DHS, HCI, RI, TMJC 2026 Prelim P1

1 © HCI 2026 HWA CHONG INSTITUTION 2026 JC2 Preliminary Examination Higher 2 9649/01 17 September 2026 3 hours Total marks: 100 Candidate Name CT Group 25 READ THESE INSTRUCTIONS FIRST Do not write anything on the List of Formulae (MF27). Answer all the questions. Write your answer to each question in the space provided in the Printed Answer Booklet. Follow the instructions on the front cover of the answer 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. 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 must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part of question. FURTHER MATHEMATICS Question Paper This document consists of 7 printed pages and 1 blank page. Remarks for Candidates: a) INSTR: Did not follow instructions stated in the question (e.g. correct significant figures, exact values, coordinates, required form, etc.). b) NOT: Incorrect or inappropriate use of mathematical notation. c) ACC: Inaccurate answer (e.g. due to premature rounding, omission of “+C” in indefinite integrals, etc.).

2 © HCI 2026 1 It is given that πi12 1 2iez = and 2 ππcos isin88z    = + −       . Let ( ) * 12w z z= . (a) Find the modulus and argument of w . [3] (b) Find the set of integers n in the range of 1 20n such that the point representing nw lies strictly in the second quadrant of the Argand diagram. [3] 2 The surface S has equation 221z x y= − − . Using calculus, show that the point on S that is nearest to the point ( )0,0, L , for some 1L , is ( )0,0,1 . [7] 3 The sequence of numbers 1 2 3, , ,...x x x is such that ( )1 Fkkxx+ = , where ( ) ( )Ff x x x x = + − with  being a non-zero constant. (a) Show that, if the sequence converges, its limit is a root of the equation ( )fxx= . [2] The first derivatives of the functions ( )f x and ( )F x are denoted by ( )f' x and ( )F' x respectively. It is known that the speed and likelihood of convergence of the sequence depends on the value of ( )F' x near the root. Ideally, ( )F' x should be as close to zero as possible. (b) It is given that the first term 1x of the sequence is near a root of the equation ( )fxx= , and ( )f ' 1x  for all x. Find  in terms of ( )1f' x such that ( )1F' 0x = . [2] (c) Given that the root  of the equation 1cosxx −= is near 0.7, use 1 0.7x = as an initial approximation to find an appropriate value of  such that the iteration ( ) 1 1 cosk k k kx x x x  −+ = + − is expected to converge to  . [2] (d) Use the iterative formula to calculate the value of  correct to 3 decimal places. [2] 4 (a) Solve the equation 11 1 0.z −= [2] Let 1 2 10, ,....,z z z and 1 be the roots found in part (a) where 1z is the root with the smallest positive argument. (b) Find the modulus of 11 z− , leaving your answer in a single trigonometric ratio. [3] (c) Show that 1z is a root to the equation 2 101 ... 0.z z z+ + + + = Hence show ( )( ) ( )1 2 101 1 .... 1 11.z z z− − − = [3] (d) Deduce the value of π 2π 3π 10πsin sin sin ...sin .11 11 11 11 [2]

3 © HCI 2026 [Turn over 5 The planes 1π and 2π are given by the equations 1π: ax y z b+ + = and 2π: x y z c+ + = . (a) Determine the necessary and sufficient condition s on b and c for which 12ππ is a subspace of 3  . [3] (b) Assume these necessary and sufficient conditions on b and c hold. Find the necessary and sufficient condition on a for which 12ππ is a subspace of dimension 1. [2] The conditions found in part (b) are to be used for parts (c) and (d). A third plane 3π is given by the equation 3π : 2x y z+ + = . (c) Is ( )1 2 3π π π a subspace of 3  ? Justify your answer. [2] (d) Is ( )1 2 3π π π a subspace of 3  ? Justify your answer. [2] 6 For a cubic equation 32 0ax bx cx d+ + + = with real coefficients, the discriminant D is given by 2 2 3 3 2 24 4 27 18D b c ac b d a d abcd= − − − + . If 0D , then 32 0ax bx cx d+ + + = has three distinct real roots. If 0D= , then 32 0ax bx cx d+ + + = has a repeated root and all its roots are real. If 0D , then 32 0ax bx cx d+ + + = has a real root and a pair of complex roots. Let A be a square matrix in 33  with a characteristic equation ( )f0 = , where ( ) ( ) ( ) 32f 5 5 6 6 k k k   = − + + + − and k  . (a) By considering the discriminant of ( )f0 = or otherwise, determine whether A is always diagonalizable when , 2,3kk  . [3] (b) By considering the eigenvalues of A, determine the range of values of k such that A is invertible. [2] For the remaining part of the question, let k = 4 and 1 13 19 2 0 6 2 4 10 −− =−−−A . (c) Evaluate ( ) 1 1 r r  − =  A , leaving your answer as a single matrix in exact form. [6]

4 © HCI 2026 7 The equatio n 3 2 2 0xx− − = has a root  in the interval  1, 2 . To find the root, three rearrangements are suggested for an iteration of the form ( )1 gnnxx+ = : • ( ) 3 2g 2xx −= ; • ( ) ( ) 1 3g 2 2xx=+ ; • ( ) 3 2 22g 32xx x += − . (a) Identify one of the above rearrangements that will fail to converge to  with the initial approximation close to  . Justify your answer with the aid of diagram(s). [3] (b) Use the other two rearrangements, with initial approximation 0 1.5x = , to find the value of  correct to 4 decimal places. For each iterative method, state the least value of n such that nx = correct to 4 decimal places. [4] (c) Suppose an iterative method using ( )1 gnnxx+ = produces a sequence  nx that converges to the root  . The error term for each round of iteration ne is defined as nx − . If there exists a constant 0p and a number 0  such that 1lim n pn n e e + → = , then p is called the order of convergence for the iterative method. (i) Show that ( ) ( )1 ggnnex + =− . [1] (ii) It is given that ( ) ( ) ( )( ) ( )( ) ( )( ) () 2g '' gg g g ' ... ... 2! ! r rn n n nx x x x r    − = − + − + + − + . Show that, for the iterative methods used in part (b), exactly one of them has first order of convergence. [4]

5 © HCI 2026 [Turn over 8 Let ( ) ( ) ( ) ( ) ( ) 22 2 , 0,0 f, 0 , 0,0 xy xyxyxy xy   +=  = be a function of two variables. (a) Find the directional derivative in the direction of the unit vector ab=+u i j at ( ) ( )00, 0,0xy  . Express your answer in terms of 00, , and x y a b . [3] (b) Find the vector equation of the tangent line at ( )( )0 0 0 0, ,f ,x y x y in the direction of u, where ( ) ( )00, 0,0xy  . Express your answer in terms of 00, , and x y a b . [2] A plane π is defined by ( )( ) ( )( )0 0 0 0 0 0 0π : f , f , xyz z x y x x x y y y− = − + − . (c) Let ( )0 0 0,,P x y z be the point 41, 2,5 on the surface of f. Show that π at P contains tangent lines at P in all directions. [3] It is defined that ( ) ( ) ( ) 0 f , f ,f , limx h x h y x yxy h→ +−= and ( ) ( ) ( ) 0 f , f ,f , limy k x y