*X7477711*©

NationalQualications2016 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, 12 MAY

9:00 AM – 12:00 NOON

A/HTP

Page 02

FORMULAE LIST

Standard derivatives Standard integrals

f x( ) ′ ( )f x f x( ) f x dx( )∫

sin−1 x2

1

1 x−sec2 ax( )

1a

ax ctan( ) +

cos−1 x 2

1

1 x−

−1

2 2a x−sin−

+1 xa

c

tan−1 x1

1 2+ x1

a x2 2+1 1

axa

ctan−

+

tan x sec2 x1x ln| |+x c

cot x cosec2− x axe1a

e cax +

sec x sec tanx x

cosec x cosec cot− x x

ln x1x

ex ex

Summations

(Arithmetic series) S n a n dn = + −( ) 12

2 1

(Geometric series) Sa r

rn

n

=−( )−

1

1

r n n r n n n r n n

r

n

r

n

r

n

= = =∑ ∑ ∑= + = + + = +

1

2

1

32 2

1

12

1 16

14

2( ) , ( )( ), ( )

Binomial theorem

a bnr

a bn

r

nn r r+( ) =

=

−∑0

where nr

C nr n r

nr

= =−!

!( )!

Maclaurin expansion

f x f f x f x f x f xiv

( ) ( ) ( ) ( )!

( )!

( )!

...= + ′ + ′′ + ′′′ + +0 00

2

0

3

0

4

2 3 4

Page 03

FORMULAE LIST (continued)

De Moivre’s theorem

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

Vector product

ˆ× sin 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) Differentiate tan .12y x x−=

(b) Given ( )2

2

1

1 4

xf xx

−=+

, find f ′(x), simplifying your answer.

(c) A curve is given by the parametric equations

x = 6t and y = 1 − cos t.

Find dydx

in terms of t.

2. A geometric sequence has second and fifth terms 108 and 4 respectively.

(a) Calculate the value of the common ratio.

(b) State why the associated geometric series has a sum to infinity.

(c) Find the value of this sum to infinity.

3. Write down and simplify the general term in the binomial expansion of 133

2xx

⎛ ⎞−⎜ ⎟⎝ ⎠

.

Hence, or otherwise, find the term in x9.

4. Below is a system of equations:

x y zx y z

x y λz

+ + =− + =

− + =

2 3 3

2 4 5

3 2 2

Use Gaussian elimination to find the value of λ which leads to redundancy.

5. Prove by induction that

( ) ( )2

1

3 1 1n

rr r n n

=

− = +∑ , n∀ ∈ .

3

3

2

3

1

2

5

4

4

Page 05

MARKS 6. Find Maclaurin expansions for sin 3x and e4x up to and including the term in x3.

Hence obtain an expansion for e4x sin 3x up to and including the term in x3.

7. A is the matrix 2 0

1λ⎛ ⎞⎜ ⎟−⎝ ⎠

.

(a) Find the determinant of matrix A.

(b) Show that A2 can be expressed in the form pA + qI , stating the values of p and q.

(c) Obtain a similar expression for A4.

8. Let 3z i= − .

(a) Plot z on an Argand diagram.

(b) Let w = az where a > 0 , a ∈ .

Express w in polar form.

(c) Express w8 in the form ( )nka x i y+ where k, x, y ∈.

9. Obtain ( )ln 27x x dx∫ .

10. For each of the following statements, decide whether it is true or false.

If true, give a proof; if false, give a counterexample.

A. If a positive integer p is prime, then so is 2p + 1.

B. If a positive integer n has remainder 1 when divided by 3, then n3 also has remainder 1 when divided by 3.

11. The height of a cube is increasing at the rate of 5 cm s−1.

Find the rate of increase of the volume when the height of the cube is 3 cm.

[Turn over

6

1

3

2

1

2

3

6

4

4

Page 06

MARKS

12. Below is a diagram showing the graph of a linear function, ( )y f x= .

y

xc

−c

O

On separate diagrams show:

(a) ( )y f x c= −

(b) ( )y f x= 2

13. Express ( )( )

3 32

4 6

xx x

++ −

in partial fractions and hence evaluate

( ) ( )x dx

x x+

+ −⌠⎮⌡

4

3

3 32

4 6.

Give your answer in the form ln pq

⎛ ⎞⎜ ⎟⎝ ⎠

.

2

2

9

Page 07

MARKS 14. Two lines L1 and L2 are given by the equations:

, ,L x λ y λ z λ

x y zL

= + = + = −

− − += =−

1

2

4 3 2 4 7

3 8 11 32

:

:

(a) Show that the lines L1 and L2 intersect and find the point of intersection.

(b) Calculate the obtuse angle between the lines L1 and L2.

15. Solve the differential equation

22

2 5 6 12 2 5d y dy y x xdx dx

+ + = + −

given y = −6 and 3dydx

= , when x = 0.

16. A beaker of liquid was placed in a fridge.

The rate of cooling is given by

( ), 0,FdT k T T kdt

= − − >

where TF is the constant temperature in the fridge and T is the temperature of the liquid at time t.

• The constant temperature in the fridge is 4 °C.

• When first placed in the fridge, the temperature of the liquid was 25 °C.

• At 12 noon, the temperature of the liquid was 9·8 °C.

• At 12:15 pm, the temperature of the liquid had dropped to 6·5 °C.

At what time, to the nearest minute, was the liquid placed in the fridge?

[END OF QUESTION PAPER]

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