Trial Mid-Year Examination (Form 4) Answer all questions below. Marks are given for your clear steps to your solution. 1. The following information refers to the function p and q. : 3 2p x x : 6 4q x x Find 1( )pq x (3marks) 2. Solve the quadratic equation 2(3 2 ) 2x x x . Give your answer correct to 4 significant numbers. (3 marks) 3. Solve the equation 5 5log (7 4) log ( 2) 1x x (3 marks)

4. Given that log 3 and log 5k kp q , express 2

125log9k k

in terms of p and q. (4 marks)

5. The variables x and y are related by the equation py kx ,

where k and p are constants. When this equation is converted to a linear form, the straight line graph obtained from it is as shown in diagram 1.

a) Find the equation of the linear graph b) Hence, find the value of i) p, ii) k. (4 marks) 6. The following information refers to the equations of two straight lines PQ and ST, which are

perpendicular to each other. : ( 1)PQ y k x : ( ) 5ST ky p k x , where k and p are constants. Express p in terms of k. (2 marks) 7. Diagram 2 shows a semi-circle ACB, with its

centre at the point O. The diameter of the semi-circle is of length 2r cm. The ratio of the length of arc AC to the length of arc BC is 2: 1

a) Show that 13

BOC radians (2 marks)

b) In the diagram, the segment enclosed by the arc AC and the chord is marked as region 1R and the segment enclosed by the arc BC and the chord BC is marked as region 2R .

If the length of the chord AC is 5 3 cm, show that r = 5. (2 marks) Hence, by using 3.142 , calculate, correct to 3 decimal places,

i) the perimeter, in cm, of region 1R (2 marks) ii) the area, in cm2, of the region 2R (4 marks)

8. Given the function f(x) = 3x − 4 and the composite function fg(x) = 5 − 6x, find a) g(x) b) the value of x when f 2 (x) = 2 (4 marks)

(0, 5)

(2, 1)

2log y

2log xO

1diagram

1R2R

2Diagram

9. Form the quadratic equation which has the roots 1 and 24

. Give your answer in the

form 2 0ax bx c , where a, b and c are integers. (2 marks) 10. Find the ranges of values of k such that k (k − 2) ≥ k + 4 (3 marks)

11. Solve the equation 71 168

xx

(3 marks)

12. Diagram 3 shows the function ( )2

x kg xx

where x ≠ 2

and k is a constant. Find the value of k. (2 marks) 13. The function h maps x in set A to h(x) in set B. If 2: 5h x x and A = {−1, 0, 1, 2, 3}, find the range of the function h. (2 marks) 14. A quadratic equation 2 23 2 12x mx m has two equal roots. Find the possible values of m (3 marks) 15. Diagram 4 shows a trapezium ABCD, with right angles

at C and D. BE is a circular arc with its centre at A. Given that AB = 10 cm, BC = 8 cm, CD = 6 cm and

BAE radians, find a) the value of b) the area of the region enclosed by the arc BE with the straight lines BC, CD and DE. (4 marks) Suggested Answers: 1. 2x − 8 2. 1.457 or − 0.4574 3. 3 4. 3q − 2p − 2 5. 2 2log 2log 5y x ; i) − 2 ii) 32 6. p = k − 1 7. b) i) 19134 cm; ii) 2.266 cm2 8. a) 3 − 2x; b) 2 9. 4x2 + 7x − 2 = 0 10. k ≤ − 1 or k ≥ 4 11. 4 12. 1 13. {−5, −4, −1, 4} 14. ± 3 15. a) 0.6435 b) 39.82 cm2

Prepared by Mr. David Ch’ng YS / PFS / 2 May 2012

2

x kxx

1

− 2 Diagram 3