SAJC 9758 2025 Prelim P1

[Turn Over ST ANDREW’S JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 MATHEMATICS 9758/01 Paper 1 1 September 2025 (Monday) 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 7 printed pages and 1 blank page.

2 [Turn Over 1 A function f is defined by 32f( )x ax bx cx d= + + + where a, b, c and d are constants. The graph of f( )yx= has a turning point at 1x=− and passes through the points (2,10) and ( 2, 2)−− . Given also that 2 2 f ( ) d 16xx − = , find the values of a, b, c and d. [4] 2 A closed cylinder is designed to have a fixed external surface area of p cm2 such that its volume is maximum. Find the radius of the cylinder in terms of p. [5] 3 The lines 1l and 2l have equations 1 2 12 : 2 1 , 02 02 : 5 2 , 83 l l   −       =+                 = + −      −    r r where  and  are parameters. (i) Show that 1l and 2l are skew lines. [2] The point P has coordinates ( )1 ,5,0 and the plane 1 has equation 20yz− − = . (ii) Find the coordinates of 'P , the reflection of P in 1 . [4] (iii) Find the cartesian equation of the plane 2 containing P and 1l . [3]

3 [Turn Over 4 (a) The function f is defined by 22 , 0 , f ( ) ( ), 2 , a x x a x a x a a x a  −   =  −   and f(x) = f(x + 2a) for all real values of x, where a is a positive constant. (i) Sketch the graph of y = f (x) for 23a x a−   . [3] (ii) Find 101f 2 a  in terms of a. [2] (b) The function g is defined by g : ln ,where , 0 e.x x x x    The diagram below shows the graph of the quadratic function h with domain defined where 04 x . The graph has a maximum point at x = 1 and has end - points e0, 2   and (4,0). (i) Determine whether hg exist. Justify your answer. [2] (ii) State the other value of x for which h(x) = e 2 . Gi