*X7477711*©

NationalQualications2018 AH

Total marks — 100

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 answer booklet provided. In the answer booklet 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.

X747/77/11 Mathematics

THURSDAY, 3 MAY

9:00 AM – 12:00 NOON

A/PB

page 02

FORMULAE LIST

Standard derivatives Standard integrals

( )f x ( )f x′ ( )f x ( )f x dx∫

sin 1 x−2

1

1 x− ( )sec2 ax tan( )1 ax ca

+

cos 1 x− 2

1

1 x−

− 2 2

1

a x−sin 1 x c

a− ⎛ ⎞ +⎜ ⎟⎝ ⎠

tan 1 x−2

11 x+ 2 2

1a x+

tan 11 x ca a

− ⎛ ⎞ +⎜ ⎟⎝ ⎠

tan x sec2 x1x ln | |x c+

cot x cosec2 x− axe 1 axe ca

+

sec x sec tanx x

cosec x cosec cotx x−

ln x1x

xe xe

Summations

(Arithmetic series) ( )12 1

2nS n a n d= + −⎡ ⎤⎣ ⎦

(Geometric series) ( )

,1

11

n

n

a rS r

r−

= ≠−

( ) ( )( ) ( ), ,2 2

2 3

1 1 1

1 1 1 12 6 4

2n n n

r r r

n n n n n n nr r r= = =

+ + + += = =∑ ∑ ∑

Binomial theorem

( )0

nn n r r

r

na b a b

r−

=

⎛ ⎞+ = ⎜ ⎟

⎝ ⎠∑ where

!!( )!

nr

n nCr r n r

⎛ ⎞= =⎜ ⎟ −⎝ ⎠

Maclaurin expansion

( ) ( ) ( )( ) ( ) ( ) ...! ! !

2 3 40 0 00 0

2 3 4

ivf x f x f xf x f f x′′ ′′′′= + + + + +

page 03

FORMULAE LIST (continued)

De Moivre’s theorem

[ ] ( )(cos sin ) cos sinn nr θ i θ r nθ i nθ+ = +

Vector product

ˆsin 2 3 1 3 1 21 2 3

2 3 1 3 1 21 2 3

a a a a a aθ a a a

b b b b b bb b b

× = = = − +i j k

a b a b n i j k

Matrix transformation

Anti-clockwise rotation through an angle, θ, about the origin,cos sinsin cos

θ θθ θ

−⎛ ⎞⎜ ⎟⎝ ⎠

[Turn over

page 04

MARKSTotal marks – 100

Attempt ALL questions

1. (a) Given ( ) sinf x x−= 13 , find ( )f x′ .

(b) Differentiate xey

x=

+

5

7 1.

(c) For cosy x y x+ =2 6 , use implicit differentiation to find dydx

.

2. Use partial fractions to find x dx

x x−

− −⌠⎮⌡ 2

3 7

2 15.

3. (a) Write down and simplify the general term in the binomial expansion of

xx

⎛ ⎞+⎜ ⎟⎝ ⎠

9

2

52 .

(b) Hence, or otherwise, find the term independent of x.

4. Given that z i= +1 2 3 and , ,z p i p= − ∈2 6 find:

(a) z z1 2;

(b) the value of p such that z z1 2 is a real number.

5. Use the Euclidean algorithm to find integers a and b such that a b+ =306 119 17.

2

2

4

4

3

2

2

1

4

page 05

MARKS

6. On a suitable domain, a curve is defined parametrically by x t= +2 1 and ( )lny t= +3 2 .

Find the equation of the tangent to the curve where t = − 1

3.

7. Matrices C and D are given by:

C−⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟−⎝ ⎠

2 1 2

1 1 0

1 0 1

and D k⎛ ⎞⎜ ⎟= +⎜ ⎟⎜ ⎟⎝ ⎠

1 1 2

3 0 2

1 1 1

, where k∈ .

(a) Obtain C D−′2 where C′ is the transpose of C.

(b) (i) Find and simplify an expression for the determinant of D.

(ii) State the value of k such that 1D− does not exist.

8. Using the substitution sinu θ= , or otherwise, evaluate

sin cosθ θ dθ

π

π∫2

4

6

2 .

9. Prove directly that:

(a) the sum of any three consecutive integers is divisible by 3;

(b) any odd integer can be expressed as the sum of two consecutive integers.

[Turn over

5

2

2

1

4

2

1

page 06

MARKS

10. Given z x iy= + , sketch the locus in the complex plane given by z z i= − +2 2 .

11. (a) Obtain the matrix, A, associated with an anticlockwise rotation of π3

radians about the origin.

(b) Find the matrix, B, associated with a reflection in the x-axis.

(c) Hence obtain the matrix, P, associated with an anticlockwise rotation of π3

radians about the origin followed by reflection in the x-axis, expressing your answer using exact values.

(d) Explain why matrix P is not associated with rotation about the origin.

12. Prove by induction that, for all positive integers n,

( )n

r n

r

−

== −∑ 1

1

13 3 1

2.

3

1

1

2

1

5

page 07

MARKS 13. An engineer has designed a lifting device. The handle turns a screw which shortens

the horizontal length and increases the vertical height.

The device is modelled by a rhombus, with each side 25 cm.

The horizontal length is x cm, and the vertical height is h cm as shown.

h

x

25 cm

(a) Show that h x= − 22500 .

(b) The horizontal length decreases at a rate of 0·3 cm per second as the handle is turned.

Find the rate of change of the vertical height when x = 30.

[Turn over

1

5

page 08

MARKS

14. A geometric sequence has first term 80 and common ratio 1

3.

(a) For this sequence, calculate:

(i) the 7th term;

(ii) the sum to infinity of the associated geometric series.

The first term of this geometric sequence is equal to the first term of an arithmetic sequence.

The sum of the first five terms of this arithmetic sequence is 240.

(b) (i) Find the common difference of this sequence.

(ii) Write down and simplify an expression for the nth term.

Let Sn represent the sum of the first n terms of this arithmetic sequence.

(c) Find the values of n for which nS = 144.

15. (a) Use integration by parts to find sinx x dx∫ 3 .

(b) Hence find the particular solution of

sin ,dy y x x xdx x

− = ≠323 0

given that x = π when y = 0.

Express your answer in the form ( )y f x= .

2

2

2

1

3

3

7

page 09

MARKS 16. Planes 1π , 2π and 3π have equations:

:::

1

2

3

2 4

3 5 2 1

7 11 11

π x y zπ x y zπ x y az

− + = −− − =

− + + = −

where a∈ .

(a) Use Gaussian elimination to find the value of a such that the intersection of the planes 1π , 2π and 3π is a line.

(b) Find the equation of the line of intersection of the planes when a takes this value.

The plane 4π has equation x y z− + + =9 15 6 20.

(c) Find the acute angle between 1π and 4π .

(d) Describe the geometrical relationship between 2π and 4π .

Justify your answer.

17. (a) Given ( ) xf x e= 2 , obtain the Maclaurin expansion for ( )f x up to, and including,

the term in x3.

(b) On a suitable domain, let ( ) tang x x= .

(i) Show that the third derivative of ( )g x is given by

( ) sec tan secg x x x x= +′′′ 4 2 22 4 .

(ii) Hence obtain the Maclaurin expansion for ( )g x up to and including the term in x3.

(c) Hence, or otherwise, obtain the Maclaurin expansion for tanxe x2 up to, and including, the term in x3.

(d) Write down the first three non-zero terms in the Maclaurin expansion for tan secx xe x e x+2 2 22 .

[END OF QUESTION PAPER]

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