RI Yr 5 H2 Math TP 2025 (Qn)

_________________________________________________________________________ This document consists of 4 printed pages. RAFFLES INSTITUTION  RI2025 Mathematics Department [Turn over RAFFLES INSTITUTION 2025 YEAR 5 TIMED PRACTICE Higher 2 MATHEMATICS 9758 2 hours Additional Materials: Pr inted Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Book let. Follow the instructions on the front cover of the answer booklet. Give non-exact numerical answers correct to 3 significant figur es, 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question.

2 H2 MA 9758/2025 RI Year 5 Timed Practice 1 Given that the curve ya x b x passes through (1 , 4 ) and intersects another curve 3yc x at 3x and 3,x find the values of a, b and c. [4] 2 Referred to an origin O, the position vectors of three points A, B and C are 325 ,ij k 42ij k and 2+ 4 ij k respectively. (a) Determine the value of  for which points O, A, B and C are coplanar. [2] (b) Given that OD OA OB      , find the value of  for which OD   is perpendicular to the y – axis. [2] (c) Find the exact area of triangle .OAD [2] 3 The function f is defined by   f : 3 1 , where , 1x xx x x  . (a) Find   1f x and state the domain of 1f. [4] (b) Sketch on the same diagram the graphs of fyx and  1f,yx  giving the coordinates of any points where the curves cross the axes. [3] The function g is defined by g : e , where , 0.xxx x   (c) Show that the composite function fg exists and state the range of fg. [2] 4 It is given that 2i is a root of the equation 43221 5 0 .za zb za z   (a) Find the values of the real numbers a and b, and without the use of a calculator, find the other roots of the equation. [6] (b) Deduce the roots of the equation 43 22i i 1 5 0 .wa wb wa w    [2]

3 H2 MA 9758/2025 RI Year 5 Timed Practice [Turn over 5 Do not use a calculator in answering this question. The curve C has equation 22 34 0xx y y   . (a) Find the exact x-coordinates of the stationary points of C. [4] (b) For the stationary point with 0,x determine, by differentiation, whether it is a maximum or minimum. [4] 6 (a) Express 3 8x  in the form    22x ax bx c  , where a, b and c are constants to be determined. [1] (b) Without using a calculator, solve the inequality  2 8 11 x xx x  . [4] (c) Hence, solve   2 2 8 .21 22 1 q q qq     [3] 7 A particle moves on the cartesian plane in such a way that at time t, its position is given by the parametric equations cos , sinx tt y t t . The particle starts moving from the origin. (a) Find the equation of the normal to the path of the particle at 2t  , leaving your answer in exact form. [5] (b) It is given that the instantaneous speed of the particle at time t is defined as 22 dd dd x y tt   . (i) Find an expression for 22 dd dd x y tt   in terms of t, giving your answer in its simplest form. [2] (ii) Find the exact instantaneous speed of the particle when it crosses the positive x-axis for the first time. [2] (iii) Explain if the particle will ever come to rest. [1]

4 H2 MA 9758/2025 RI Year 5 Timed Practice 8 A home interior designer is looking to desi gn a pull-down wardrobe lift. To test his concept, the designer built a small-scale model of it, in which points  ,,x yz are defined relative to a corner of the model at 0, 0, 0O . The rods used in the model can be modelled by equations of straight lines. The main rod can be modelled by the line 1l with equation 1 1, . 0 5 5 8          r  Another rod, line 2 ,l passes through the points A 5, 5, 8 and B 8, 4, 3 . (a) A supporting wire connects B to a point F on 1l such that BF   is perpendicular to 1l . Find the position vector of F. [3] (b) A retractable rod, line Rl , is placed such that it can be modelled by reflecting 2l about 1l . Find a vector equation of Rl . [3] The panels of the small-scale model can be modelled by equations of planes. The main panel, plane 1p , contains the main rod 1l and an attachment point C 0, 0, 6 . (c) Show that 1p can be represented by the cartesian equation 53 0 .xy z  [3] (d) A decorative plank, plane 2 ,p has equation 2 ,:5 31 ypz x   where ,.   If the distance between 1p and 2p is exactly 3 3 units, find the possible values of  and . [3]