NYJC 2022 FM CT2 P2 - modified

This document consists of 7 printed pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2022 [Turn Over NANYANG JUNIOR COLLEGE JC2 COMMON TEST 2 Higher 2 FURTHER MATHEMATICS 9649/02 Paper 2 6th July 2022 3 hours Additional Materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. 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. The number of marks is given in brackets [ ] at the end of each question or part question.

2 NYJC 2022 JC2 Common Test 2 9649/02 [Turn Over Section A: Pure Mathematics [50 marks] 1 The curve with equation sin 2 ,y x x=+ for 0 2 ,x  is rotated completely about the y-axis. The volume of the solid of revolution is denoted by V. (a) Explain with the aid of a sketch, why it is more appropriate to use the method of shells than the method of discs in calculating V. [2] (b) Verify that V is just under 500 cubic units. [3] 2 [Do not use a calculator in answering this question] Let 2π 2πcos isin99    =+        . (a) (i) Write down, in terms of , the roots of 9 10z −= . [1] (ii) Show that 2 3 4 5 6 7 810       + + + + + + + + = . [1] The complex numbers 1z and 2z are given by 234 1z    = + + + and 1 2 3 4 2z    − − − −= + + + . (b) (i) Find the exact value of 12zz+ , justifying your working. [2] (ii) Deduce the exact value of 2 4 6 8cos cos cos cos9 9 9 9    + + + . [3] (iii) Express 12zz in the form 2 4 6cos cos cos9 9 9a b c d   + + + , where a, b, c and d are integers to be determined. [3]

3 NYJC 2022 JC2 Common Test 2 9649/02 [Turn Over 3 The linear transformation 44:T →  is represented by the matrix 0 1 2 1 0 0 22 1 0 1 2 k k k k k k −  = − − − A . Let N and R denote the null space and the range space of T respectively. (i) If dim(N) < dim (R), what can you say about the possible value(s) of k ? [2] It is given that k = 1. (ii) Find a basis for NR . [3] (iii) Let m be the least positive integer such that mA represents the zero transformation (i.e. the transformation that maps all elements into the zero element). By first writing Ax as a linear combination of appropriate basis vectors and considering 2Ax , find m. [4] 4 The diagram below shows an equilateral triangle OAB of side length 1 in a cartesian plane. The centroid of a triangle is the point in the triangle that minimises the sum of squares of distances from the point to each vertex. In the diagram above, the centroid of OAB is the point P that minimises the quantity 2 2 2S OP AP BP= + + . (i) By considering ( ) ( ), cos , sinP x y P r r = , write down S in terms of r and  , and use differentiation to find the exact coordinates of the centroid of OAB. You should prove that this gives a minimum value of S. [10] Let G be the midpoint of AB, and denote the centroid of OAB by M. (ii) Show that O, M and G are collinear. [2] x O A (1,0) B y P (x, y)

4 NYJC 2022 JC2 Common Test 2 9649/02 [Turn Over 5 A nn matrix is called a Markov matrix if all entries are non -negative and the sum of each column is equal to 1. Prove that a Markov matrix always have an eigenvalue 1. [2] A caravan company has three locations from which it rents caravan for a day. These locations are Airport, Dandeton and Morningong. The daily proportion of migration of caravans is shown in the table below: Rented from Airport Dandeton Morningong Returned to Airport 0.95 0.02 0.05 Dandeton 0.03 0.90 0.05 Morningong 0.02 0.08 0.90 (i) Write down a matrix that represents the migration of caravan. [1] (ii) Find the characteristic polynomial of the matrix found in (i). [3] It is given that the initial fraction of caravans at Airport, Dandeton and Morningong are 0.35, 0.35 and 0.3 respectively. (iii) Find the distribution of caravans after a day. [1] (iv) Due to space constraints in the Airport, the fraction of caravans in the Airport cannot exceed 0.4. Using eigenvalues and eigenvectors, find the long-term distribution of caravans and state the implications on the business. [7] Section B: Probability and Statistics [50 marks] 6 Let 12, , ... , nX X X be n independent observations of a continuous random variable X which is uniformly distributed between 0 and m. Another random variable W is defined by  12max , , ... , nW X X X= . (i) Find an expression for P( )Ww in terms of m, n and w. [2] (ii) Find the probability density function of W. [1] (iii) Find ( )E W . [2]

5 NYJC 2022 JC2 Common Test 2 9649/02 [Turn Over 7 Most students would probably argue that they failed a test due to careless algebraic errors only. However, considerable errors from misconception and misinterpretation leads to uncertainty to this claim. As such, a study is conducted to compare the differentiated performances of students of those who failed, the average and the high performers, with their total number of conceptual and interpretation errors made in a lecture test on Integration. The following table summarises the finding. Grades Total number of Conceptual and Interpretation Errors made Total ≤ 4 5 – 6 ≥ 7 A 6 0 0 6 B 5 1 0 6 C 3 1 0 4 D 4 6 2 12 E 0 3 1 4 S 1 1 1 3 U 1 6 15 22 Total 20 18 19 57 Carry out a suitable test at 1% significance level to ascertain the students’ claim that their performances in the lecture t est on Integration is independent of the total number of conceptual and interpretation errors made. [7] 8 The continuous random variable X has probability density function given by ( ) 23 (4 ) if 0 2,22 f 3 if 222 0 otherwise 4, . xx x x  −   =     (i) Given P(X > a) 0.1= , find a. [3] (ii) Given E(X) = 15 ,11 find Var(X). [2] (iii) 55 independent observations of X are taken. Find the probability that the sum of these observations is between 56 and 78. [2]

6 NYJC 2022 JC2 Common Test 2 9649/02 [Turn Over 9 During recess, Mitch and Megan play a game by tossing a coin. The probability that the coin lands head up is p and tail up is 1qp=− . Mitch repeatedly tosses the coin until two consecutive heads or two consecutive tails are obtained. If two consecutive heads are obtained, the game ends and Mitch wins. If two consecutive tails are obtained, the game ends and Megan wins. (i) Find the probability that Mitch wins the game, simplifying your answer. [2] Let X be the number of tosses needed for a winner to emerge. (ii) Show that 1 2 2P( 2 ) ( ) ( ) kX k pq p q −= = + . Find P( 2 1)Xk=+ . [4] (iii) Hence find E( )X when 0.4p= . [3] [You may assume that 12 1 (1 ) for < 1r r rt t t  −− = =− .] 10 The scores for a group of twelve randomly selected golfers in the first two days of a golf tournament are summarised in the following table: Golfer 1 2 3 4 5 6 7 8 9 10 11 12 Day 1 69 70 76 69 70 70 69 72 72 70 68 73 Day 2 70 68 74 72 73 75 71 70 76 70 72 71 (i) State, with a reason, whether a two-sample t-test or a paired-sample t-test is more appropriate to justify the claim that