2025 JPJC H2 Math Prelim P1 QP

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2025 MATHEMATICS 9758/01 Higher 2 3 Sept 2025 Paper 1 3 hours Additional materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST This document consists of 7 printed pages and 1 blank page. [Turn over 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 co rrect 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 by [ ] at the end of each question or part question.

2 1 The curve C has equation 2 1 4y xx= − . (i) Find d d y x . Hence, find the x-coordinate, 1xx= , of the turning point on C and determine its nature. [3] (ii) Using calculus, find the exact area of the region between C, the x-axis and the lines with equations 1x = and 1xx= . [3] 2 (i) Find, in terms of a, the roots of the equation ( ) 2 1 xa xa =− − . [3] (ii) On the same axes, sketch the curves with equations ( ) 2 1y xa = − and y x a=− , where 1a  . Hence solve the inequality ( ) 2 1 xa xa − − . [3] 3 A ladder of length 3.12 m is sliding down a vertical wall such that the foot of the ladder is moving along the floor at a constant rate of 0.2 m/s (see diagram). Find the rate at which the top of the ladder is sliding down the wall when it is 1.2 m above the floor. [4] Top of ladder Foot of ladder wall floor floor ladder

3 4 Functions f and g are defined by   2 2 f : ln( 1) 2, , 3g : 4 3 , . 2 x x x a x x x x − +  + −  (i) It is given that the function 1f− exists. State the smallest value of a. [1] (ii) Find an expression for 1g ( )x− , stating its domain. [3] Using the value of a found in part (i), (iii) determine whether the composite function 11gf−− exists. [1] 5 (a) The curves 1C and 2C have equations 22 194 xy −= and 2 2 2y x k+= respectively, where k is a positive constant. (i) Sketch 1C and 2C on the same diagram, stating the coordinate