ASRJC 9758 2024 Prelim P2

[Turn Over ANDERSON SERANGOON JUNIOR COLLEGE H2 MATHEMATICS JC2 Prelim Paper 2 (100 marks) 9758/2 16 Sept 2024 3 hours Additional Material(s): List of Formulae (MF26) CANDIDATE NAME CLASS / READ THESE INSTRUCTIONS FIRST Write your name and class in the boxes above. Please write clearly and use capital letters. Write in dark blue or black pen. HB pencil may be used for graphs and diagrams only. Do not use staples, paper clips, glue or correction fluid. Answer all the questions and write your answers in this booklet. Do not tear out any part of this 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. 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. All work must be handed in at the end of the examination. If you have used any additional paper, please insert them inside this booklet. The number of marks is given in brackets [ ] at the end of each question or part question. Question number Marks 1 2 3 4 5 6 7 8 9 10 11 Total This document consists of 22 printed pages and 6 blank pages.

2 Section A: Pure Mathematics [40 marks] 1 By using the substitution cot , for 0 , 4x=    find 22 1 d . 1 x xx + [4] 2 (a)(i) Find 2 d98 49 uu u − + . [3] (ii) The curve C is given by the parametric equations 2 1x u u= + + , 2 9 , where 0.49 uyu u= + Find the exact area bounded by C, the x-axis and the line 3.x= [4] (b) Find the volume of the solid formed when the shaded region bounded by the lines x = – 4, y = 2 and the ellipse 22( 2) 4( 1) 4xy+ + − = is rotated through 2π radians about the y-axis. Give your answer correct to 1 decimal place. [3] 3 With respect to the origin O, the points A, B, C, D and E have coordinates A ( )2,3, 4 , B ( )6,5,7 , C ( )8,9,6 , D ( )4,7,3 and ( )5,6,10E . (a) Show that the cartesian equation of the surface containing the points A, B and E is 5 2 5x y z− + =− . [2] A line passes through the point D and the midpoint M of the line segment EC. (b) Find the vector equation of the line DM. [3] (c) Find the exact coordinates of the foot of the perpendicular from the point M to the surface found in part (a). [3] (d) Hence find the exact shortest distance from the point M to the surface found in part (a). [2] (e) Verify the line DM intersects the surface found in part (a) at the point P with coordinates ( )9,8,13 . Hence find the vector equation of the reflection of the line DM about this slant surface. [4] y 1 x –4 –2 2 O

3 [Turn Over 4 A popular toy company is designing a new water play feature for children. The toy consists of a cylindrical water container that will hold water for various playful activities. This cylindrical container is designed to be inscribed within a fixed, rigid hemispherical shell made of durable plastic of negligible thickness. The shaded region in the diagram above shows the cross -sectional view of the upright cylindrical container that is inscribed in a hemisphere with fixed radius 2a cm. (a) If the radius of the cylindrical water container is r cm, show that the volume V cm3 of the water container is given by 2 2 24V r a r=− . [1] The unique feature of this toy is that the height of the cylindrical container is adjustable, allowing it to expand or contract while always touching the inner surface of the hemisphere. (b) Water is pumped into the container at a rate of 31100 cm s − while the adjustment is taking place. If a = 50, find the exact rate of change of the radius of the container at the instant when the height of the water container is 80 cm. [5] (c) Using differentiation, find in terms of a, the value of r which gives a maximum value of V. Justify that this value indeed gives a maximum V. Hence write down the exact maximum volume of the cylinder in terms of a. [4] (d) Sketch the graph showing the volume of the cylinder as the radius of the cylinder varies. [2] r cm 2a cm

4 Section B: Probability and Statistics [60 marks] 5 An amateur music composer is arranging a sequence of four musical notes followed by three beats. There are 7 possible notes (labeled A to G) and 5 possible beats (labeled 1 to 5). The order of the notes and beats is important in the composition. Find the probability that a randomly chosen sequence has (i) the third beat being a higher number than the second beat, [1] (ii) exactly two notes the same or exactly two beats the same, but not both. [3] 6 Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered 0, 1, 2, and 3 respectively. The dice are rolled, and the numbers faced down on the two dice are recorded. The random variable T is defined as the score on the red die multiplied by the score on the blue die. (i) Find the probability distribution of T. [3] (ii) Find E(T) and show that Var(T) 115 16= . Show your workings clearly. [2] (iii) Evaluate ( )P2T − , where = E(T) and 2 = Var(T). [2] 7 The masses, in grams, of the packets of semolina flour follow the distribution 2N(225, 25 ) and the masses, in grams, of the packets of millet flour follow the distribution 2N( , ). (a) Find the probability that 4 times the mass of a packet of semolina flour is between 0.85 kilograms and 1.05 kilograms. [2] (b) Let M be the mean mass of 3 packets of semolina flour and 2 packets of millet flour. Given that P( 125) P( 265) 0.02MM =  = , show that the value of  is 150. Hence, by finding an equation involving  , find the value of  . [5]

5 [Turn Over 8 An office team of 10 people includes 7 men and 3 women named Anne, Beth, and Cathie. For an upcoming fire drill exercise, 5 individuals will be chosen, each assigned a unique role, to carry out the drill. Determine the number of possible ways to select 5 people from this group of 10 (i) to conduct the fire drill, [1] (ii) such that at most 1 woman is selected to conduct the fire drill. [2] (iii) After the fire drill exercise, the 10 people are to hold a discussion at a round table with 10 identical seats. Determine the number of ways in which Beth is seated between Anne and Cathie. [1] (iv) A group photo of the 10 people, arranged in two rows of five, was taken after the discussion. Determine the number of ways in which Beth is not standing beside Anne or Cathie. [4] 9 A bakery produces batches of cookies. On average, the proportion of flawed cookies produced is p, where 0 < p < 1. The cookies are packed in boxes of 20. The number of flawed cookies in a box of cookies is denoted by C. (a) State, in context, one assumption needed for the number of flawed cookies in a box to be well-modelled by a binomial distribution. [1] (b) Given that P(C = 0 or 1) = 0.15, write down an equation for the value of p, and find this value numerically. [2] For (c) and (d), take p = 0.08. (c) Ten boxes of cookies are randomly chosen. As part of the bakery’s quality control process, a box of cookies will be accepted if it contains fewer than 4 flawed cookies, otherwise it will be rejected. Find the probability there are at least 2 but no more than 5 rejected boxes. [4] (d) A random sample of 15 boxes of cookies is taken and 3 of the boxes are found to be rejected. Find the probability that the third rejected box occurs on the fifteenth box. [3]

6 10 (a) Observations of