Preliminary Examination: Mathematics Paper II

PAPER 2

TOTAL: 150 TIME: 3 HOURS

SECTION A

QUESTION 1:

(a) In the diagram, B is the midpoint of line segment OA.

Determine:

(1) the co-ordinates of B. (2)

(2) the gradient of OA. (2)

(3) the inclination of OA. (2)

(4) the gradient of PQ. (2)

(b) Would it be possible to find a circle centre (3; -2) that would have points (1; 2) and (1,5; 1,5) on its circumference? Explain why or why not. (3)

at (4)

QUESTION 2:

ANSWER THIS QUESTION ON THE SEPARATE SHEET PROVIDED AND HAND THIS IN WITH YOUR ANSWER BOOKLET.

ΔABC has co-ordinates A(-2,-3), B(-1,-6), C(2,-2) as shown on the next page.

For each of the following transformations, draw the image of ABC and describe the transformation in words.

(a) if ) (5)

(b) if (5)

QUESTION 3:

(a) Simplify without the use of a calculator

(1) (7)

(2) (5)

(b) Prove the following identity: (3)

(4)

(d) If and , find without using a calculator, the value of (5)

(e) Sketched below are the functions and

° ° ° ° ° ° ° °

(1) Determine the values of b and c. (4)

(2) What is the period of g ? (1)

(3) What is the amplitude of f ? (1)

(4) Write down the new equation of f if it is shifted 30° horizontally to the right in the form … (1)

QUESTION 4:

ANSWER THIS QUESTION ON THE SEPARATE SHEET PROVIDED AND HAND THIS IN WITH YOUR ANSWER BOOKLET.

On the beach you find ten shells and measure their lengths. These lengths are given in the table below:

Length (cm) / /
3,2
3,6
5,0
4,1
4,3
4,7
3,4
5,2
4,6
4,3

(a) Calculate the mean length of the shells. (1)

(b) Complete the table above and use it to calculate the standard deviation of the length of your sample of shells. (6)

SECTION B

QUESTION 5:

(a) A circle M(a;b) is inscribed in .

The circle touches OA at N(6,8).

Determine:

(1) The equation of OA, the tangent to the circle. (2)

(2) The equation of NM, the normal at N. (3)

(3) The co-ordinates of M, the centre of the circle (6)

(4) The equation of the circle. (3)

(Remember: The tangent to the circle is perpendicular to the radius at the point of contact.)

(b) The two circles x2 + y2 - 2x - 2y = 2 and x2 + y2 - 8x - 10y = k, where k is a constant are given. Find the value(s) of k if the circles touch each other externally. (8)

QUESTION 6:

(a) Calculate the co-ordinates of B, C and D if A(1; 5). (7)

(b) Describe the following transformations in words:

(1) (2)

(2) (2)

QUESTION 7:

(a) If , express each of the following in terms of m:

(1) (2)

(2) (2)

(3) (3)

(b) (1) Solve for θ if (7)

(2) Hence determine the values of if (2)

(c) In the diagram , P, Q and R are three points in the same horizontal plane and SR is a vertical tower of height h metres. The angle of elevation of S from Q is ,, and PQ = 6 metres.

(1) Express QR in terms of h and a trigonometric function of . (2)

(2) Express in terms of . (2)

(3) Hence show that (7)

QUESTION 8:

The ogive (cumulative frequency) below represents the finishing times of the 590 runners who completed the RAC Old Parks 10km race on 1 June 2008

(a) Estimate in how many minutes a runner would have had to complete the race in order to be placed in the 15th percentile or better. (2)

(b) If a silver medal is awarded to all runners completing the race in under 40 minutes, assuming that the first ten runners are awarded gold medals, estimate the number of runners who would receive a silver medal. (3)

SECTION C:

QUESTION 9:

(a) In the diagram below, a cross ABCDEFGHIJKL is sketched. The cross has rotational symmetry of about the origin.

If it is further given that the equation of line FG is given by y = -3x – 6, determine:

(1) the equation of JI. (5)

(2) the length of AL, correct to one decimal digit. (7)

TOTAL: 150

MEMORANDUM: PAPER 2

QUESTION 1:

a) (1) B(3; 2)ü ü (2)

(2) üü (2)

(3) ü

ü (2)

(4) (Ext angle triangle) ü

ü (2)

b) Let A(3;-2); B(1;2); C(1,5;1,5)

AB = AC =

= ü = ü

No, circle cannot be drawnü (3)

c) Centre of circle: (0;3) ü

ü (4) [15]

QUESTION 2:

(9)

a) Translation 3 units to the left and 2 units up.ü ü (2)

b) Reflection about the line y = xüü (2)

c) Rotation of about the origin in a anti-clockwise direction üü(2) [15]

QUESTION 3:

a) (1) üü ü ü (2) üüü

= üü = ü

= ü (7) = -sin Aü (5)

b) ü

= ü

= 1ü (3)

LHS = RHS

c) ü

ü (4)

d) ü

r = 5ü

üü

= ü (5)

e) (1) ü c = üü (2)

= 3ü (2)

(2) ü (1) (3) 1 (1) (4) ü (1)

[31]

QUESTION 4:

a) ü (1)

Length (cm) / /
3,2 / -1,04 / 1,0816
3,6 / -0,64 / 0,4096
5,0 / 0,76 / 0,5776
4,1 / -0,14 / 0,0196
4,3 / 0,06 / 0,0036
4,7 / 0,46 / 0,2116
3,4 / -0,84 / 0,7056
5,2 / 0,96 / 0,9216
4,6 / 0,36 / 0,1296
4,3 / 0,06 / 0,0036
4,0964

(4)

ü (2) [7]

QUESTION 5:

a) (1) ü (2) ü

ü (2) ü

ü (3)

(3) ü

(4) r = 5ü

ü üü (3)

a = 10ü

b = 5ü

M(10;5) (6)

üü üü

Centre (1;1) r = 2 Centre (4;5) r =

5 – 2 = ü

k + 41 = 9ü

k = -32 ü (8) [22]

QUESTION 6:

a) B(1;-5)

C(-5;1) üü

D rotation of üanti-clockwise of Bü

D: x = ü

y = ü

D(3) ü (7)

b) (1) Rotation of about the originüü

(2) Rotation of about the originüü (4) [11]

QUESTION 7:

a) (1) ü (2) ü

= -mü (2) = mü (2)

(3) ü

ü (3) ü

b) (1) ü

ü or ü

ü ü (7)

(2) üü (2)

c) (1) In (2) ü

ü ü (2)

ü (2)

(3) In

ü (7)

[27]

QUESTION 8:

a) 40 min or less. üü (2)

b) 15% or the runners 89 runnersüü

Number of silver medals awarded = 89 – 10 = 79 runners. ü (3)

(5)

[10]

QUESTION 9:

(1) JI FGü

Passes through J(6;0) ü

ü (5)

(2) A(0;6) ü

Co-ordinates of F:

By symmetry: ü

AL = ü (7)

= 3,6 unitsü [12]