2020 HCI H2 Prelim P2 Qn

HWA CHONG INSTITUTION 2020 JC2 Preliminary Examination Higher 2 9758/02 14 September 2020 3 hours Candidate Name CT Group Centre Number S Index Number Write here how many additional pieces of writing paper you have used (if any). For Examiner’s Use Qn Marks Total Remarks 1 6 2 7 3 8 4 9 5 10 6 6 7 7 8 9 9 11 10 12 11 15 100 READ THESE INSTRUCTIONS FIRST 1. Write your name and class on this Cover Page and any additional writing paper you hand in. 2. Write in dark blue or black pen. 3. You may use an HB pencil for any diagrams or graphs. 4. Do not use staples, paper clips, highlighters, glue or correction fluid. 5. Answer all the questions and write your answers in the spaces provided in the Question Paper. 6. 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. 7. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. 8. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. 9. You are reminded of the need for clear presentation in your answers. 10. The number of marks is given in brackets [ ] at the end of each question or part question. MATHEMATICS Candidates answer on the Question Paper. Additional Materials: List of Formulae 26 (MF26) This document consists of 31 printed pages and 1 blank page.

2 © HCI 2020 Section A: Pure Mathematics [40 marks] 1 The diagram below shows the graph of  fyx for 3 2x . The graph has stationary points at ,24  and 2,45   , axial intercept at 10, 2   and asymptotes 0y and 3 2x . Sketch, on separate diagrams, the graphs of the following, showing clearly the coordinates of stationary points, axial intercepts and equations of asymptotes where applicable. (i)   1 fy x , [3] y x Nothing is to be written on this margin Nothing is to be written on this margin

3 © HCI 2020 [Turn over (ii)  f 'yx . [3] Nothing is to be written on this margin Nothing is to be written on this margin

4 © HCI 2020 2 (a) 6 equilateral triangles each with sides 5 units are arranged to form a regular hexagon as shown in the diagram below. By expressing OQ in terms of OP and OR , or otherwise, find  OP OP OQ OR . [3] O P Q R Nothing is to be written on this margin Nothing is to be written on this margin

5 © HCI 2020 [Turn over (b) ,OE OF and OG are non-zero position vectors relative to a fixed origin O for points E, F and G respectively. It is given that 2 5 3OE OF OG   0 . Show that E, F and G are collinear. State the ratio of :EF EG and find EF EG . [4] Nothing is to be written on this margin Nothing is to be written on this margin

6 © HCI 2020 3 (i) It is given that 1 f ( ) ( 2)2 r rr   . Show that f ( 1) f ( ) 2 r r r ar   , where a is a constant to be determined. [2] (ii) Hence find 1 2. n r r r   [3] Nothing is to be written on this margin Nothing is to be written on this margin

7 © HCI 2020 [Turn over (iii) Using the result in part (ii), find 3 ( 2)2 n r r r   in terms of n. [3] Nothing is to be written on this margin Nothing is to be written on this margin

8 © HCI 2020 4 The variables x and y are related by the equation 1cosy x x  , where 01 x and 10 cos 2x  . (i) Prove that 32 2 dd 1dd yy xxx   . [4] Nothing is to be written on this margin Nothing is to be written on this margin

9 © HCI 2020 [Turn over (ii) Find the values of y , d d y x and 2 2 d d y x at 0x . [1] (iii) By further differentiation, find the series expansion of y in ascending powers of x up to and including the term in 3x . [3] (iv) Deduce the series expansion of 2 11 1 x   up to and including the term in 2x . [1] Nothing is to be written on this margin Nothing is to be written on this margin

10 © HCI 2020 5 The complex number a has modulus r and argument  , where r  and 0 2  .The complex number b is such that 2iba and the complex number c is purely imaginary with negative imaginary part. It is given that bc . Let the points A, B, C and D represent the complex numbers a, b, c and bc respectively, and O be the origin. (i) On a single Argand diagram, illustrate the points A, B, C and D, indicating clearly the modulus and argument of a, b and c. [4] Nothing is to be written on this margin Nothing is to be written on this margin