TJC_9758_2025_Prelim_P1

1 9758/01/TJC/25 TEMASEK JUNIOR COLLEGE 2025 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9758/01 1 Sep 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 6 printed pages and 2 blank pages. [Turn over

2 9758/01/TJC/25 1 The sequence {un} is a geometric sequence with first term a and common ratio 3 2 , where 0a  . Another sequence {vn} is defined by 9 n n v u= for all positive integers n. (a) Show that {vn} is also a geometric sequence. [2] (b) Given that 22vu= , find the value of a. [2] 2 The curve C has equation 31 ( )xy x y= + − . Find the gradient of the tangent to C at the point A where 0y= . [4] 3 A curve C has equation 22 ay bx x= − , where a, b > 0. (a) Describe the transformation that maps the graph of C onto the graph of 22 ay bx= − . [2] (b) Given that a = 1 and b = 2, s ketch C stating the equations of any asymptote s and coordinates of any stationary points and of the points where the curve crosses the axes . [2] 4 In the triangle ABC, AC = 1, angle BAC = 3  radians and angle ABC = 6   + radians. (a) Show that 3 cos 3 sin BC  = + . [2] (b) Given that θ is sufficiently small such that 3 and higher powers of  may be neglected, show that ( ) 231BC a b  + + where a and b are constants to be determined. [3] 5 (a) Without using a calculator, solve the inequality 2 2 25 223 x xx − −− . [4] (b) Hence solve the inequality ( ) 2 2 1 25 2 1 2 1 3 x xx +−  + − + − . [3]

3 9758/01/TJC/25 6 The points A and B lie on a circle with center O and radius  unit. With reference to the origin O, points A and B have position vectors a and b respectively. The point X on the line segment AB is such that AX : XB = 1 : 3 and the point Y is the foot of perpendicular of X on OB. (a) Find the position vector of X. [1]