ADDITIONAL MATHEMATICS

Form 5

PAPER 02 2 Hours 30 Minutes

INSTRUCTION TO CANDIDATES 1. Do not open this examination paper until you are instructed to do so. 2. This paper consists of six (6) questions. Answer all questions 3. Write all your solutions with full working in the booklet provided

1. (a)

(b)

A function f is defined on domain by i. State the value of x for which the function f is not defined. ii. Find the value of x for which ( ) . -1 iii. Determine the inverse function f . iv. State the domain of f -1.

[1 mark] [2 marks] [3 marks] [2 marks]

The expression – has a factor x – 2 and leaves a remainder of 75 when divided by x + 3. Find the value of a, and of b. [5 marks] Total 13 marks

2. (a)

Let ( ) i.

Express f(x) the in the form (

)

Determine ii. The turning point and state whether it is a maximum or minimum iii.

The maximum or minimum value of f(x)

[3 marks]

[1 mark] [1 mark]

(b)

Find the values of x for which

[4 marks]

(c)

Given the series i. Show that this series is geometric,

[3 marks]

ii.

Hence find the sum to infinity of this series

[2 marks] Total 14 marks

3. (a)

(b)

P and Q are the points (2,4) and (8,7) respectively i. ii.

State the position vector of the point P and the point Q. Find the displacement vector PQ .

[2 marks] [2 marks]

iii.

Calculate the magnitude of the displacement vector PQ .

[2 marks]

iv.

Find the unit vector in the direction PQ

[5 marks]

the equation of a circle is: (x + 5)² + (y - 6)² = 50. P (2, 7) is a point on the circle. Find the equation of i. ii.

the tangent at P the normal at P

(

[8 marks] Total 19marks

)

4. (a)

Solve the equation

(b)

Solve the equation

©

Show that

(d)

Solve the simultaneous equations -x – y = -1 y = 2x² - 3x + 1

( – )

34 y 34 y 3 37 y 33 y 6

1 1 – 2cosec cot 1 – cos 1 cos

[3 marks]

[3 marks]

[4 marks]

[5marks] Total 15marks

5. (a)

the diagram below shows a circle of centre O and radius 15cm. the sector AOB subtends the angle at the centre. 3 A

O

3

B

Working in radians, calculate, giving your answer in terms of π i. the length of the minor arc AB ii. the area of the minor sector OAB

(b)

i).

( ii).

√

Given that √

)

where x is acute, show that

(

Using the fact that find the exact value of in your working.

©

Show that

–

–

[1 mark] [2 marks]

) √

and

[4 marks] ,

showing ALL steps [3 marks]

[4 marks] Total 14 marks

6. (a)

Differentiate the following with respect to x, simplify your result as far as possible

y 5 2 x 1 x

(b)

Find the gradient of

4

y 1 4 x at the point

on the curve where x

3

1 . 2

[3marks]

dy for y 2sin 5 x 5cos 2 x dx

©

Find

(d)

The volume of water V(m3) flowing through a pipe is related to time t(seconds) by the equation V= 5t + 2. i. ii.

[4marks]

[4 marks]

Calculate the initial volume of water flowing through the pipe Find the rate of change of V

[1 marks] [3 marks]

Total 15 marks

End of Examination Total 90 marks

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