SAJC 2017_H2_MATHS_Qns Prelims

SAINT ANDREW’S JUNIOR COLLEGE Preliminary Examination MATHEMATICS Higher 2 9758/01 Thursday 24 August 2017 3 hours Additional materials : Answer paper List of Formulae (MF26) Cover Sheet 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 : 100 Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of agles 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 allowed unless a question specifically states otherwise. Where unsupported answers from a graphic calculator are not allowed in a question, you are required to present the mathematical 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 including this page. [Turn over

2      [Turn over   1 The volume of a spherical bubble is increasing at a constant rate of  cm3 per second. Assuming that the initial volume of the bubble is negligible, find the exact rate in terms of  at which the surface area of the bubble is increasing when the volume of the bubble is 20 cm3. [5] [The volume of a sphere, 34 3Vr  and the surface area of a sphere, 24A r where r is the radius of the sphere.] 2 The diagram shows the triangle ABC. It is given that the height AD is h units, 3ABD  and  ACD =  4  x . Show that if x is sufficiently small for x 3 and higher powers of x to be neglected, then  2 + + 3 tan 4 hhBC h p qx rx x    , for constants p, q, r to be determined in exact form. [5] 3 It is given that 2 2 2 2 1 for f( ) (2 )1 for 3 xba x aax xaaa x a a        and that f( 4 ) f( )x ax for all real values of x, where a and b are real constants and 0. ab (i) Sketch the graph of f( )yx for 8.ax a  [3] (ii) Use the substitution cosxa  to find the exact value of 4 3 f( ) d a a x x in terms of a, b and . [5]      A B D C h

3      [Turn over   4 (i) State a sequence of transformations that would transform the curve with equation 2 exy onto the curve with equation f( )yx , where 2 f( ) e axx b , 0a and 1b . [2] (ii) Sketch the curve f( )yx and the curve 1 .f( )y x You should state clearly the equations of any asymptotes,