RI DHS HCI TMJC 9649 2023 Prelim P1

This document consists of 8 printed pages. RAFFLES INSTITUTION Mathematics Department  RI2023 [Turn over FURTHER MATHEMATICS 9649/01 Paper 1 September 2023 3 hours Additional materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet and a graph paper booklet will be provided with this question paper. You should follow the instructions on the front cover of both booklets. If you need additional answer paper or graph paper ask the invigilator for a continuation booklet or graph paper booklet. Write your name and CT group on all 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. The number of marks is given in brackets [ ] at the end of each question or part question. RAFFLES INSTITUTION 2023 YEAR 6 PRELIMINARY EXAMINATION

2 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 1 1 (a) Show, without sketching a graph, that there is exactly one root,  , of the equation 32 e 2 0xx    in the interval [1, 2]. [2] (b) Use linear interpolation once, on the interval [1, 2] , to find an approximation, 1 , to  , giving one decimal place in your answer. [2] (c) Use the Newton-Raphson method, with the starting value 1 , to find the value of  correct to 3 decimal places. [2] 2 A sequence of real numbers 1,u 2,u 3,u satisfies the recurrence relation 1ua and 1 8 21 2 n n n uu u    for 2a and 1n . (a) Prove algebraically that, if the sequence converges, then it converges to either 3 or 7. [2] (b) Use an algebraic method to prove that 1 7nnuu  if 37 nu . [4] (c) Describe the behaviour of the sequence for 1 4u  . [2] 3 The terms in the sequence 0x , 1x , 2x , ... satisfy the recurrence relation 2122 n n nax x a x where a is a positive constant and 0n . (a) Find the general solution of this recurrence relation. [3] (b) It is given that 3a , 0 3x  and 1 1 3 x  . Show that    f ( ) cos g( ) sin g( )nx n C n D n  where C and D are constants, and f ( )n and g( )n are expressions of n, to be determined. [3] (c) Determine, with reasons, the range of positive values of a such that nx tends to zero as n tends to infinity. [2]

3 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 1 4 It is given that A is a 4  4 matrix with eigenvalues , 1, 2, 3, 4,i i  such that 1 2 3 4 0       , and that xi is an eigenvector of A associated with i . It is known that if the eigenvalues are distinct, then the set { 1 2 3 4, , ,x x x x } is linearly independent. (a) Show clearly why, for any 4x and positive integer k, 324 1 1 1 2 2 3 3 4 4 111 A x x x x x kkk kk c c c c                         , where 1 2 3 4, , ,c c c c are constants. [3] (b) Explain why 1 1 1 kk cA x x for large values of k. [1] The matrix M is defined to be 0 0 1 1 1 0 1 0 1 1 0 0 1 0 0 0       and its eigenvalues are 1 , and corrected to 5 decimal places, 1.24698, 0.44504, 1.80194 . (c) Find an eigenvector of M associated with the eigenvalue 1 . [1] (d) Let the eigenvalues of M be , 1, 2, 3, 4,i i  such that 1 2 3 4 0       . Find the least integer value of k such that 1 0.0001 k i     for 2, 3, 4i . Use this value of k to find 1 0 0 0 k       M , and deduce an approximate eigenvector of M associated with 1 in the form 1 2 4 1 u u u       , where 1 2 4,,u u u are positive constants to be determined, correct to 2 decimal places. [3] [Turn over

4 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 1 5 It is given that A a b a b a a   where a and b are positive real constants. (a) Show that b is an eigenvalue of A and find an eigenvector of A associated with b. [3] (b) Find A2 and A–1. [3] (c) Let n be a positive integer. Conjecture an expression for An when  n is odd,  n is even. Use mathematical induction to prove the correctness of your conjecture. [5] 6 Let  be a real constant such that sin 0  . (a) Show that the polar equation  1 cos cos , (*)ABr      represents a line with cartesian equation 1mx ny , where m and n can be expressed in terms of A, B and  . [2] (b) A conic E, with focus at the origin O, has polar equation 1 cos ar e   where e is the eccentricity and a is a nonzero constant. Points P and Q lie on E corresponding to    and    respectively, for 0 2  . (i) Using (*) in part (a), show that the polar equation of the chord PQ may be expressed as  1 sec cos cose r a a      . [4] (ii) Find the polar equation of the tangent to E at the point where  . [1]

5 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 1 7 (a) Given that 1zw w , where  cos sinw k i  as  varies, and k is a positive constant, express the real part and the imaginary part of z in terms of k and  . [2] Hence show that the point representing z in an Argand diagram lies on a conic section and state the eccentricity of this conic section in terms of k. [3] (b) On a single Argand diagram, sketch the loci (i) 1zw w , where  2 cos sinwi  as  varies, [2] (ii) 52. 2z [2] Given that the complex number v satisfies parts (b)(i) and (b)(ii), deduce the possible exact value(s) of (iii) 3arg , 2v  in the interval ( , ] , [2] (iv) 2 2 .vv   [1]

6 H2 FM 9649/2023 RI Year 6 Preliminary Examination Paper 1 8 The matrix A is defined by 11 13 01 a b c    where a, b and c are constants, and 1ab . It is given that the determinant of ATA is 0. (a) Show that the dimension of the null space of A is 1 and that 13 1 ac ab   . [4] In the rest of the question, it is given further that 1a and 0b . (b) Give a geometric interpretation of the column space of A. [1] In the rest of the question, let x be a 31 matrix and b = 1 0 1    . (c) It is given that Ax = b has no solution. Without any computation, justify whether 2 0 2    belongs to the column space of A. [1] When the equation Ax = b has no solution, we can use the Least Squares Method to find an approximate solution. This approximate solution is called a least squares solution. A least squares solution to Ax = b is v such that x = v is a solution to ATAx = ATb. (d) By considering a system of linear equations, find all the possible least squares solutions, v, to Ax = b. [3] (e) By considering the scalar product of Av and Av b , draw a diagram to illustrate the relation between b and Av. [3] (f) With reference to the