ASRJC Prelims 2025 H2 Math P1 Questions

ANDERSON SERANGOON JUNIOR COLLEGE H2 MATHEMATICS Preliminary Examination Paper 1 9758/01 3 hours Additional Material(s): Printed Answer Booklet 25 Aug 2025 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.

2 O B C A r This document consists of 6 printed pages and 2 blank pages. 1 A cubic curve passes through the points (2, 3) and ( 3, 22)−− . Find the equation of the curve if it has a stationary point at (1, 6)− . [4] 2 (a) Show that 3cos3 4cos 3cosθ θθ= − . [3] (b) Hence, or otherwise, evaluate 6 0 sin 2 cos3 d π θ θθ∫ exactly. [4] 3 (a) Given that θ is small, show that 21sec 1 2θθ≈+ . [2] (b) The diagram below shows a circle, centre O and radius r , with points A and B on the circumference such that radians AOB θ∠= , where θ is small. AC is a tangent to the circle at A and OBC is a straight line. (i) Show that the length of chord AB can be approximated by rθ . [2] (ii) Hence show that the perimeter of triangle ABC can be approximated by ( )r abθθ + , where a and b are constants to be determined. [4] 4 The Folium of Descartes is a curve, defined by the equation 33 3x y axy+= , where a is a real constant. It is given that a ≠ 0 for this question. (a) Show that 2 2 d d y ay x x y ax −= − . [2] (b) The point 33 ,22 aa  lies on this curve. Show that the equation of the normal at this point on the curve is independent of the value of a. [2] (c) Given that the curve has a stationary point at (,)pa qa , where p and q are positive constants, find the exact values of p and q. You need not determine the nature of this stationary point. [4]

3 [Turn Over 5 The curve C has equation given by y = 21 3 18 93 xx x +− + . (a) Without the use of a calculator, find the range of values that y can take. [4] (b) Sketch the graph of C indicating clearly its asymptotes and stationary points. [3] (c) State a sequence of transformations that will transform the curve C to the curve with equation y = 21 3(2 1) 18(2 1) 9(2 1) 3 xx x + −− − −+ . [2] 6 It is given that 1 74 9 2 5 ( 1)( 2) 2 1 2 n r r rr r n n= + = −−++ + +∑ . (a