SAJC_H2_Math_P1_Question

[Turn Over ST ANDREW’S JUNIOR COLLEGE PRELIMINARY EXAMINATION MATHEMATICS Higher 2 9740/1 Monday 31 AUG 2015 3 hours READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Answer all the questions. Total marks is 100. Give non-exact numerical answers correct to 3 signi ficant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphic calculator. Unsupported answers from a graphic calculator are a llowed unless a question specifically state otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematic steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, fasten all your work securely together. This document consists of 6 printed pages includi ng this page.

2 [Turn Over 1 Without using a calculator, solve the inequality 2 2 14 32 3 5 x x x x + − − ≤− . [4] 2 Prove by mathematical induction that, where sin5 0 x ≠ , for non-negative integers n. [5] 3 The function f is defined by (i) Show by differentiation that f is a decreasing function on any interval in the domain. [2] (ii) Find ( ) 1f x− , stating its domain. [3] 4 The diagram shows the curve with equation 3 2 2 3 1x y x y + = . The curve is symmetrical about the line y x = and has a stationary point A . (i) Find the exact coordinates of A . [5] (ii) The point B is the point of the curve at which the tangent is parallel to the y -axis. Hence, or otherwise, state the exact coordinates of B . [1] ( )cos 4 cos 10 6 sin sin11 sin 21 ... sin(10 1) 2sin 5 x n x x x x n x x − + + + + + + = ( ) 3 5 for f , 2. 2 x xx x x= − ∈ ≠ − R O

3 [Turn Over 5 The ellipse 1C with the equation 2 2 2 2 16 3 x y + = is transformed by a stretch with scale factor k parallel to the x -axis, followed by a translation of m units in the positive x -direction, followed by a translation of n units in the negative y -direction, to form 2C . (i) Find the equation of 2C in terms of k, m , and n. [3] (ii) Given that 2C is a circle centred at ( )4, 7 − , find the values of k, m and n. [2] 6 The variables u and t are related by ( ) 2 d1 5 d ut t t+ = . (i) Find the particular solution of t