EJC QP 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 that 7 ln 212 . State your reasoning clearly. [3] (b) By considering nS for a suitable value of n, find a rational number q such that 7 ln 212 q . [2] (c) Express nS in the form 1 g( ) n r r =  , where the function g is to be determined. [1] (d) By considering another suitable set of n rectangles, show with the aid of a diagram that ln 2 1 2 n nS+ where n is any given positive integer. [3] y x 2 … 1

5 6 There are three distinct points A, B and P. The point P lies on the circle with centre O and diameter AB. Relative to the origin O, the position vectors of points A and P are a and p respectively. (a) By expressing AP  and BP  in terms of a and p, show that 0AP BP =.  . [3] The point C with position vector c lies on AB produced such that AC : AB is :1 . (b) Find c in terms of  and a. [2] (c) It is given that PC is a tangent to the circle and that angle AOP is 120 . Using a suitable scalar product, find the value of  . [4] 7 Do not use a calculator in answering this question. An arithmetic sequence 1 2 3, , ,u uu  has 1 9u = and 3u b= , where b is a constant. (a) Find 2u in terms of b . [2] A geometric sequence 1 2 3, , ,v vv  has 1 9v = and 3v b= . (b) Find the possible values of 2v in terms of b . [2] It is now given that 22 8u v−= , and the geometric sequence 1 2 3, , ,v vv  is convergent. (c) Find the value of b . Hence find 2u and 2v . [5] 8 Do not use a calculator in answering this question. The complex number w is such that 2 2iw = and 0 arg( ) 2w  . (a) Find w. [3] (b) Given that one of the roots of the equation 32 (1 i) 0z z z s− + − + = is w, find the other roots of the equation and the value of s. [5] (c) Hence, find the roots of the equation 32i (1 i) 0v v v s− + + + + = . [2]

6 9 (a) (i) State the coordinates of the turning points of the graph of 32 3 9 5y x x x= − − + . [1] (ii) The function f is defined by 32f ( ) 3 9 5x x x x= − − + , x  , x a where a is a constant. Find the largest possible value of a such that 1f − exists, and state the domain of 1f − in this case. [2] (b) The function g is defined by 12g( ) 2 xx x −= + , x  , 2x− . (i) Find 2g ( )x . [2] (ii) Hence find the possible expressions of g ( )n x , where n is a positive integer. [2] (c) The function h is defined by 2h( )xx= , x  , 0x . (i) Explain why the composite function gh exists. [2] (ii) Find the range of gh. [2] 10 (a) Find 4 1 0 edx x−  , leaving your answer in exact form. [4] (b) Find 22sin 3 tan 3 dx x x+ . [3] (c) Use the substitution 6xu= , where 0u , to find 3 1 dx xx+  . [5]

7 11 A cylindrical container of height h metres is originally completely filled with water. The water is flowing out through an opening at the base of the container into a pipe. The rate of flow of water through the opening at time t seconds is proportional to the square root of the height of the water in the container, y metres. Since the container is cylindrical, the amount of water in the container is also proportional to the height of water, so the differential equation relating y and t can be written as d d y t ky=− where k is a positive constant. (a) Solve this differential equation to find y in terms of t, h and k. [4] (b) Hence find the time taken for the container to empty, in terms of h and k. [2] It is found that the water was draining too fast. To regulate the flow of water, a mechanical device is placed at the opening. This device slows the flow of water into the pipe by covering all or part of the opening, where the area covered varies with time. It is proposed to model this situation using the revised differential equation od s d 1c 2 t yy kt = +−   , where k is the same constant as before. (c) With reference to the possible values of cos t , explain how the revised differential equation corresponds to a model with slower flow of water. [1] (d) Given again that the container is originally completely filled with water, solve the revised differential equation to find y in terms of t, h and k. [3] (e) In the case where 10h= and 0.1k= , find the ratio of the time taken in the revised model to the time taken in the original model for the container to empty. Give your answer to 3 significant figures. [2] END OF PAPER