SAJC 9758 2025 Prelim P2

[Turn Over ST ANDREW’S JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 MATHEMATICS 9758/02 Paper 2 16 September 2025 (Tuesday) Preliminary Examination 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) ______________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Answer all questions. Total marks: 100 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 document consists of 8 printed pages.

2 [Turn Over Section A: Pure Mathematics [40 marks] 1 (i) Solve the inequality 23 4 2xx−  + . [4] (ii) Hence, solve 2 2 3 1 214 xxxx +− . [2] 2 (a) It is given that 64f ( )x rpx qx++= , where p, q, and r are real constants. (i) Show that if x = is a root of f ( ) 0x = , then x =− is also a root. [1] (ii) Given now that 5x= and x = are roots of f ( ) 0x = , where Re( ) 0  and Im( ) 0  , write down all the remaining roots. [3] (b) The complex number z satisfies the equation 32 (3 ) (2 6i) 6 0z a z z+ − − + − = , where a is a complex number. It is given that one of the roots is 1 + i. Find a and the other roots of the equation. [5] 3 The diagram below shows  ABC and  AED.  ABC is an equilateral triangle with each side measuring 2 units and  AED is a right-angled triangle with 4ADE x  = + . Points C and E both lie on the line BD. (a) Show that 31 tan 4 BD x=+  + . [2] (b) Given that x is sufficiently small for 3x and higher powers of x to be neglected, show that 2BD a bx cx + + where a, b and c are exact constants to be determined. [4] A B C D E 4 x + 2

3 [Turn Over 4 A sequence 1 2 3, , , ...u u u is such that 123 19 16, for 0rru u r+ = +  and 1 2u = . (i) Write down the value of 10u , giving your answer correct to 4 decimal places. [1] (ii) It is given that as r→ , rul → . Show that 4l = . [1] (iii) Hence, find the smallest integer r for which ru exceeds 99.9% of l . [3] It is known that the kth term of this sequence is given by 1 1942 23 k ku − =−  . (iv) Given that a new series nS is defined by ( ) 1 n nk k S