2024 JC2 H2 Math prelim - ACJC

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME TUTORIAL/ INDEX FORM CLASS NUMBER MATHEMATICS 9758/01 Paper 1 20 August 2024 3 hours Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your index number, class and name on all 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. The use of an approved graphing calculator is expected, where appropriate. 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. This document consists of 24 printed pages and 2 blank pages. [Turn Over Question Marks 1 /4 2 /5 3 /6 4 /8 5 /9 6 /9 7 /10 8 /11 9 /12 10 /12 11 /14 Total 100 /100

2 ANGLO-CHINESE JUNIOR COLLEGE 2024 H2 MATHEMATICS 9758/01 1 The complex numbers z and w satisfy the following equations. 2i 2 2i i 2 i w wz z z w += + = + Find z and w, giving your answers in the form of iab+ where a and b are real numbers. [4] 2 Rectangular Built-in cupboard compartment An interior designer designed a built-in cupboard for his client as shown above. The built-in cupboard of length 9 a metres, a > 0, has three equal sections and each section has a semi - elliptical hole in the centre. The designer wants to fit a hollow rectangular compartment, for storage into each of the elliptical hole. Each rectangular compartment with negligible thickness, has a length of l metres, where 2la , a height of y metres, and a fixed depth. The cross-section for part of the built-in cupboard is shown in the diagram below and the elliptical holes are modelled by the equation ( ) 2 2 for ,1f for 2 ,0 x a x ax a a x a  −   −=   and ( ) ( )f 3 fx a x+= for 99 22 aa x−   , where a is a real constant. ( )fyx= (a) Write down, in terms of l and a, the value of ( )f x when 13 2x a l=+ . [1] (b) The interior designer wishes to maximi se the rectangular compartment storage space. Show that the length of the compartment l, is 2a metres, when the space is maximised. Find also the corresponding height of the compartment. (You do not need to show that the value is a maximum.) [4] x y metresl metresy 9 metresa O a a− 9 2 a 2a 9 2 a− 2a−

3 ANGLO-CHINESE JUNIOR COLLEGE 2024 H2 MATHEMATICS 9758/01 [Turn Over 3 Elly started planking as an exercise and she continues the exercise every day to build her core muscles. If she meets her target duration, she increases the target duration of the exercise by an additional 4 seconds on the next day. On any day, she will stop her exercise once she meets her target duration for the day. However, Elly does not always meet her target. Each day when Elly misses her target, she decreases her target duration by 5% on the following day. On Day 1, Elly carries out 20 seconds of planking, and she hopes to reach her target of 2 minutes by the end of 30 days. (a) Assume that Elly met her targets for the first 11 days but missed her target duration from Day 12 to Day 15. Determine whether Elly will be able to reach her target of 2 minutes by the end of 30 days, if she met all her targets from Day 16 onwards. [3] Due to the difficulty level, Elly decides to restart the programme by increasing the target duration of the exercise by a% each day, regardless of whether she meets her target. (b) Find in terms of a, the total target duration Elly has completed by the end of 30 days if she carries out 20 seconds of planking on Day 1. [2] [You may assume that on any day, she will stop her exercise once she meets her target duration for that day.] (c) If the total target duration she has completed by the end of 30 days is at least 30 minutes, find, to the nearest integer, the least value of a. [1] 4 (i) Using standard series from the List of Formulae (MF26), show that for 4x and higher powers to be neglected, ( ) 31 2 16f ln 4 1 2 3 xx x x x +=  + − . [3] (ii) Use your series from part (i) to estimate 0.04 0 12ln d12 x xx +  − , correct to 8 decimal places. [1] (iii) Use your calculator to find 0.04 0 12ln d12 x xx +  − , correct to 8 decimal places. [1] (iv) Comparing your answers to parts (ii) and (iii), and with reference to the value of x, comment on the accuracy of your approximations. [2] (v) Explain why a Maclaurin series for ( ) 2g ln 2 xx x +=  − cannot be found. [1]

4 ANGLO-CHINESE JUNIOR COLLEGE 2024 H2 MATHEMATICS 9758/01 5 A curve has equation 5e 4e 3 x x y= − . The line 5y= intersects the curve at points A and B. (i) Find the exact x-coordinates of the points A and B. [3] (ii) Using the substitution exu= , find the exact volume generated when the area bounded by the curve and the line 5y= is rotated about the x-axis through 360 . Give your answer in the form ( )25 ln 38 ab − , where a and b are constants to be determined. [6] 6 Do not use a calculator in answering this question. (a) (i) One of the roots of the equation 4 3 216 21 5 0aw w w aw− + − + = , where a is real, is 2i− . Find the value of a and the other roots. [4] (ii) Hence solve 4 3 25 21 16 0w aw w w a− + − + = . [2] (b) The complex number z is given by 4 cos isin33 cos isin12 12 z k       −       =     −+         , where k is a positive real constant. Find z and arg z . [3] 7 (i) Show that 2 3 1 1 2 1 ! ( 2)! ( 1)! ! rr r r r r −+ = − +−− . Hence find 2 3 31 ! n r rr r= −+  in terms of n. [3] (ii) It is given that ( ) 251 32 31 23! a rr rr rr + == −+ =− . Find the value of a. [3] (iii) State the value of 2 3 31 !r rr r  = −+  . Hence evaluate ( ) 2 7 1 1!r rr r  = −− + . [4]

5 ANGLO-CHINESE JUNIOR COLLEGE 2024 H2 MATHEMATICS 9758/01 [Turn Over 8 The curve C is defined by the parametric equations 2cosx =− and siny =− where 0  . (a) Show algebraically that the gradient of C is never negative for all points on C. [2] (b) Find the equation of tangent that is parallel to y – axis. [2] (c) If  is sufficiently small for 3 and higher powers to be neglected, show that 2d d y a a bx  + + , where a and b are constants to be determined. [3] The line D has cartesian equation 1 4yx=+ . (d) Find the exact x-coordinates of the point(s) of intersection(s) of curve C and line D. [4] 9 The function f is given by ( ) ( ) 1f ,for , ,x x a x x a xa= − +  − where a is a positive constant. (i) Using differentiation, (a) find, in terms of a, the coordinates of the stationary point(s) of ( )fyx= for .xa [2] (i) (b) show that ( )fyx= has no stationary points for .xa [2] (ii) Sketch the curve of ( )fyx= , showing clearly the equations of asymptotes, the coordinates of the points where the curve crosses the axes and coordinates of any turning point(s). [3] (iii) Describe a sequence of transformations which transforms the curve of ( )fyx= on to the curve of 122 2y x a x= − + . [3] The function g is given by ( )