RVHS 9758 2025 Prelim P2

1 ©RIVER VALLEY HIGH SCHOOL 9758/02/2025 RIVER VALLEY HIGH SCHOOL 2025 JC2 Preliminary Examination Higher 2 NAME CLASS INDEX NUMBER MATHEMATICS Paper 2 Additional Materials: Printed Answer Booklet List of Formulae (MF27) 9758/02 19 Sep 2025 3 hours READ THESE INSTRUCTIONS FIRST This document consists of 8 printed pages and 0 blank page. Write your class, index number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. 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 are required to 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.

2 ©RIVER VALLEY HIGH SCHOOL 9758/02/2025 Section A: Pure Mathematics [40 marks] 1 A sequence is such that 1uN= where N is a constant, and 1 38nnuu+ =− , for 1n . (a) Describe how the sequence behaves when (i) 4N = , [1] (ii) 3N = . [1] (b) Find the value of N for which 5 409u = . [2] (c) The sequence ( ) ( )1 n nnv u k= − + for 1n where k is a constant, is convergent when 1 4u = . State the value of k . [1] 2 The points A , B and C represent the complex numbers 5 6iAz =+ , 9 3iBz =+ and Cz respectively. ABC is an isosceles triangle labelled in a clockwise direction where 90CAB =  . (a) Find Cz . [3] (b) The point D representing the complex number Dz , is such that ABDC is a parallelogram. Find Dz . [2] 3 (a) The region A is bounded by the curves 22 5xy+= and 2 1yx=+ for 1y . Find the volume when A is rotated π radians about the y-axis. [2] (b) The region B is bounded by the curve 2 2 47 y xx = −+ , the line 2yx=− , the line 1x = and the x-axis. Find the exact volume when B is rotated 2π radians about the x-axis. Give your answer in the form 23ππab + , where a and b are constants to be determined. [6]

3 ©RIVER VALLEY HIGH SCHOOL 9758/02/2025 4 The dosage of medicine given to a patient needs to be carefully managed in order to achieve the desired result. (a) To give the patient time to adapt to a particular medicine, the dosage is slowly increased over time. The patient is initially given a dosage of 0.75 units. The dosage is increased by 0.5 units each day until a total of at least 10 units of medicine has been taken. Find the least number of days it takes to achieve this, and the dos age on that day. [4] (b) The doctor prescribed a new medication to another patient. After n days, the total dosage of the new medicine administered to the patient is given by 1 3 5 4 20 5 nn n−  −  units. (i) Show that the increase in dosage of the new medicine administered per day follows a geometric progression. [3] (ii) Find the total dosage of the new medicine administered to the patient after a long period of treatment. [2] (c) Doctors can manage the medication so that the daily dosage of medicine increases as an arithmetic progression. Explain in context why the arithmetic progression is not a preferred model for the dosage of medication. [1]

4 ©RIVER VALLEY HIGH SCHOOL 9758/02/2025 5 The Planetary Defense Coordination Office at NASA observes and tracks Near Earth Objects (NEOs) that could be potentially hazardous in a collision with Earth. Over short periods of time, vectors can be used to model the trajectories of the Earth and NEOs. They use coordinates (x, y, z) with units in millions of kilometers relative to the sun which is at position (0, 0, 0). Earth’s orbit is contained by the plane  with equation 1 10 5  • − = r . (a) A satellite moves along a path with equation 32 2 3 , 31  −       = +           r . Determine whether the satellite crosses Earth’s orbital plane. [3] Over a short period of time, Earth’s motion can be modelled by the equation 15 4 20 1 , 11         = + −       −−    r . An asteroid moves along a path with equation 10 0.25 22 xz y k −− = + = where k is a positive constant. (b) Determine the possible values of k if the paths of the Earth and the asteroid do not intersect. [3] It is now given that k = 3. (c) Find the acute angle between the paths of the Earth and the asteroid. [2] The asteroid is closest to the Earth when they are at points ( )12, 18, 3− and ( )12, 1 , 2.75−− on their respective paths. (d) (i) Find the distance between the Earth and the asteroid at this instant. [2] (ii) By considering the vector between the asteroid and the Earth at this instant, determine whether the distance found in part (d)(i) is the shortest distance between the two paths. [2]

5 ©RIVER VALLEY HIGH SCHOOL 9758/02/2025 Section B: Statistics [60 marks] 6 A Pizzeria sells pizza by individual slices. 8 slices forms a full pizza. (a) A group of three friends bought 4 slices of Hawaiian, 1 slice of pepperoni, 1 slice of seafood, 1 slice of cheese, and 1 slice of vegetarian pizzas. Find the number of ways the pizza slices bought could be arranged in a circle such that the 4 slices of Hawaiian pizzas are placed together. [1] (b) The three friends each eat a slice of Hawaiian pizza. They then want to distribute the remaining 5 slices amongst the three of them. Find the number of ways this can be done such that everyone gets at least one more slice of pizza. [3] 7 In a game, a fair six -sided die is rolled. If a 1 or a 2 is rolled, then the die is rerolled and the score is the result of the reroll. Otherwise, the score is the number rolled originally. (a) Give the probability distribution function of the scores in the form of a table. [2] (b) Show that the expected score is 25 6 . Explain the meaning of this value in this context. [2] (c) Find the exact variance of the scores. [1] 8 A game at a funfair involves pulling balls of different colours out of a bag. A bag contains 1 white ball, 2 blue balls, 3 green balls and 4 red balls. The aim of the game is to pull the white ball from the bag as many times as possible. Players get three attempts. If a red ball is drawn, it is discarded, but any other ball is replaced in the bag. (a) Find the probability that a player draws a white ball and two green balls on their three attempts. [2] (b) Find the probability that exactly one white ball was drawn, given that a red ball was drawn on the second attempt. [3] Events A and B are defined as A: A white ball is drawn on the first attempt B: A white ball is drawn on the second attempt (c) Determine with reason whether events A and B are independent. [2]

6 ©RIVER VALLEY HIGH SCHOOL 9758/02/2025 9 A machine produces a large number of camera lenses daily. During the process, lenses could get scratched and become defective. (a