END OF QUESTION PAPER]

2

1

1

[BLANK PAGE]

DO NOT WRITE ON THIS PAGE

Write your answers clearly in the spaces provided in this booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space.

Additional space for answers is provided at the end of this booklet. If you use this space you must clearly identify the question number you are attempting.


Before leaving the examination room you must give this booklet to the Invigilator; if you do not you may lose all the marks for this paper.

FOR OFFICIAL USE

X747/76/01 Mathematics Paper 1 (Non-Calculator)Answer Booklet

Fill in these boxes and read what is printed below.

Full name of centre Town

Forename(s) Surname Number of seat

Day Month Year Scottish candidate number

*X7477601*

*X747760101*©

Mark

Date of birth

H NationalQualications2017

FRIDAY, 5 MAY

9:00 AM – 10:10 AM

A/PB

*X747760102*Page 02

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1.(a)

1.(b)

2.

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7.

8.

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9.(a)

9.(b) (i)

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10.(a)

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10.(b)

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11.

12.

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13.

14.(a)

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14.(b) An additional diagram, if required, can be found on Page 13.

30 x

y

4

3

2

1

0

−1

−2

−3

−4

60 90 120 150 180 210 240 270 300 330 360

*X747760112*Page 12

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15.(a)

15.(b)

15.(c)

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QUESTIONNUMBER ADDITIONAL SPACE FOR ANSWERS

30 x

y

4

3

2

1

0

−1

−2

−3

−4

60 90 120 150 180 210 240 270 300 330 360

Additional diagram for Question 14(b).

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For Marker’s Use

Quest ion No Marks/Grades

*X7477612*©

NationalQualications2017 H

Total marks — 70

Attempt ALL questions.

You may use a calculator.

Full credit will be given only to solutions which contain appropriate working.

State the units for your answer where appropriate.

Answers obtained by readings from scale drawings will not receive any credit.

Write your answers clearly in the spaces provided in the answer booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space.

Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting.


Before leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper.

X747/76/12 MathematicsPaper 2

FRIDAY, 5 MAY

10:30 AM – 12:00 NOON

A/PB

FORMULAE LIST

Circle:

The equation 2 2 2 2 0x y gx fy c+ + + + = represents a circle centre (−g, −f ) and radius 2 2g f c+ − .

The equation ( ) ( )2 2 2x a y b r− + − = represents a circle centre (a, b) and radius r.

Scalar Product: cos , where is the angle between and θ θ=a.b a b a b

or 1 1

1 1 2 2 3 3 2 2

3 3

where and .a b

a b a b a b a ba b

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= + + = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

a.b a b

Trigonometric formulae:

sin (A B) sin A cos B cos A sin B± = ±

cos (A B) cos A cos B sin A sin B± = ∓

sin A sin A cos A2 2=

cos A cos A sin A2 22 = −

cos A22 1= −

sin A21 2= −

Table of standard derivatives:( )f x ( )f x′

sin ax

cosax

cosa ax

sina ax−

Table of standard integrals:( )f x ( )f x dx∫

sin ax

cosax

cos1 ax ca

− +

sin1 ax ca

+

MARKSAttempt ALL questions

Total marks — 70

1. Triangle ABC is shown in the diagram below.

The coordinates of B are (3,0) and the coordinates of C are (9,−2).

The broken line is the perpendicular bisector of BC.

y

x

A

B

C

O

(a) Find the equation of the perpendicular bisector of BC.

(b) The line AB makes an angle of 45° with the positive direction of the x-axis.

Find the equation of AB.

(c) Find the coordinates of the point of intersection of AB and the perpendicular bisector of BC.

2. (a) Show that ( )1x − is a factor of ( ) 3 22 5 2f x x x x= − + + .

(b) Hence, or otherwise, solve ( ) 0f x = .

[Turn over

4

2

2

2

3

MARKS

3. The line y = 3x intersects the circle with equation ( ) ( )2 22 1 25x y− + − = .

y

O x

Find the coordinates of the points of intersection.

4. (a) Express 23 24 50x x+ + in the form ( )2a x b c+ + .

(b) Given that ( ) 3 212 50 11f x x x x= + + − , find ( )f x′ .

(c) Hence, or otherwise, explain why the curve with equation ( )y f x= is strictly increasing for all values of x.

5

3

2

2

MARKS

5. In the diagram, PR 9 5 2= + +i j k and RQ 12 9 3= − − +i j k.

Q

S

P

R

(a) Express PQ in terms of i, j and k.

The point S divides QR in the ratio 1:2.

(b) Show that PS 4= − +i j k.

(c) Hence, find the size of angle QPS.

6. Solve sin cos5 4 2 2x x− = for 0 2x≤ < π .

7. (a) Find the x-coordinate of the stationary point on the curve

with equation 36 2y x x= − .

(b) Hence, determine the greatest and least values of y in the interval 1 9x≤ ≤ .

[Turn over

2

2

5

5

4

3

MARKS 8. Sequences may be generated by recurrence relations of the form

1 20n nu k u+ = − , 0 5u = where k∈R.

(a) Show that 225 20 20u k k= − − .

(b) Determine the range of values of k for which 2 0u u< .

9. Two variables, x and y, are connected by the equation ny kx= .

The graph of log2 y against log2 x is a straight line as shown.

O

3

−12

log2 y

log2 x

Find the values of k and n.

2

4

5

MARKS

10. (a) Show that the points A(−7, −2), B(2, 1) and C(17, 6) are collinear.

Three circles with centres A, B and C are drawn inside a circle with centre D as shown.

AB

CD

The circles with centres A, B and C have radii rA, rB and rC respectively.

• A 10r =

• B A2r r=

• C A Br r r= +

(b) Determine the equation of the circle with centre D.

11. (a) Show that sin sin cos sincos

2 32

2

x x x xx

− = , where 02

x π< < .

(b) Hence, differentiate sin sin coscos

22

2

x x xx

− , where 02

x π< < .

[END OF QUESTION PAPER]

3

4

3

3

[BLANK PAGE]

DO NOT WRITE ON THIS PAGE

Write your answers clearly in the spaces provided in this booklet. The size of the space provided for an answer should not be taken as an indication of how much to write. It is not necessary to use all the space.

Additional space for answers is provided at the end of this booklet. If you use this space you must clearly identify the question number you are attempting.


Before leaving the examination room you must give this booklet to the Invigilator; if you do not you may lose all the marks for this paper.

FOR OFFICIAL USE

X747/76/02 Mathematics Paper 2Answer Booklet

Fill in these boxes and read what is printed below.

Full name of centre Town

Forename(s) Surname Number of seat

Day Month Year Scottish candidate number

*X7477602*

*X747760201*©

Mark

Date of birth

H NationalQualications2017

FRIDAY, 5 MAY

10:30 AM – 12:00 NOON

A/PB

*X747760202*Page 02

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1.(a)

1.(b)

1.(c)

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4.(a)

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ational 4XDOLÛFDWLRQV 2017 - Scottish … 2017 Total marks — 60 Attempt ALL questions. You may...

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