NYJC VJC TJC 9649 2025 Prelim P1

This document consists of 7 printed pages and 1 blank page. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2025 [Turn Over NANYANG JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 15th September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all the questions. Write your answers on 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 must 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 question.

2 NYJC 2025 JC2 Preliminary Examination 9649/01 1 Consider the differential equation d 36 1d 35 180 yy yhx = −−  , where h is a positive constant. (a) Given that d 0d y x ≥ for some values of y , find the range of values of h. [2] (b) For 25h= , find the equilibrium values of y , stating whether they are stable or unstable. [3] 2 The curve C has equation cos 2yx x= , 0 2x π<≤ , and has a single turning point at P. (a) Show that the x -coordinate of P is a solution of the equation 111tan 022x x − −=  . [2] (b) Show that the equation in part (a) has a root α between 0.3 and 0.5. [1] The iterative formula 1 1 11tan22 n n x x − + =   with 1 0.3x = is used to find α . (c) Find, correct to 5 decimal places, the values of 2x , 3x and 4x . [2] The diagram below shows the graphs of yx= and 111tan22y x − =  for 0x> . (d) Using the sketch in your answer booklet, show how the convergence to the root α takes place. Indicate clearly on your diagram the positions of 1x , 2x , 3x and 4x in relation to α . [2] O y x

3 NYJC 2025 JC2 Preliminary Examination 9649/01 [Turn Over 3 It is known that the equation f ( ) 0,x = where 2 ,f) 22( xxx += − has a root β . To approximate β , linear interpolation is used on the interval [ ]0,k to get an initial approximation 0x , followed by the Newton-Rhapson method. (a) If 4k = , find 0x . [1] (b) Use the Newton -Raphson method repeatedly with the initial approximation found in part (a) to determine β correct to 3 decimal places, justifying the correctness of the result obtained. [3] (c) For k β> , linear interpolation is used to obtain 0x as in part (a). Find the range of values of k such that the Newton-Raphson method would fail. [3] 4 The prices per unit weight, in dollars , of two commodities, coffee ( )nC and tea ( )nT , in the month n are modelled by the following recurrence relations: 11 11 0.8 0.2 5 0.1 0.7 3 nn n nn n CC T TC T −− −− =++  =++ where 0 20C = and 0 15T = . (a) Denoting the equilibrium prices of coffee and tea by *C and *T respectively, find the values of *C and *T . [2] (b) Using the substitution *nnX CC= − and *nnYTT= − , derive the second order recurrence relation for nX . [3] (c) Hence find nC in terms of n. [3]

4 NYJC 2025 JC2 Preliminary Examination 9649/01 5. A solid of revolution is formed when a circle of radius r , with centre at ( ),0R , is rotated about the y -axis through 2π radians, where 0Rr>> . (a) Using the shell method and the substitution uxR= − , show that the volume V of the solid of revolution can be expressed as ( ) 224d r r V Ru r u uπ − = +−∫ . [3] (b) (i) Use Simpson’s rule with five equally spaced ordinates to estimate V, expressing your answer in the form ( ) 24 3 Rr ab cπ + , where a, b, and c are constants to be determined. [4] (ii) Without further calculation, explain how your answer in part(b)(i) would change if r is halved while R is doubled. [1] 6 (a) By considering Euler’s formula, find the exact value of 1 0 2 πcos n k k nθ − =  +∑ where 1n> . [4] (b) A regular n-sided ( 3n≥ ) polygon with vertices 01 1,,, nAA A − is inscribed in a circle with centre O and radius r. The point P is on the plane of the polygon such that OP p= . By defining 2 π , 0,1, 2, , 1k kPOA k n nθ∠= + = −  , prove that ( ) 2 2 2 22 01 1 nPA PA PA n r p −+ ++ = + . [3] (c) The value of 22 2 01 1 nPA PA PA −+ ++  is denoted by • X when P is the centre O of the circle, • Y when P lies on the circumference of the circle. Find the value of Y X . [3]

5 NYJC 2025 JC2 Preliminary Examination 9649/01 [Turn Over 7 The curve C is defined by the equation ( ) 223ay x a x= − , where a is a positive constant. The finite region R is bounded by the curve C. (a) Show that ( )( ) 3d d6 axa xy x ay −−= . [2] (b) Sketch C, indicating clearly the behaviour of the curve near the origin. Show the coordinates of the x- intercepts and turning points. [2] (c) Show, with full working, that the length of the perimeter of region R can be expressed in the form pq ar , where p, q and r are positive integers. [6] (d) Find, in terms of a, the surface area generated when region R is rotated through 2π radians about the y-axis. [3] 8 (a) Use the substitution y ux= to find the particular solution of the differential equation 2 d2 , 0, 0d yyx xyxx y = + >> , given that the solution curve passes through the point (1,1). [5] (b) Use two steps of Euler method, find an approximation to the value of y when 2x= . [3] (c) F ind the percentage error in the approximation of y obtained above, giving your answer correct to 3 decimal places. [2] (d) Suggest two ways in which the percentage error in the approximation of y can be reduced. [2]

6 NYJC 2025 JC2 Preliminary Examination 9649/01 9 A linear transformation 33T: → is given by matrix A 5 12 11 12 5 aa aa − = − − A such that ( )T =x Ax , 3∈x  . (a) (i) It is given that a is an integer. By using Gaussian elimination, show that a = 5 when dim (T) = 2. Hence, find a basis for the range space R of T. [3] (ii) Given that 3 :2 3 5 1 x S y xyz z  = ∈ +−=  , find SR∩ . Give the geometrical interpretation of SR∩ and state whether SR∩ is a subspace of 3 . [4] (b) Given that A has an eigenvalue 3− , show that 2 12 32 0aa− += . [2] Using 4a= for the rest of this question. (i) Find the remaining eigenvalue(s) of A and for each eigenvalue, give the full set of eigenvectors.[4] (ii) Give a geometrical interpretation of each eigenvalue’s set of eigenvectors in relation to T. [2] (iii) State, with justification, whether the set of all eigenvectors of A forms a basis for 3 . [1] (iv) Hence find a matrix P and a diagonal matrix D such that A 3 = 1−PDP . (You do not need to evaluate 1−P ). [2]

7 NYJC 2025 JC2 Preliminary Examination 9649/01 10 A marine biologist monitors a large saltwater aquarium where the dissolved organic waste concentration (measured in milligram/litre) changes daily. The waste is reduced by two processes: • Natural breakdown by microbes in the aquarium • Probiotic treatment added every morning Such processes convert harmful substances (e.g. ammonia) into harmless compounds that do not threaten aquatic life. The waste concentration nW measured at the end of day n is modelled by the following recurrence relation: ( ) 2 1n dn pW k W kT −= + , 1n≥ where • T is the daily probiotic treatment dose • dk is the natural breakdown rate due to microbial activity • pk is the treatment efficacy due to the probiotic treatment • 0W is the initial waste concentration m