NYJC 2024 FM CT2 - modified

This document consists of 7 printed pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2024 [Turn Over NANYANG JUNIOR COLLEGE JC2 COMMON TEST Higher 2 FURTHER MATHEMATICS 9649 26th June 2024 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 2024 JC2 Common Test 9649/02 [Turn Over Section A: Pure Mathematics [75 marks] 1 Let l be a line in the xy-plane that passes through the origin and makes an angle  with the positive x- axis where 22  −   . Let A be the matrix representation of the transformation 22:T →  that maps every vector in 2  onto its vector projection on l. (a) By considering the standard basis vectors for 2  under the transformation T, show clearly that 2 2 cos sin sin cos cos sin       =  A . [3] (b) Find a basis for the nullspace of T. [2] (c) Find, in exact form, the angle  such that the vector projection of 1 5   onto l is 3 5 53 4 3 4     +  +  . [3] 2 Let A be the matrix representation of the transformation 33:T →  . A basis for the nullspace of T is given by 1 1 1   . (a) State, with a reason, the value of an eigenvalue of A and a corresponding eigenvector. [1] It is also given that 5 2 3 2 0 2 422 − =−− A and the other eigenvalues of A are 2− and 1− with corresponding eigenvectors 1 0 1    and 1 2 0    respectively. (b) Without performing row operations on A , explain why these two eigenvectors form a basis of the range space of T. [2] (c) Hence, express p q r    A in the form of 11 02 10         +          , where  and  are to be expressed in terms of p, q and r. Deduce a similar form for n p q r    A . [4] The matrix B is such that ( ) 1 4 − =−B A A I . (d) Without evaluating B, find the eigenvalues of B. [3]

3 NYJC 2024 JC2 Common Test 9649/02 [Turn Over 3 Let f ( , ) 3 ln( 1)x y xy y x= − + . (a) Find all the first and second partial derivatives of f ( , )xy . [2] (b) Find all the stationary point(s) of f ( , )xy for 0x . For each of the stationary point(s), determine whether it is a local maximum, local minimum, or saddle point. [5] (c) Using local linearisation of f ( , )xy at the point (2, 1) , find an approximate value of f (2.05, 0.98) . [3] 4 Given that 3 2 13iz+ −  , 5 2 1 4iizz+ +  + − and 0arg( 6)z + , illustrate the locus of the points representing the complex number z in the Argand diagram. [3] (a) Find the exact complex number z , in cartesian form ixy+ , which gives the minimum value of 7 iz+− . [3] (b) Determine the maximum value of ( )arg 8 3 iz+− , giving your answer in the form of 1tan p− , for some rational number p. [4] 5 A large tank initially contains 100 gallons of fresh water. Brine flows in at a constant rate of 2 gallons/minute containing 1 kg of salt/gallon. The mixture is constantly stirred and flows out at a constant rate of w gallon/minute. Let x be the amount of salt (in kg) in the tank at time t (in minute). It is given that 2w= . (a) Find x in terms of t. and sketch the graph of x against t. [4] It is given instead that 1w= . (b) Find x when the tank contains 150 gallons of mixture, giving your answer to 2 decimal places.[6] (c) Find the time taken for the salt concentration in the mixture to reach 0.9 kg/gallon. [2]

4 NYJC 2024 JC2 Common Test 9649/02 [Turn Over 6 Alan downloaded a new mobile game FM: Battlegrounds which was developed by No Sine Studios. In the game, players form alliances, grow a population of a certain species and invade the territories of other alliances on a map with finite area. The species in Alan’s alliance are classified as ‘children’, ‘adult’ or ‘ageing adult’. The population of the species in the alliance after k months is given by kP where .k k k kP A B C= + + In this equation, kA is the number of ‘ageing adults’ (that is, species that are too old to breed), kB is the number of ‘adults’ of breeding age, and kC is the number of ‘children’ in the alliance (that is, those too young to breed). For the positive constants , , and ,    each between 0 and 1, the numbers in each group are modelled by the following system of recurrence relations. 1 1 1 k k k k k k k k k k k k A A B A B B C B C C B C    + + + = + − = + − = + − (a) (i) Give an interpretation of each of , , and .    [4] (ii) Hence give one criticism of this population model. [1] In a beta version of the game, in Alan’s alliance invasion to other territories, it has to attack in the north direction to advance its position. Similarly, other alliances can also attack his alliance and his alliance can retreat in the south direction. T he distance due North of Alan’s alliance from the East - West latitude line, in units, after n weeks is denoted by nx . Initially, his alliance is 2 units due North from the latitude line. After 1 week, there is no change in the position of his alliance. It is given that nx satisfies the recurrence relation 1245n n nx x x −−=− , 2n . (b) Solve the recurrence relation for nx , giving your answer in the form of ( ) ( )ii nn A x y B x y+ + − where ,AB   and ,xy   are to be determined. Hence show that 3 222 5 cos 4 n nxn  =+  , where 1 1tan 2 −= . [6] (c) Give one criticism of modelling nx in this way. [1]

5 NYJC 2024 JC2 Common Test 9649/02 [Turn Over 7 A curve D is given by the equation 2cos cos 2 1, 2 .sin si 2 , 0 2n x t tt y t t  =−  − =− (a) The arc-length between two points on D, where t = and t = , is given by the formula 22 dd ddd xy ttt         +       . Using integration, show that the length of D is 16. [3] (b) The surface area generated when the arc between two points on D, where t = and t = , is rotated fully about the x-axis is given by the formula 22 dd2d dd xy ttty          +       . When D is rotated through  radians about the x-axis, a surface of revolution is formed with area A. Using calculus, determine the exact value of A. [4] (c) Show that the equation of D can be expressed in polar form ( )2 1 cosr =− . [3] (d) Hence, using calculus, find the exact area enclosed by the curve D. [3] Section B: Probability and Statistics [25 marks] 8 The number of thunderstorms, x, reported in a particular month by 100 meteorological stations are as follows: x 0 1 2 3 4 5 Number of stations 22 37 20 13 6 2 By considering the mean number of thunderstorms for the month and its variance, explain why a Poisson distribution would be a suitable model for the above data. Test at the 10% significance level whether the number of thunderstorms in the particular month has a Poisson distribution. [6]

6 NYJC 2024 JC2 Common Test 9649/02 [Turn O