NYJC_9758_2025_prelim_P1

This document consists of 6 printed pages and 2 blank pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2025 [Turn Over NANYANG JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 CANDIDATE NAME CT CLASS 2 4 Centre Number/ Index Number / MATHEMATICS 9758/01 Paper 1 2 September 2025 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. The total number of marks for this paper is 100.

2 NYJC 2025 JC2 Preliminary Examination 9758/01 1 Curve C has equation 2 ,1 qxy rpx x += − + where p, q and r are real numbers. Given that C has a turning point at (3, 7) and an oblique asymptote parallel to the line ,53 2yx=+ find p, q and r. [4] 2 An ornament consists of two identical solid right circular cones whose flat bases are separated by a solid sphere, such that the sphere is in contact with the two bases. The axis passes through the vertex of each cone and the centre of the sphere. It is given that the distance between the vertices of the cones is 2 l, and the radii of the bases of the cones are 3 2 l , where l is a constant. Find the radius of the sphere in terms of l if the total volume of the ornament is to be a minimum. [The volume of a sphere with radius r is given by 34 3Vr = .] [6] 3 (a) A triangle ABC is such that 2 2 2AB BC AC+= . By considering AB BC AC+= and using the fact that 2 =v v v for any vector v, prove that ABC is a right-angle. [4] (b) In a triangle ABC, the point D divides AC in the ratio :1 − , where 01  . Let the position vectors of A, B, C and D be denoted by a, b, c and d respectively. Show that the area of triangle ABD is given by k  +  + a b b c c a where k is to be determined in terms of  . [4]

3 NYJC 2025 JC2 Preliminary Examination 9758/01 [Turn Over 4 (a) The graph of ( )fyx= has turning points at ( ),0Aa− , ( ),B a b and ( ),C c b− and intersects the x-axis at the points ( ),0