2019 SH2 NJC H2 Math Prelim (with ans)

1 [Turn over NATIONAL JUNIOR COLLEGE SENIOR HIGH 2 Preliminary Examination NAME SUBJECT CLASS 2ma2 REGISTRATION NUMBER H2 MATHEMATICS 9758 / 01 28 August 2019 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) Writing Paper (1 sheet) READ THESE INSTRUCTIONS FIRST Write your name, class and registration number in the boxes above. Please write clearly and use capital letters. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use 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. The use of an approved graphing calculator is expected, where appro priate. Unsupported answers from a graphing calculator are allowed unle ss a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathemat ical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. Up to 2 marks may be deducted for improper presentation. The number of marks is given in the brackets [ ] at the end of each quest ion or part question. Question Number Marks Possible Marks Obtained 16 26 37 47 58 6(a) 3 6(b) 6 71 0 81 0 91 2 10 12 11 13 Presentation Deduction –1 / –2 TOTAL 100 This document consists of 29 printed pages and 3 blank pages. www.KiasuExamPaper.com 426

2 © NJC 2019 1 The complex number z has modulus r and argument T , where 01 r and π π2 T . The complex number w is such that 1w z and arg( ) arg( ) πwz . Let the points A, B, C, D and E represent the complex numbers z, w, zw , *z and *zz respectively, where *z denotes the conjugate of z. (i) On a single Argand diagram, illustrate these five points. [5] (ii) Identify the shape of the quadrilateral ACDE. [1] www.KiasuExamPaper.com 427

3 [ Turn over STAPLE HERE STAPLE HERE www.KiasuExamPaper.com 428

4 © NJC 2019 2 Do not use a calculator in answering this question. Solve the inequality 225 9 47 xx xx d . [4] www.KiasuExamPaper.com 429

5 [ Turn over STAPLE HERE STAPLE HERE Hence, solve the inequality 225 e 9 e 4e 17e 1 xx x x t . [2] www.KiasuExamPaper.com 430

6 © NJC 2019 3 Two sightseeing points A and B are located offshore. A cruise ship S travels along a straight path passing through A and B. A lighthouse is located at the shore. If the lighthouse’s position is taken as the origin O, the position vectors of A and B are a and b respectively. (i) I f :: 1AS SB OO , find the position vector of S, giving your answer in terms of O , a and b. [1] When the cruise ship is closest to the lighthouse, (ii) show that

O ˜ ˜ aab ba ba , [2] www.KiasuExamPaper.com 431

7 [ Turn over STAPLE HERE STAPLE HERE (iii) find its position vector in terms of a and b, if sightseeing point A is three distance units away from the light house, sightseeing point B is two distance units away from the lighthouse and the angle between a and b is 60q. [4] www.KiasuExamPaper.com 432

8 © NJC 2019 4 (i) Show that 2 22 44 112 2 px p q x p q pq xxx x . [1] (ii) Find the values of p, q and r such that the series expansion of 2 2 44 ln 1 3 12 px p q x p q rx xx

up to and including the term in 2x is 211 3 xx . [4] www.KiasuExamPaper.com 433

9 [ Turn over STAPLE HERE STAPLE HERE (iii) Find the range of the values of x for which the expansion is valid. [2] www.KiasuExamPaper.com 434

10 © NJC 2019 5 The diagram shows a sketch of the curve f( )yx . The curve cuts the x-axis at (–2, 0) and it has a stationary point at (4, 0). The equations of the asymptotes are 0x , 2y and 2yx . Sketch, in separate diagrams, the graphs of (i) 1 f( )y x , [3] (ii) fyx c , [3] stating the equations of any asymptotes and coordinates of axial intercepts. www.KiasuExamPaper.com 435