VJC_9758_2025_Prelim_P1

2025/VJC/Math Dept [Turn over VICTORIA JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION 2025 H2 MATHEMATICS 9758/01 PAPER 1 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (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. The total number of marks for this paper is 100. This document consists of 6 printed pages and 2 blank pages.

2 2025/VJC/Math Dept 1 Using an algebraic method, solve the inequality 2 43 14 3 1 x xx − +− . [3] Hence, find the set of values of x that satisfy 2 4ln 3 14(ln ) 3ln 1 x xx − +− . [2] 2 The function f is defined by 1f : , xx x − x , 0,x 1x . (a) Show that ( ) ( ) 21ff xx −= . [3] (b) Find ( ) 3f x in simplified form. [1] (c) Find ( ) 2030f5 . [2] Functions g and h are defined by 1g: xx x − , x , 1x , h : sinx ax− , x , where a is a positive constant. (d) Find the value of a given that the range of hg is ( 1,0− . [2] 3 Referred to the origin O, points A and B have position vectors a and b respectively. Point C lies on OA, such that : 2:1OC CA = . Point D lies on OB, such that ::OD DB = . It is given that the area of triangle ABD is half the area of triangle ABC. (a) Show the area of triangle ABD is given by ( )2   + ab . Hence find the ratio : . [4] (b) The point E has position vector 15 48 +ab . Show that A, E and D are collinear. [3] It is further given that the angle AOB is π 4 and O lies on the perpendicular bisector of the line segment AB. (c) Find the length of projection of a on b , giving your answer in terms of b . Hence find the position vector of the point F, the foot of perpendicular from A to OB. [3]

3 2025/VJC/Math Dept [Turn over 4 (a) A sequence is such that 1up= , where p is a constant and 1 5 81 n n n uu u + = + , for 1n… . (i) Describe how the sequence behaves when 1p= . [2] (ii) Find the value of p for which 6 3125 6253u = . [2] (b) Another sequence 1v , 2v , 3v , … is such that for all 1n… , 21 2n n nv v v k++− + = , where k is a constant. Let 1n n nw v v +=− for 1n… .