2024 JC2 H2 Math prelim - DHS

© DHS 2024 This document consists of 22 printed pages and 2 blank pages. Name: Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination Year 6 MATHEMATICS (Higher 2) 9758/01 Paper 1 10 September 2024 3 hours Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your Name, Index Number and Class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. 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. 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. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. For teachers’ use: Qn Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Total Score Max Score 5 5 7 8 7 9 9 14 12 12 12 100

2 DHS 2024 Year 6 H2 Mathematics Preliminary Examination Paper 1 1 (a) Without the use of a calculator, solve the inequality 2 2 2 2 11 .21 1xx xx−+ +− [3] (b) Hence solve the inequality 2 2 11 2 2 121 xx xx − −+ − − . [2]

3 DHS 2024 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over Do not use a calculator in answering this question. 2 The complex number z is given by 2 ππ 12 12 ππ 66 cos isin cos isin z  −= + . (a) Find z in the form ier  where 0r  and π π.−   [2] (b) Show that 3(1 ) i,zp+= where p is a real constant to be determined. Hence or otherwise, find 33(1 ) (1 *) .zz+ + + Show your working clearly. [3]

4 DHS 2024 Year 6 H2 Mathematics Preliminary Examination Paper 1 3 It is given that 2 2 d d ln dd y y xx x x x+= where 0,x and d 1d y x = at 1.x= Use the substitution d d yzx x= to show that ( ) 2 lnd 1 1 .d2 xy x x x=+ Hence find the exact equation of the tangent to the curve f( )yx= at ( )7 6e, . [7]

5 DHS 2024 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 4 (a) Find 2 .14 3 d 98 x xx x+ −−  [4] (b) Find 2 2 0 3 e dkxxx in terms of k, where k is a positive constant. Explain whether there exist solutions for k satisfying the equation 2 2 30 63 e d .kxxx k=− [4]

6 DHS 2024 Year 6 H2 Mathematics Preliminary Examination Paper 1 5 The function f is defined by 2 f: 4 xx x− , x , 4 8.x (a) Find 1f ( )x− and write down the domain of 1f.− On the same axes, sketch the graphs of f( )yx= and 1f ( ).yx −= [4]

7 DHS 2024 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over (b) The region R is bounded by the curve 1f ( ),yx −= the lines 5, 8yy== and the y-axis. Find the exact area of R. [3]

8 DHS 2024 Year 6 H2 Mathematics Preliminary Examination Paper 1 6 (a) The diagram shows the curve h( ).yx= The curve has maximum points at ( )6, 4− and the origin, and crosses the x-axis at ( )5, 0 .− The lines y = 0, x = −4 and y = −x + 3 are the horizontal, vertical and oblique asymptotes to the curve respectively. (i) On the diagram given above, sketch the graph of ( ) ( ) 22 26 9 ,x y r+ + − = where r is a positive constant. State the range of values of r for the equation ( ) ( )( ) 22 26 h 9x x r+ + − = to have at least one real root. [3] (ii) On a separate diagram, sketch the graph of ( ) 1 .hy x= [3] y = h(x) x = −4 y = −x + 3 (−5, 0) (−6, 4) y = 0 y O

9 DHS 2024 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over (b) The graph of 10 1yx= − + undergoes a sequence of transformations which transform its equation into 1.yx=− Describe and write down the transformations. [3]

10 DHS 2024 Year 6 H2 Mathematics Preliminary Examination Paper 1 7 The points A, B and C represent the complex numbers a, b and c respectively, such that 0,a= 3b= and 2 i.c=− + The three complex numbers are roots to the equation f( ) 0z = where f(z) is a quartic polynomial with real coefficients and z is a complex variable. (a) Express f(z) as a product of two quadratic factors with real coefficients. [3] (b) Sketch an Argand diagram showing the roots of the equation f( ) 0z = . [2]