DHS Prelims 2025 H2 Math P1 Questions

© DHS 2025 This document consists of 7 printed pages and 1 blank page. Name: Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination Year 6 MATHEMATICS (Higher 2) 9758/01 Paper 1 16 September 2025 3 hours Additional Materials : Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all the 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 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 1 1 It is given that 22d2 d yxy x yx = − where 0, 0.xy>< Using the substitution ,yu x= show that the differential equation can be transformed to 2 2d1 .13 d uu uxx =− Hence find the general solution of y in terms of x. [6] 2 A series is given by 1 2(4 3 ) n r r x = −∑ where x is constant. (a) Explain why this is a geometric series . Determine the range of values of x for the sum to infinity of this series to exist. [3] (b) Using 1,x= and given that find ( )( ) 1 0 2(4 3 ) 1 2 5 , n r r xr r − =  − ++ +∑ leaving your answer in the form of 2( ),n an bn c++ where a, b and c are constants to be determined. [4] 3 (a) Find the first three non-zero terms in the Maclaurin series for e sin( ).x x π+ [3] (b) It is given that the three terms found in part (a) are equal to the first three terms in the series expansion of ( )1 c ax bx+ for small x, where a, b and c are constants. Find the exact values of a, b and c. Use these values to find the coefficient of x 4 in the expansion of ( )1, c ax bx+ giving your answer as a simplified rational number. [5] 4 A sequence of real numbers x 1 , x2 , x3 , … satisfies the recurrence relation 1 52 23 n n n xx x + += + for all 1.n≥ (a) Given that the sequence converges to l, find the possible exact values of l. [3] (b) Describe how the sequence behaves when 1 3.x = [1] (c) Given that 5 3503 ,2158x = find the value of 1.x [3] 2 1 ( 1)(2 1),6 n r nr nn = = ++∑

3 DHS 2025 Year 6 H2 Mathematics Preliminary Examination Paper 1 [Turn over 5 The graphs of f '( )yx= and f( )yx= are shown below. (a) State the nature of all turning point(s) of the