2019 J2 YIJC H2 Math Prelim (with ans)

This paper consists of 20 printed 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 4 September 2019 3 hours READ THESE INSTRUCTIONS FIRST Write your CG and name on the work you hand in. Write in dark blue or black pen. You may use a 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 significa nt figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is spec ified in the question. The use of an approved graphing calculator is expected, where appropriate . Unsupported answers from a graphing calculator are allow ed unle s s a q u e s t i o n s p e c i f i c a l l y s t a t e s 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 ans wers. 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 Question 8 9 10 11 Marks www.KiasuExamPaper.com 893

2 ©YIJC 9758/01/Prelim/19 1 (i) Expand sin 24 xS§· ¨¸©¹ in ascending powers of x , up to and including the term in 3x . [3] (ii) The first two non-zero terms found in part (i) are equal to the first two non-zero terms in the series expansion of 1()ab x in ascending powers of x. Find the exact values of the constants a and b. Hence find the third exact non-zero term of the series expansion of 1()ab x for these values of a and b. [3] www.KiasuExamPaper.com 894

3 ©YIJC 9758/01/Prelim/19 2 (a) Vectors a and b are such that 00zza , b and ab ab . Show that a and b are perpendicular. [2] www.KiasuExamPaper.com 895

4 ©YIJC 9758/01/Prelim/19 (b) Referred to the origin O, points C and D have position vectors c and d respectively. Point P lies on OC produced such that :1 : 1OC CP O , where O >1. Point M lies on DP, between D and P, such that :2 :3DM MP . Write down the position vector of M in terms of O , c and d. Hence, find the area of triangle OPM in the form kO ucd , where k is a constant to be found. [4] www.KiasuExamPaper.com 896

5 ©YIJC 9758/01/Prelim/19 3 The function f is defined by

2 2 for 0 2 ,f( ) for 24 4, 2 2 xa a xax axxa a ­ d° ® d

° ¯ where a is a positive real constant and that f4 fxa x for all real values of .x (i) Sketch the graph of fy x for 83 aaxd d . [3] (ii) Hence find the value of 2 8 f| |d a a xx ³ in terms of a. [3] www.KiasuExamPaper.com 897

6 ©YIJC 9758/01/Prelim/19 4 A curve C has parametric equations . (i) The curve intersects C at point A. Without using a calculator, find the coordinates of A. [2] (ii) The tangent at the point on C meets the x-axis at point D and the y-axis at point E. The point F is the midpoint of DE. Find a cartesian equation of the curve traced by F as p varies. [5] 2 1 , , 0xt y t t ! 8y x 2 1,Pp p §· ¨¸¨¸©¹ www.KiasuExamPaper.com 898

7 ©YIJC 9758/01/Prelim/19 5 The equation of a curve is

2 21 .xy y x (i) Find the equations of the two tangents which are parallel to the y-axis. [4] www.KiasuExamPaper.com 899

8 ©YIJC 9758/01/Prelim/19 (ii) The normal to the curve at the point A 1, 0meets the y-axis at the point B. Find the area of the triangle OAB. [3] 6 The sum of the first n terms of a sequence is a cubic polynomial, denoted by nS . The first term and the second term of the sequence are 2 and 4 respectively. It is known that 5 90S and 10 830S . (i) Find nS in terms of n . [4] www.KiasuExamPaper.com 900

9 ©YIJC 9758/01/Prelim/19 (ii) Find the 54th term of the sequence. [2] (b) (i) Given that cos(2 1) cos(2 1) 2sin sin 2nn nDD D D and α is not an integer multiple of π, show that 1 11sin 2 cot cosec cos(2 1)22 N n nNDD D D ¦ . [3] www.KiasuExamPaper.com 901

10 ©YIJC 9758/01/Prelim/19 (ii) Explain whether the series 1 2sin 3n nSf ¦ converges. [1] 7 (a)(i) Find cos (ln )dxx³ . [3] (ii) A curve C is defined by the equation

31 22cos ln , for e e .yx x SS d d The region R is bounded by C, the lines 2ex S , 2ex S and the x-axis. Find the exact area of R . [3] www.KiasuExamPaper.com 902