NJC_9758_2025_Prelim_P2

NATIONAL JUNIOR COLLEGE SENIOR HIGH 2 PRELIMINARY EXAMINATION Higher 2 NAME CLASS 2ma2 REGISTRATION NUMBER MATHEMATICS 9758/02 Paper 2 22 September 2025 3 hours Candidates answer on the Printed Answer Booklet. Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Write your name, class and registration number on the work you hand in. Write in dark blue or black pen. Answer all the 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. You are reminded of the need for clear presentation in your answers. Up to 2 marks may be deducted for improper presentation. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 7 printed pages.

2 © NJC 2025 Section A: Pure Mathematics [40 marks] 1 Do not use a calculator in answering this question. The complex number w is such that i3w=− . (i) Find w and ( )arg w in exact form. [2] (ii) Represent w , w− and 2 w− in the same Argand diagram, illustrating clearly the geometrical relationship between them. [3] (iii) Hence, find ( )arg 2 w− exactly. [2] 2 The curve E has equation ( ) 2 24 4,x a y− + = where 02 a . (i) Sketch E. [2] You are now given that 1a = . (ii) Show that the equation of a normal to E at 0x= is 323 2yx=− . [4] (iii) The region R is bounded by E, the normal in part (ii) and the y-axis. Find the volume of the solid generated when R is rotated through 2π radians about the y-axis. [5] 3 With respect to an origin O, points P and Q have variable position vectors p and q respectively, given by ( ) ( )sicos nt tp k i j= +− , ( ) ( )sin 2cos2t ti j kq −= + , where  is a real parameter and 0 πt . (i) By considering the cross product or otherwise, find the exact values of t and  if ,OP and Q are collinear. [5] It is given that 0.5. = (ii) Show that .cos3 0.5tp q• −= Find the greatest length of projection of q onto p . [4]

3 © NJC 2025 [Turn_over 4 The rate of decrease of atmospheric pressure P Pascals (Pa) with respect to altitude h kilometres above sea level, is proportional to P T , where T is the temperature meas