NYGH 2015-S3EOY-IM2

Class Register Number Name 南洋女子中学校 NANYANG GIRLS' HIGH SCHOOL End-of-Year Examination 2015 Secondary Three INTEGRATED MATHEMATICS 2 2 hours Monday 5 Oct 2015 0845 – 1045 READ THESE INSTRUCTIONS FIRST INSTRUCTIONS TO CANDIDATES 1. Write your name, register number and class in the spaces at the top of this page. 2. Answer questions 1 - 11 before attempting question 12 (Bonus Question). 3. Write your answers and working on the separate writing paper provided. 4. Omission of essential working will result in loss of marks. 5. 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 1. The number of marks is given in brackets [ ] at the end of each question or part question. 2. The total number of marks for this paper is 80. 3. You are reminded of the need for clear presentation in your answers. Setter: OLH This document consists of 6 printed pages. NANYANG GIRLS' HIGH SCHOOL

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0 , x = a acbb 2 42 −±− . 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A Formulae for ∆ABC C c B b A a sinsinsin == a2 = b2 + c2 − 2bc cos A ∆ = 2 1 ab sin C

3 [ Turn over 1 (a) Find the range of values of x for which (6 – x)2 > x. [3] (b) The roots of the quadratic equation x2 – 3x + 5 = 0 are α and β. Find a quadratic equation with integer coefficients and with roots βα 12 − and αβ 12 − . [4] 2 Solve each of the following equations. (a) e x – 7 = 2e–x [4] (b) yy 39 log2log =− [3] 3 (a) The function y = f(x) undergoes two transformations: I: Translation in the positive x-direction by 5 units, II: Scaling in the y-direction by a factor of 2. The final function is y = 2x2 – 16x + 32. Find the function f(x). [3] (b) (i) Sketch the graph of y = ln (x – 1), labelling clearly the intercept(s) and asymptote. [2] (ii) By adding a suitable straight line to the graph in part (b)(i), find the number of solutions to the equation 1 e 1 3 += − xx . [2] 4 Solve the simultaneous equations 27x ÷ 3y = 9, 22x × 41–y = 64. [4] 5 The value, V dollars, of a vehicle depreciates over time. Given that V = 84000e kt, where t is the time in years since it was bought and k is a constant, calculate (i) the initial value of the vehicle, [1] (ii) the value of k if, after 3 years, the value of the vehicle has halved, [2] (iii) the value of t when the value of the vehicle is one-fifth its original value? [2]

4 6 (a) A curve has the equation y2 + (x + p)2 = 8, where p is a constant. Find the range of values of p for which the line y = x + 4 meets the curve. [4] (b) Given that y = x2 – 4x + c, find the value of the constant c for which the minimum value of y is 3. [3] 7 Answer the whole of this question on the INSERT provided. (a) The graph of y = h(x) is given in the diagram. On the same axes shown on the INSERT, sketch the graph of h –1(x). [3] (b) The graph of y = g(x) is given in the diagram. On the same axes shown on the INSERT, sketch the graph of y = g(2x) + 1. [2] y = h(x) 2 1 (2, –1) (–2, 3) y x 0 y = g(x) 2 –1 (2, –2) y x 0 (3, 1)

5 [ Turn over 8 Given the function 12sin −  −= xy for x ≥ 0 radian. (i) State the maximum and minimum value of y. [2] (ii) State the period of y. [1] (iii) State the amplitude of y. [1] (iv) Find the smallest value of x such that y = 0. [2] (v) Sketch the graph of 12sin −  −= xy for 0 ≤ x ≤ 2π. [2] 9 (a) Express the following in terms of sin θ, cos θ or tan θ, where θ is an acute angle. (i) tan (2π + θ ), [1] (ii) sin (2π – θ ), [1] (iii)    − 2 π cos θ . [1] (b) Solve the equation tan x + 2 sec2 x – 5 = 0, for 0 ≤ x ≤ 2π. [5] 10 (a) Prove the identity A AAA cos1 sincot cosec +≡− . [3] (b) Given that A and B are in different quadrants, 4 3tan −=A , 13 5cos −=B , 0° ≤ A ≤ 270° and 0° ≤ B ≤ 270°. Without using a calculator, find the value of (i) cos A, [2] (ii) tan B, [2] (iii) cosec A sec B, [2]

6 11 The functions f and g are defined by f : x 12 3 +x for all values of x except 2 1−=x , g : x 132 +− xx . (i) Find the values of x which map onto themselves under the function f. [3] (ii) Find, in similar form, f 2. State the domain clearly. [3] (iii) Express g(x) in the form of h(x + k)2 + n where h, k and n are constants. Hence, deduce the range of the function g. [3] (iv) If the domain of g is x ≤ c where c is a constant, state the maximum value of c for which the function g –1 exists. Hence, find the function g –1 in similar form. [4] Bonus Question 12 If a > b > 1 and 293log 1 log 1 =+ ab ba , find the value of ab abab log 1 log 1 − . [3] END OF PAPER