ASRJC Prelims 2025 H2 Math P2 Questions

ANDERSON SERANGOON JUNIOR COLLEGE H2 MATHEMATICS Preliminary Examination Paper 2 (100 marks) 9758/02 3 hours Additional Material(s): Printed Answer Booklet 01 September 2025 MF27 Formula Booklet CANDIDATE NAME CLASS / 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 booklet contains 8 printed pages.

2 Section A: Pure Mathematics [40 marks] 1 Relative to the origin O , the position vectors of two points A and B are a and b respectively. The length of a is k units and b is a unit vector. The angle between a and b is 6 π radians. It is g iven that the point M lies on the line segment AB such that AM :AB is 3:4, find exact area of OAM∆ , giving your answer in terms of k. [4] 2 (a) Without the use of a graphing calculator, solve the inequality 42 12 xx x ++ ≥+ . [3] (b) Hence solve 3 e21 e1 2 x x ++≥ + . [2] 3 (a) Given that y = 1 1tan 5 x− , show that ( ) 2 2 2 dd25 2 0 dd yyxx xx+ += . [2] (b) By further differentiation of this result, find the Maclaurin’s series of y up to and including the term in x3. [3] 4 The curve C is defined by the parametric equations x = cos t + 1 2 cos 5t, y = sin t + 1 2 sin 5t for 0 ≤ t ≤ π 2 . (a) Find the coordinates of the point on the curve C corresponding to t = 0. [1] Another curve D has the following parametric equations. x = 2 + cosk θ , y = 3 sin2 θ for 0 ≤ θ ≤ 2π and k > 0. (b) Find the cartesian equation for curve D. [2] (c) Sketch the curves C and D on the same diagram. [2] (d) Hence determine the range of values of k such that the equation ( ) ( ) 22 cos 0.5cos5 2 4 sin 0.5sin 5 19 t t tt k +− + += has no real solutions. [1]

3 [Turn Over 5 (a) Solve the following integral. (i) 2 12 d 32 x x xx − +− ⌠ ⌡ . [2] (ii) ( ) 2ln 2 dx xx −∫ , where 22 x− << . [3] (b) A function f is defined by f( ) e 2 xx = − . (i) Sketch the graph of f( )yx= , indicating clearly the equation(s) of asymptote(s), if any. [2] (ii) Hence find ln 3 0 f( ) dxx∫ , giving your answer in exact form. [3] 6 The points P and Q are represented by the complex numbers p and q respectively, where