EJC_9758_2025_Prelim_P1

[Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2025 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. MATHEMATICS Paper 1 9758/01 01 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae (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. This document consists of 7 printed pages and 1 blank page.

2 1 (a) Solve the inequality 11 1x x+ − giving your answer in exact form. [3] (b) Hence solve, in exact form, 1e1 e1 x x+ − . [2] 2 Pete has 180 cm of wire. He bends it to form the outline of his brand logo, which consists of a semicircle centered on top of a rectangle as shown in the diagram below. The width and length of the rectangle are 3a cm and b cm respectively. The diameter of the semicircle is one-third the width of the rectangle. Find the maximum possible area enclosed by the wire, showing that it is a maximum value. Give your answer correct to 2 decimal places. [6] b 3a

3 3 In the right -angled triangle ABC, angle C is a right angle. 13 cmAB= , 5 cmBC= and X is a point on BC such that angle BAX is  radians. (a) Find the exact value of cos BAC . [1] (b) By considering triangle AXC, or otherwise, show that 156 12cos 5sinAX = + . [3] (c) Given that  is a sufficiently small angle, show that 213AX p q  + + , where p and q are constants to be determined exactly. [4] 4 The diagram below shows the graph of ( )f2yx= . The graph has a turning point at ( )2, 0 , and asymptotes with equations 0x= and yk= . (a) State a single transformation that will transform the graph of ( )f2yx= onto the graph of ( )f 2 4yx=+ . Hence sketch the graph of ( )f 2 4yx=+ . [3] On separate clearly labelled diagrams, sketch the graphs of (b) ( ) 1 f2y x= , [3] (c) ( )fyx=− . [3] x y (2, 0) O  A B C X

4 5 The diagram shows the curve with equation 1y x= . n rectangles of equal width are drawn under the curve between 1x= and 2x= . Let nS be the total area of the n rectangles. (a) Let 2n= . By considering 2S , show