HCI Paper D Sec 3 Higher Math

1 Name: __________________________________ ( ) Class: _____ HWA CHONG INSTITUTION Paper D HIGHER MATHEMATICS Level : Secondary Three Duration : 1 hour Do not open this booklet until you are told to do so. INSTRUCTIONS TO CANDIDATES Write your name, class and index number on all the work you hand in. Answer 4 out of 7 questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 40. You are expected to use an electronic calculator to evaluate explicit numerical expressions. The use of a graphing calculator is not permitted. You are reminded of the need for clear presentation in your answers. Omission of essential working will result in loss of marks. This question paper consists 4 printed pages, including this page.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c   2 4 2 b b acx a    2. TRIGONOMETRY Identities 22sin cos 1AA 22sec 1 tanAA 22cosec 1 cotAA Formulae for ABC sin sin sin a b c A B C 2 2 2 c 2 cosa b bc A   1 = sin2 ab C

3 Answer 4 out of 7 questions 1 (a) (i) By considering   222 2xy , factorise 44 4xy completely. [3] (ii) Hence, determine all distinct prime factors of 8652 . [3] (b) (i) State the range of values of cos x for 360 360x    . [1] (ii) Hence or otherwise, solve the equation 661 tan secxx for 360 360x    . [3] 2 (a) Simplify  24 8 4 3 8 4 3   . [3] (b) The parabola 2 1yx  is transformed into   213 1 1 4yx   under three successive transformations. (i) Describe the transformations in order. [3] (ii) Given that the area enclosed by the parabola 2 1yx  and the x-axis is 4 3 square units. By considering (i), calculate the area enclosed by   213 1 1 4yx   and the x-axis. Explain your working clearly. [4] 3 (a) A point P moves along the parabola 2 4 2 5 0y x y    so that its x-coordinate is always equal to its distance from a fixed point Q. Find the coordinates of point Q. [3] (b) (i) Find the range of values of x satisfying    2 112 x xx  . [4] (ii) Hence or otherwise, solve the inequality    2e 1 e 1 1 2e x xx     , leaving you answer in the exact form. [3]

4 4 (a) By using a suitable substitution, solve the equation 2 4 2 3 22 2 4 xx xx      . [5] (b) Prove the identity sin cos sin cos 1 1 tan cot sin cos x x x x x x x x     . [5] 5 A quartic polynomial   4 3 2P x x ax bx cx d     leaves a remainder of 34x when divided by 2 2xx and a remainder of 4x when divided by 2 2xx . (i) Prove that   2 26P x x x   has both factors 2 2xx and 2 2xx . [6] (ii) Hence or otherwise, find  Px . Explain your working clearly. [4] 6 (a) Given that 45 75 15a b c , find the value of cc ab . [4] (b) Two functions f and g are defined by: 3f : 2 1xx  and 3 1g: 2 xx  . (i) Show that functions f and g are inverse functions. [2] (ii) Hence, solve the equation     1 2017 terms of "fg" fgfg........fgfg gxx  . [4] 7 (i) In the Cartesian plane, a circle passes through the origin O, point  1,1X  and point  7,7Y . Find the equation of the circle. [5] (ii) Calculate the greatest distance between (2, 6) and a point on the circumference of the circle. Leave your answer in exact form. [2] (iii) Two tangent lines are drawn from point  2, 8Z  to the circle. Find the size of the acute angle formed by the two tangents drawn. [3] End of Paper