2020 HCI H2 Prelim P1 Qn

HWA CHONG INSTITUTION 2020 JC2 Preliminary Examination Higher 2 9758/01 28 August 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 4 2 5 3 6 4 7 5 7 6 8 7 8 8 9 9 10 10 10 11 12 12 14 100 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 c orrect 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 27 printed pages and 1 blank page.

2 © HCI 2020 1 A curve has parametric equations 12xt t and 1yt t where t is a parameter and 0t  . (i) Show that   2 d 1 3 .d2 2 2 1 y x t   [2] (ii) Find the range of values that d d y x can take. [2] Nothing is to be written on this margin Nothing is to be written on this margin

3 © HCI 2020 [Turn over 2 The coefficient of 3x in the expansion of   5 1 2 ax is 5 times the coefficient of 2x in the expansion of 4 ax where a is a non -zero constant. Find the exact value of a . [5] Nothing is to be written on this margin Nothing is to be written on this margin

4 © HCI 2020 3 A curve C has equation  fyx , where     2 21f 3 xx x   . (i) Sketch C, showing clearly the equations of the asymptotes and coordinates of any stationary points and axial intercepts. [3] Nothing is to be written on this margin Nothing is to be written on this margin

5 © HCI 2020 [Turn over (ii) Sketch the graph of  fyx on a separate diagram. It is given that the equation   22 22 21 3 xx k a x      , where 0a and 0k  , has an odd number of roots. By adding a suitable curve to the graph of  fyx , find a in terms of k. [3] Nothing is to be written on this margin Nothing is to be written on this margin

6 © HCI 2020 4 (a) Find the exact solution of the inequality 2 21 1.1 x x   [3] Nothing is to be written on this margin Nothing is to be written on this margin

7 © HCI 2020 [Turn over (b) A curve G has equation 23 2,ay bx cx   where , and a b c are constants. It is given that G passes through the point  1, 3 and the gradient at the point  1,1 is 3 .2 Find the equation of the curve G. [4] Nothing is to be written on this margin Nothing is to be written on this margin

8 © HCI 2020 5 A closed right circular cone of base radius r and height h has a fixed total surface area k 2units , where k is a positive constant. Show that   2 2 2 2 kr hk   . Given that r and h vary, find the greatest possible volume of the cone, giving your answer in exact form in terms of k. [7] [The volume of a cone of base radius r and height h is given by 21 3V r h  and the curved surface area of the cone is given by , where is the slant heightA rl l .] Nothing is to be written on this margin Nothing is to be written on this margin

9 © HCI 2020 [Turn over Nothing is to be written on this margin Nothing is to be written on this margin

10 © HCI 2020 6 Find (a) 3e d5 0.3e x x x , [2] (b)  cos ln dxx , where 0x , [3] Nothing is to be written on this margin Nothing is to be written on this margin