XJC H2 Math - Set 3 - P2 (QP)

X Junior College [Turn over This document consists of 6 printed pages and 2 blank pages. X JUNIOR COLLEGE YEAR 6 PRELIMINARY EXAMINATION Mock Arrangement in preparation for Candidates’ Examination Higher 2 MATHEMATICS Paper 2 Additional Materials: Answer Paper Graph paper List of Formulae and Results (MF27) 9758/02 Set III 3 hours READ THESE INSTRUCTIONS FIRST Write your name on the work you hand in. Write in dark blue or black pen on both sides of the paper. 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. 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 are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. ABOUT THIS PAPER X Junior College (XJC) is an unofficial initiative aimed at preparing pre-university and/or junior college students in Singapore for school-level and/or national-level H2 Mathematics examinations through self-prepared mock papers. It has no affiliation with any existing institution in Singapore or worldwide. This mock paper follows closely the 9758 H2 Mathematics GCE Advanced Level syllabus, most suitable for preparation towards preliminary examinations and A–Levels. The paper intends to explore the unconventional ways and/or applications in which topics within the syllabus can be tested, which may affect the difficulty of this paper to varying degrees. While it is ideal to attempt this paper under examination constraints, prospective candidates are reminded not to use this potentially non-conforming paper as a definitive gauge for actual performance. (For enquiries, mail to: xjuniorcollege@gmail.com)

© XJC 9758/02/III 2 Section A: Pure Mathematics [40 marks] 1 The complex numbers 𝑢, 𝑤 and 𝑧 satisfy the equations 𝑢−𝑤+𝑧=−6+6i, 3𝑢+i𝑤−2i𝑧=16+19i, i𝑢∗−2𝑤∗−3𝑧=4−8i. Find 𝑢, 𝑤 and 𝑧, giving your answers in the form 𝑎+i𝑏, where 𝑎 and 𝑏 are real constants. [5] 2 A 1.8-metre man walks on a straight path at a constant speed of 2 ms−1. Nearby, a 9-metre lamp is positioned 4 metres away from a point 𝑋 on the path which is nearest to the lamp. The lamp creates a lit area which illuminates part of the man’s path (see diagram). When the man walks in the lit area, a shadow of 𝑠 metres is cast onto the ground when he is 𝑥 metres away from 𝑋. At one point during the walk in the lit area, the length of the shadow decreases at a rate of 0.3 ms−1, and 𝑇 seconds later increases at a rate of 0.4 ms−1. Find a relationship between 𝑠 and 𝑥 and use it to find the exact value of 𝑇. [6] 3 (a) Use the substitution 𝑥=sin2𝜃 to find ∫√(1−𝑥𝑥)d𝑥. [4] (b) Find the exact volume of the solid formed when the region bounded by the curve 𝑥𝑦4+𝑥=1 and the line 𝑥=34 is rotated completely about the 𝑥–axis. [3] MAN LAMP LIT AREA 𝑋

