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 measured in Kelvin (K). The temperature T decreases linearly by 6.5 K for every one-kilometre increase in the altitude above sea level. It is given that the temperature is 293 K at sea level. (i) Find a differential equation for d d P h in terms of P and h. [2] (ii) Given that 0P , solve this model in part (i) to show that 293 6.5 b P A h=− , where A and b are constants. Explain whether b is positive or negative. [4] It is given that the atmospheric pressures are 101 300 Pa at sea level and 80 000 Pa at 2h= . (iii) Find the atmospheric pressure at the top of Mount Everest with an altitude of 8848 metres. [4] (iv) For 0h , sketch the graph of P against h. Explain a limitation of this model. [3]

4 © NJC 2025 Section B: Probability and Statistics [60 marks] 5 The random variable Y has distribution ( )B 12, p for 0 1.p (i) Show that ( ) ( ) ( ) ( ) P 13 P 1 1 Y k k p Y k k p =− == − − . [2] It is given that the mode of the distribution is 4. (ii) Find the exact range of values of p. [3] 6 A B C D E F A circular board has 6 sectors labelled A, B, C, D, E and F. A game is played by placing a pawn on sector A. It is then moved clockwise around the board according to the number shown on the top face of a fair die when it is tossed. The die has faces labelled 2, 2, 3, 4, 4, 5. For example, an initial toss of 4 would move the pawn from sector A to sector E. The player continues tossing the die and moving the pawn accordingly. When the pawn lands on sector F, the game ends. Let T be the number of tosses required to move the pawn until the game ends. (a) Show that ( ) 5P3 36T == . Hence, copy and complete the probability distribution table for T. t 1 2 3 4 or more ( )P Tt= [4] The score S for each game is given by 5,ST= for 1 ,2,3t= and 0s = for 4.t  (b) Find ( )E S and ( )Var .S [3]

5 © NJC 2025 [Turn_over 7 A toy train is made up of 3 types of train units: engine, carriage, and wagon. There are 4 identical engines (E), 3 identical carriages (C), and 2 identical wagons (W). (a) All the train units are placed in a line such that either the first unit is an engine or the last two units are engines, or both. Find the number of ways to form the toy train. [4] (b) For another kind of toy train, all the train units are placed in a line such that the first train unit is an engine and all the carriages are in front of the wagons. An example of such a toy train would be: E C C E E C W E W. Find the number of ways to form the toy train. [3] 8 The Ministry of Health declared "War on Diabetes" in 2016 to rally a whole-of-nation effort to tackle diabetes. A study was conducted to investigate the effects of diabetes on one’s health, specifically to determine if the blood pressure of an adult aged between 50 and 60 years old is dependent on one’s haemoglobin level. Data from six patients in this age group from a hospital was collected. Their haemoglobin level, s mmol/mol, and blood pressure, p mmHg, are shown in the table. (i) Draw a scatter diagram of these data. [1] It is thought that blood pressure p can be modelled by one of the formulae 2 or dp a bs p c s= + = + where a, b and c are positive real numbers and d is a negative real number. (ii) By using your diagram, explain which of 2p a bs=+ or dpc s=+ is the better model. [2] (iii) Using the model you chose in part (ii), write down the equation for the relationship between s and p, giving the numerical values of the coefficients. State the product moment correlation coefficient for this model. [2] (iv) Estimate the value of p when 45s= . Determine whether this estimate is reliable. [3] s 32 37 41 46 50 54 p 120 150 168 172 179 183

6 © NJC 2025 9 In a sales campaign, a supermarket gives each shopper a receipt printed with a cartoon character when a shopper pays for his or her purchase. There are 10 different cartoon characters available, character A, B, C, …, J. The printed cartoon character is equally likely to be any one of the 10 different cartoon characters. A shopper received three receipts after having made three purchases. (i) Find the probability that only one receipt has the character A printed on it. [2] (ii) Given that exactly one of the cartoon characters printed is A, find the probability that the other two characters printed are B and C. [3] To redeem a gift, exactly 8 distinct cartoon characters need to be collected. After some time, Tom collected 7 distinct characters. Let np be the probability that Tom can redeem the gift on exactly the thn additional purchase. (iii) Find 1p and the recurrence relation in the form ( )1fnnpp −= , for 2n . [2] (iv) Find the probability , in terms of n, that at most n additional purchases are needed to redeem the gift. [2] 10 The production manager of a food container manufactur