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 ( ),0Dd and ( ),0Ee . The graph of ( )fyx= is as shown in the diagram below. Sketch the graph of ( )fy a x=− , labelling the coordinates of the corresponding points of A, B, C, D, and E clearly. [3] (b) The graph of ( )gyx= has asymptotes y = 0, y = – m and x = – n, and intersects the y-axis at ( )0,Pp , where p < m. The graph of ( )gyx= is as shown in the diagram below. Sketch on separate diagrams the graphs of (i) ( )gyx= and [3] (ii) ( )gyx= , [2] labelling clearly the asymptote(s) and coordinates of P on each graph. y = f(x) y x C(c, – b) B(a, b) A(– a, 0) D (d, 0) E(e, 0) P(0, p) y y = – m y = 0 x x = – n y = g(x)

4 NYJC 2025 JC2 Preliminary Examination 9758/01 5 The nth term of a series G is given by 2 1 173 7ngn n=−+ , where 1n . (a) Given that 1 2 ( 1)(2 1),6 n r n nnr = += + find 1 . n r rm g =+  (You need not simplify your answer.) [4] The nth term of a series H is given by 3) 107(5n nh −=+ , where 1n . (b) State the smallest number that can be found in both series H and G. [1] The sum of the first n terms of a series J is given by ,2 (3 1)nb − where b is a positive odd integer. (c) By finding the nth term of series J, explain why each of the terms in series J is a term of the asdvasarithmetic progression with first term 1 and common difference 2. [3] 6 The points P, Q and R representing the complex numbers p, q and r on an Argand diagram are such that ( )arg p = and ( )arg q = , where 0 2   ,  >2 and r p q=+ . (a) If pq= , describe the shape of the quadrilateral of OPRQ. Hence find ( )arg r in terms of  and . [3] (b) The point Q’, representing the complex number q’, is the reflection of the point Q in OP. State the angle POQ’. [1] By leaving your answers in terms of ,  and q where applicable, hence, or otherwise, (i) find the argument of the complex number q’, [1] (ii) find the real and imaginary parts of q’ and write down q’ in iab+ form. [3] 7 (a) Use the substitution 2sinx = , where 0 2  , to find 1 2 0 16 d1 x xx− exactly. [4] (b) Find 2 1 d 1 12 b a x x xx − −+  in terms of a and b, where 1ab . [4] 8 A sequence is such that 1 1.5,nnu u+ =+ for .1, 2, 3,n=  (a) Given that 150 99 ,uu = show that 1 0.75.u = [2] (b) Find the least value of n such that 2 4 6 2 ... nu u u u+ + + + is more than 2025. [2] (c) It is given that 25, ku u and 5u are the first three terms of a geometric progression. (i) Find k. [2] (ii) Find the sum of the first 13 terms of the geometric series, giving your answer to 3 decimal places. [2]

5 NYJC 2025 JC2 Preliminary Examination 9758/01 [Turn Over 9 The function f is defined by ( ) 1f 23 xx x += − for , x x k . (a) State the value of k and explain why this value has to be excluded from the domain of f. [2] (b) Determine, with a reason, if 2f exists. [2] (c) Find ( )1f x− . [2] (d) Hence find ( )h x for which ( ) 3 7 1fh , 2 1 2 xxx x −= − . [3] 10 In the question you may use expansions from the List of Formulae (MF27). (a) Find the Maclaurin expansion of 3cos( )t in ascending powers of t, up to and including the term in t12. Hence, find the Maclaurin series of 23 0 cos )d( x t tt up to and including the term in 15x given that x is small. [4] (b) Use your expansion from part (a) to find an approximate value for 0.1 23 0 cos( ) dt tt , correct to 5 decimal places. [1] (c) Find 23 0 cos )d( x t tt in terms of x. Hence evaluate 0.1 23 0 cos( ) dt tt , correct to 5 decimal places. [3] (d) Comparing your answers to parts (b) and (c), and with reference to the value of x, comment on the accuracy of your approximations. [1] 11 The curve C has equation 22 16xy−= , where 0y . The line L has equation 1 11 22yx=− + . (a) Find the area enclosed by C, L and the line 8x= . [3] (b) For 0x , the region bounded by C, L and the x-axis is rotated about the x-axis through 2 radians. Find the exact volume generated. [4] (c) The region bounded by C, the line 5x= and the x-axis is rotated about the y-axis through 2 radians. Find the exact volume generated. [4]

6 NYJC 2025 JC2 Preliminary Examination 9758/01 12 Scientists are studying the growth of a newly discovered biomass that entered Earth’s atmosphere together with an asteroid. A sample of 100 grams of the biomass was collected initially . It was found that on the next day, the sample grew to 110 grams. The scientists observed that the rate of growth of biomass was proportional to the amount of biomass present. The amount of biomass at time t days is denoted by B grams. (a) Write down a differential equation relating B and t. [1] (b) Solve the differential equation obtained in part (a), expressing B in terms of t. [4] (c) Sketch the graph of B against t, and explain what