© XJC 9758/02/III [Turn over 3 4 Two non-overlapping circles 𝐶1 and 𝐶2 have radius 𝑎 and 𝑏 respectively, where 𝑎≠𝑏, and centre 𝑃 and 𝑄 with position vector 𝐩 and 𝐪 respectively, with reference to origin 𝑂. The lines 𝐿 and 𝑀 are outer common tangents of 𝐶1 and 𝐶2. The point where the tangents intersect, denoted by 𝑋, is called the focus of 𝐶1 and 𝐶2 (see diagram). (a) Show by considering similar triangles that 𝑂𝑋⃖⃖⃖⃖⃖⃑=(𝑎𝑎−𝑏)𝐪−(𝑏𝑎−𝑏)𝐩. [2] Another circle 𝐶3 has radius 𝑐, where 𝑎≠𝑐 and 𝑏≠𝑐, and centre 𝑅 with position vector 𝐫 such that 𝐶3 does not overlap either 𝐶1 or 𝐶2. The focus of 𝐶1 and 𝐶3 is 𝑌 and the focus of 𝐶2 and 𝐶3 is 𝑍. (b) Write down the position vectors of 𝑌 and 𝑍 in terms of 𝑎, 𝑏, 𝑐, 𝐩, 𝐪 and 𝐫 where applicable. Hence, show that 𝑋, 𝑌 and 𝑍 are collinear. [5] (c) Given that 𝑌 is the midpoint of 𝑋𝑍, find an expression for 𝑐 in terms of 𝑎 and 𝑏. [2] 5 The variables 𝑥 and 𝑦 are related by the recurrent differential equation (1−𝑥2)d𝑛+2𝑦d𝑥𝑛+2−(2𝑛+1)𝑥d𝑛+1𝑦d𝑥𝑛+1+(4𝑘2−𝑛2)d𝑛𝑦d𝑥𝑛=0,𝑛=0,1,2,3,… , where 𝑘>0. It is given that the line 𝑦=1 is tangent to the graph of 𝑦 against 𝑥 at its 𝑦-intercept. (a) Show that d2𝑦d𝑥2=−4𝑘2 at 𝑥=0. [2] (b) Use the recurrent differential equation to find, in terms of 𝑘, the Maclaurin expansion of 𝑦 in ascending powers of 𝑥 up to and including the term in 𝑥4. [3] It is also known that 𝑥 and 𝑦 satisfy a second differential equation given by d𝑦d𝑥=−2𝑚(1−𝑦21−𝑥2)12, where 𝑚>0. (c) Solve the second differential equation and show that 𝑦=cos(2𝑚f(𝑥)), where f is an inverse trigonometric function to be determined. [3] (d) Deduce the relationship between 𝑘 and 𝑚. [You may use expansions from the list of formulae.] [5] 𝐶1 𝐶2 𝑃 𝑄 𝑎 𝑏 𝑋 𝐿 𝑀

© XJC 9758/02/III 4 Section B: Probability and Statistics [60 marks] 6 A disc is placed on an 8×8 square grid at the upper-left corner. It can be moved one square at a time downwards or rightwards. Find the number of different ways it can reach the square at the lower-right corner without making three or more consecutive downward moves. [4] 7 A company owns a social media platform where users can upload videos of varying durations. Past records suggest that the average duration of uploaded videos is 𝜇0 seconds. For research purposes, a hypothesis test is conducted on the durations, 𝑡 seconds, of a random large sample of uploaded videos, which has been summarised as follows. Σ(𝑡−𝜇0)=−38.5 Σ(𝑡−𝜇0)2=362.869375 The test is done with appropriate hypotheses at 𝛼% significance level, for some integer 𝛼, such that the test does not reject having 𝜇0 as the current average duration of uploaded videos if the sample mean 𝑡 is within the range 29.6325<𝑡<30.3675. It is known that the sample gives a 𝑝-value of 0.0400404. (a) Give a general definition for the 𝑝-value of a hypothesis test. [1] (b) Deduce the hypotheses and the size of the sample used in the test and use them to determine the conclusion of the test in context, where values relevant to the hypotheses and conclusion are to be found. [6] (c) It is found that the durations of uploaded videos have a variance of 1.8752 seconds2. With this information, the company repeats the test above on a new sample and achieves a different conclusion from (b). What can you deduce about this new sample? [1] 8 A game is played with a fair coin and an unbiased cubical die with faces labelled 1, 1, 1, 2, 2, and 3. In one round of the game, the coin is tossed once and the die is rolled once. The outcome of the round, 𝑋, is determined by the result of the coin toss and the number 𝑑 shown on the die as follows. 𝑋={|𝑎𝑑−1|,|𝑎−𝑑|, if the result of the toss is head,if the result of the toss is tail, where 𝑎 is a known single-digit integer. It is known that E(𝑋2) and E(𝑋) are integers. (a) Show that E(𝑋2)=𝑎+136(𝑎−1)2. [3] (b) By considering E(𝑋) in terms of 𝑎, deduce the value of 𝑎. [2] The game is played in a carnival. Players pay $1 to play the game, which consists of two rounds. The sum of the outcomes of the two rounds is recorded and denoted by 𝑆. Players are paid back $𝑦 if 𝑆>15, or $0 otherwise. (c) Find the least integer 𝑦 for which players profit from the game in the long run. [5]

© XJC 9758/02/III [Turn over 5 9 A continuous random variable 𝑋 is normally distributed with mean 𝜇. The distribution curve of 𝑋 can be modelled by the equation 𝑦=f(𝑥), where f is the probability density function of 𝑋. The table below shows the values of