HCI Paper E Sec 3 Higher Math

1 Name: ______________________________ ( ) Class: ________ HWA CHONG INSTITUTION Paper E 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 5 printed pages, including this page.

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2. TRIGONOMETRY Mensuration Arc Length = r , where  is in radians Sector area = 21 2 r  , where  is in radians Formulae for 2 0ax bx c   2 4 2 b b acx a    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) Given that 1xy , find the value of 33 3x xy y . [3] (b) Solve the inequality 2 2 ( 1) ( 2) 07 12 xx xx   . Hence, solve 2 2 (sin 1) (sin 2) 0sin 7sin 12 xx xx   for 02 x  . [7] 2 (a) Given that the equation 2 2 1 ( 2)x k k x    has two real roots and the sum of squares of the two roots is 11, find the value of k. [5] (b) The graph 2 11 30y x x m    intersects the x-axis at two points, and both points are on the right of the point (5, 0). Find the range of values that m can take. [5] 3 (a) Express the following simultaneous equations in matrix form, and hence solve for the value of x and of y. [4] 4 5 2 2 7 20 xy xy   (b) A polynomial 73 4px qx rx   has a remainder of 3 when it is divided by 7x . Find the remainder when it is divided by 7x . [3] (c) A standard catenary has the equation 2 xx aaeeya    , where a is a real constant, and is symmetrical about the y-axis. Given that the equation of a transformed catenary curve is 55 24 xxeey    , find the coordinates of the catenary’s turning point. [3] [Turn Over

4 4 (a) Simplify 8 2 15 , leaving your answer in the form ab , where a and b are positive integers. [2] (b) Given that a, b and c are the sides of a triangle ABC, simplify 2 2 2( ) ( ) ( )a b c a b c a b c        . [3] (c) Solve the equation 3log 3 log 1x xx . [5] 5 Three points have coordinates A(4, 3), B(5, 2) and C(1, 0). Find (i) the equation of the circle that passes through the points A, B and C, [6] (ii) the equation of the line that is a tangent to the circle and passes through the origin. [4] 6 (a) Functions f and g are defined by . f g : 2 , :2 x x a x x b   Given that 2f (4) 1 and 1g(2) g (2) , find (i) the value of a and of b, [3] (ii) the value of fg(3). [2] (b) The function f is defined by f : , 1 1 xxx x  . On two separate diagrams, sketch the following graphs, indicating clearly the x-intercepts, the y-intercepts and the asymptotes, if any. (i) f( )yx [3] (ii)  fyx [2]

5 7 In ABC , AB = 8 cm, BC = 5 cm and AC = 7 cm. O is a point in the triangle such that 120AOB AOC BOC     . It is also given that OA = x cm, OB = y cm and OC = z cm. (i) Find the area of ABC , leaving your answer in exact form. (Hint: use Heron’s formula, Area ( )( )( )s s a s b s c    .) [1] (ii) Show that 22 64.x xy y   Hence, write down the value of 22y yz z and of 22x xz z . [3] (iii) Find the value of xy yz xz . [3] (iv) Hence, find the value of 2()x y z . [3] End of Paper z cmy cm x cm 7 cm 5 cm 8 cm A B C O 120° 120° 120°