2024 Y6 Timed Prac Rev Paper 5 (Qns)

 RI2023   [Turn over                 CANDIDATE NAME   CLASS 23   MATHEMATICS 9758 3 hours Candidates answer on the Question Paper Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class 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 calc ulator 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 EXAMINER’S USE  Section A: Pure Mathematics TOTAL Q1  Q2  Q3  Q4  Q5  Q6  Subtotal                                100  Section B: Statistics           Q7  Q8  Q9  Q10  Q11  Subtotal           5          7            7           8           13            40    This document consists of 26 printed pages.   RAFFLES INSTITUTION Mathematics Department RAFFLES INSTITUTION 2023 YEAR 6 COMMON TEST     4                10             10              11              12             13              60            2024 Y6 H2 Math Timed Practice Revision Paper 5:      The solution will be released in Ivy on 17 June (Monday)

2   H2 MA 9758/2023 RI Year 6 Common Test   Section A: Pure Mathematics [60 marks] 1 The points A, B and C have position vectors ,a b and c respectively. The points A and B are fixed while C varies. (a) Given that ,a b and c are non-zero vectors such that () ()ca b b ca , find the relationship between ()ca  and b . [2]

3 H2 MA 9758/2023 RI Year 6 Common Test [Turn over   (b) Hence describe geometrically the set of all possible positions of the point C. [2]

4   H2 MA 9758/2023 RI Year 6 Common Test   2 Complete the square for 229 5 .xx    [1] The function f is defined by 2f : 2 9 5 , for .xx x x       (a) Sketch the graph of f( ),yx stating the coordinates of a ny turning points and points of intersection with the axes. [2] (b) If the domain of f is further restricted to 1, x k state with a reason the largest value of k for which the function 1f exists. [2]

5 H2 MA 9758/2023 RI Year 6 Common Test [Turn over   In the rest of the question, the domain of f is x  , 12 x . (c) Find the solution set of 1f( ) f ( )x x . [1] The function g is defined by 1g: x x , for x  , 05 x .  (d) State whether the composite function 1fg exists, justifying your answer. [2] (e) If the domain of g is further restricted to p xq and the composite function gg exists, find an equation involving p and q. [2]

6   H2 MA 9758/2023 RI Year 6 Common Test        3 Cadence, the pedal speed in cycling, is measured in pedal stroke revolutions per minute (RPM). For example, a cadence of 60 RPM means that one pedal makes a complete revolution 60 times in one minute. Every comp lete revolution of th e pedal stroke of a stationary bicycle is equivalent to cycling an approximate distance of 0.006 km. Lisa signed up for a beginner spinning class. The workout routine requires her to start cycling at 30 RPM consistently for the first minute and progressively increase the cadence by 2.5 RPM for every subsequent minute. (a) At which minute will Lisa cycle at 120 RPM? [2] (b) Find Lisa’s average speed of cycling, in km/h, for the first 15 minutes. [3]

7 H2 MA 9758/2023 RI Year 6 Common Test [Turn over   Celina signed up for an intermediate spinning cl ass. The workout routine requires her to start cycling at 25 RPM consistently for the first minute and increase the cadence by 15% for every subsequent minute. (c) At which minute does Celina’s cadence first exceed 120 RPM? [2] (d) If Lisa and Celina start their workout rou tine at the same time, at which minute does Celina’s total distance cycled first exceed Lisa’s total distance cycled? [3]

8   H2 MA 9758/2023 RI Year 6 Common Test   4 A curve C has parametric equations  1 22 2x at , lnyt , for 0 ta  and 1.a (a) C crosses the x-axis at the point P. Find the x-coordinate of P in terms of a. [1] (b) Show that d 0d y x  for 0 ta . [3] (c) Describe the behaviour of the tangent to C as 0t . [1]

9 H2 MA 9758/2023 RI Year 6 Common Test [Turn over   (d) Sketch the graph of C, stating the equations of any asymptotes and the coordinates of the points where the curve meets the axes. [2] (e) The tangent at P cuts the y-axis at the point Q and the normal at P cuts the y-axis at the point R. Find the area of triangle PQR. [4]

10   H2 MA 9758/2023 RI Year 6 Common Test   5 It is given that d (1 ) ( 2)d w wwx   , for 12 w . (I) (a) By solving (I), show that e1 1e x x Aw A  , where A is a positive constant. [5]