2020 J2 YIJC H2 Math Prelims P1

This document consists of 28 printed pages and 4 blank pages. YISHUN INNOVA JUNIOR COLLEGE JC 2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CG INDEX NO MATHEMATICS Paper 1 Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) 9758/01 2 SEPTEMBER 2020 3 hours READ THESE INSTRUCTIONS FIRST Write your CG, index number and name on the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all the questions. Write your answers in the spaces provided in the Question Paper. 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. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. 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. 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. The total number of marks for this paper is 100. --------------------------------------------------------------------------------------------------------------------------------------- For Examiners’ Use Question 1 2 3 4 5 6 7 Marks Total marks 100 Question 8 9 10 11 Marks

2 ©YIJC 9758/01/Prelim/20 BLANK PAGE

3 ©YIJC 9758/01/Prelim/20 [Turn Over 1 Vectors a, b and c are such that b 0 and 3 7 .  a b b c (i) Show that 3 7  a c b , where  is a constant. [2] (ii) It is now given that a and b are unit vectors, that the modulus of c is 3 7 and that the angle between a and c is o60. Using a suitable scalar product, find exactly the two possible values of  . [4]

4 ©YIJC 9758/01/Prelim/20 2 (i) Show that     3 4 1 8 2 1 2 3 2 5 2 1 2 3 2 5 r k r r r r r r         , where k is a constant to be found. [2]

5 ©YIJC 9758/01/Prelim/20 [Turn Over (ii) Hence find    0 2 7 2 1 2 3 2 5 n r r r r r     . (There is no need to express your answer as a single algebraic fraction.) [3] [Question 2 continues on the next page.]

6 ©YIJC 9758/01/Prelim/20 2 [Continued] (iii) Explain why    0 2 7 2 1 2 3 2 5r r r r r       is a convergent series, and state the value of the sum to infinity. [2]

7 ©YIJC 9758/01/Prelim/20 [Turn Over 3 The function f is defined by 3 2f ( )x ax bx cx d    , where a, b, c and d are real numbers. (i) Given that 1 2i and 1 are roots of f ( ) 0,x  find b, c and d in terms of a. [4] [Question 3 continues on the next page.]

8 ©YIJC 9758/01/Prelim/20 3 [Continued] (ii) Given instead that 2, 1 and 4a b c   , find the range of values of d such that the equation f ( ) 0x  has only one real root. [2]

9 ©YIJC 9758/01/Prelim/20 [Turn Over (iii) It is now given that the curve with equation f ( )y x meets the x-axis at  ,0k and the y-axis at  0,k , where 0k , as shown in the diagram. Sketch the graph of 1 f ( )y x , stating the equations of any asymptotes and the coordinates of any points of intersections with the axes. [2] O x y

10 ©YIJC 9758/01/Prelim/20 4 (i) Find the exact roots of the equation 4 5 32 x xx   . [3]