RI 2026 H2 MATH PROMO QP

@ RI 2026 [Turn Over RAFFLES INSTITUTION 2026 YEAR 5 PROMOTION EXAMINATION Higher 2 MATHEMATICS 9758 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. 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 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. This document consists of 9 printed pages and 1 blank page. RAFFLES INSTITUTION Mathematics Department

2 H2 MA 9758/2026 RI Year 5 Promotion Examination 1 [It is given that a sphere of radius r units has surface area 24 r .] The diagram shows a sphere inscribed in a cylinder with base radius r units. The sphere remains inscribed within the cylinder as the volume of the cylinder increases at a rate of 2 units3 per minute. Find the exact rate of change of the surface area of the sphere when the volume of the cylinder is 16 units3. [4] 2 (a) Solve the inequality 22 5 7 2 7x x x    using an algebraic method. [4] (b) Hence, solve the inequality 22 5 7 2 7x x x    . [2] 3 (a) It is given that e e 4 2x y y   and that 0y when 0x . (i) Show that   2 2 2 d de e 4 e 0d dx y y y yx x       . [2] (ii) Find in exact form, the first two non-zero terms of the Maclaurin expansion of y . [2] (b) Given that  is sufficiently small, show that   2 sin 2 14 1 tan 2 a b c           , where , and a b c are real constants to be determined. [4] r

3 H2 MA 9758/2026 RI Year 5 Promotion Examination [Turn over 4 A curve C has equation 3 3 3x y xy  . (a) Use calculus to find the exact equations of tangents to C that are parallel to the x-axis. [4] (b) Show that C is symmetrical about the line y x . Hence write down the equations of tangents to C that are parallel to the y-axis. [2] (c) Describe geometrically the region enclosed by the tangents found in parts (a) and (b). [1] 5 A curve C has parametric equations sin 3 cos , 3 sin cos for .x t t y t t t      (a) Find ddyx and the exact coordinates of the stationary point(s) of C . [4] (b) By considering the expressions for 2 2x y and xy , find the Cartesian equation of C . [4]

4 H2 MA 9758/2026 RI Year 5 Promotion Examination 6 (a) A sequence 1 2 3, , , ...u u u is such that 1 1 8 11 3 n nu u   for positive integers n. (i) Describe how the sequence behaves when 1 5.u  [1] (ii) It is given that as n , nu l . Find the exact value of l . [2] (b) It is given that 2 1 1 ( 1)(2 1).6nr r n n n    Find 3 7 (6 2 )( 1) N r r r    in the form 3 2aN bN cN d   , where , , and a b c d are integers to be determined. [5] 7 A curve C has equation 2x ky x k  , where  \ 1,0k  . (a) Find the range of values of k such that C has no stationary points. [3] Let 4k  for the rest of the question. (b) Sketch C, stating the equations of any asymptotes, coordinates of any turning points and the points where C crosses the axes. [4] (c) State the range of values of m for which the graph of  2y k m x k   does not intersect with C. [1]

5 H2 MA 9758/2026 RI Year 5 Promotion Examination [Turn over 8 (a) The diagram below shows the graph of   1 fy x with stationary points at 10, 5    and 13, 2    . The lines 0y and 3x  are asymptotes of the graph. Sketch the following graphs on separate diagrams, showing clearly the coordinates of the axial intercepts, turning points as well as equations of any asymptotes: (i)   1 f 1y x  , [3] (ii)  fy x . [3] (b) Describe a pair of transformations which transforms the graph of 32e xy  onto the graph of 3 1e xy  . [2] y x 0y  O

6 H2 MA 9758/2026 RI Year 5 Promotion Examination 9 The lines 1l and 2l have equations 1 2 32 1 and 2 24 zx ya                     r respectively, where  is a real parameter and a is a real constant. (a) Given that 1l and 2l are skew lines, find the range of values of a . [3] In the rest of this question, 1a . The plane p contains the line 1l and the plane q contains the line 2l . Planes p and q are parallel and n is a vector perpendicular to both p and q. (b) Find n . [2] The points A and B have coordinates  1, 2, 4 and  2, 0, 3 respectively and point C is the foot of perpendicular from A to q . (c) By considering the length of projection of a suitable vector onto n , find the exact perpendicular distance between p and q and hence find AC   . [3] The plane m contains the point A . It is given that the point of reflection of B in m lies on q . (d) State the relationship between m and q. [1] (e) The line 3l contains the points A and B . Given that C is also the foot of perpendicular from B to m , find a vector equation of the line of reflection of 3l in m . [3]

7 H2 MA 9758/2026 RI Year 5 Promotion Examination [Turn over 10 The function f is defined by f : ln 4 ,x ax  for x  , x k , where a and k are constants with 0a . (a) Sketch the graph of  fy x , giving in terms of a , the equations of any asymptotes and the coordinates of the points where the curve meets the axes. [3] (b) If the domain of f is restricted to x b , state the least value of b in terms of a , such that the function 1f  exists. Hence find  1f x and state its domain. [5] The function g is defined by 2 1g: , 4 3xx x for x  and 43x . (c) Explain why the function gf does not exist. [1] It is now given that 3a . (d) Using the restricted domain found in part (b), find the value of x for which 1gf ( ) 1x  . [2]

8 H2 MA 9758/2026 RI Year 5 Promotion Examination 11 On 1 January 2025, Amy puts $100 into an ordinary account which pays no interest. In each calendar year, she deposits the same amount on the first day of every month. At the start of each new year, she increases her monthly deposit by $30 compared to the previous year. Thus she deposits $100 on the first day of every month in 2025, $130 on the first day of every month in 2026, and so on. (a) Find the year in which the total amount in the account will first exceed $50 000. Hence find the month in which this happens. [6] Amy’s mother, Beatrice, retired on 31 Dec 2024. On 1 Jan 2025, she puts $500 000 into a savings account that pays a monthl