NYJC, TJC, VJC 2026 Prelim P1

This document consists of 7 printed pages and 1 blank page. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2026 [Turn Over NANYANG JUNIOR COLLEGE JC2 Preliminary Examination Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 16 September 2026 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 2026 JC2 Preliminary Examination 9649/01 1 A sequence {}nu is given by 1uM= and 3 1 6 for 1,7nn uun+ += where M is a constant. It is given that the sequence converges to the limit L. (a) Find the possible values of L. [1] (b) Show that if 12 nu , then 112 nu + and 1nnuu+  . [3] (c) Given that 1.5M = , deduce the value of L. [1] 2 The complex variables w and z, represented by the points W and Z respectively in an Argand diagram, are related by the equation 4i 2 zw z+= − , 2z . (a) Given that the locus of W is the imaginary axis, find the cartesian equation of the locus of Z and give a geometrical description of this locus. [4] (b) Given instead that the locus of W is the real axis, show that the locus of Z is a straight line, and write the equation of the line in the form pz qz r += where z denote the complex conjugate of z and p, q and r are complex numbers to be determined. [3] 3 (a) It is given that for non-negative integer n, π 6 0 sin dnnI = . Show that ( ) 2 3 1 2 nn nnI n I −=− + − , for 2n . [3] (b) The region R is bounded by the curve 22 1y x x=− , the x-axis and the line 1 2x= where 10, 2x  . Using the shell method and the substitution sinx = , show that the volume of solid generated when R is rotated through 2π radians about the y-axis is given by ( ) π 3mnk + , where k, m and n are integers to be determined. [5]

3 NYJC 2026 JC2 Preliminary Examination 9649/01 [Turn Over 4 Ethan uses the expression ( ) ( )1f 0 f f 1 2E a b c = + +  to estimate the integral ( ) ( ) 1 0 1 f d ,I x x x=+ where a, b and c are constants. It is given that Ethan’s method gives the exact value of I for ( ) 2f 1, and x x x= . (a) Find the values of a, b and c. [4] (b) The Simpson’s rule with two strips is applied directly to estimate I. Show that Simpson’s approximation is identical to Ethan’s method. [2] (c) Tang applies the trapezium rule with two strips to estimate I. Show that his approximation can be written in the form ( ) ( )1f 0 f f 1 2T    = + +  where , and    are values to be determined. [2] (d) If ( )f x is a quadratic function, determine whose estimate is more accurate. [2] 5 The variables x, y and z are related by the equation ( )f,z x y= where f is a function of the independent variables x and y. We define the independent differentials dx and dy to be the small increments of x and y respectively. The total differential dz , which is the small increment of z corresponding to small increments of x and y, is given by d d d zzz x y xy=+ . A right circular cone has a circular base radius of 9 cm and a height of 20 cm. Denote • the radius of the circular base of the cone by r, • the height of the cone by h, • the volume of the cone by V. (a) Find expressions in terms of r and h where appropriate, for V r   and V h   . [2] In one instance, in measuring r and h, a man made a maximum error of 0.01 cm in each of them. (b) Use the above result to estimate the maximum possible error in calculating V, leaving your answer in the form πk where k is to be determined to two decimal places. [3] In another instance, a man used the same faulty instrument to measure r and h and gave an overestimation of each by an amount  cm but maintained that the error in calculating V is no more than π cm3. (c) Determine in fractional form, the largest possible value of  . [3]

4 NYJC 2026 JC2 Preliminary Examination 9649/01 6 A water authority is considering a few potential service reservoirs to supply water to a prospective new town for short-term usage. For a particular service reservoir, let nw denote the volumes of water (in million litres) in this reservoir at the start of month n. The volume of water in the reservoir is modelled by 120.8 0.32 ,n n nw w w −−=+ where 3n , 1 100w = and 2 80w = . (a) Find an exact expression for nw in terms of n and 3 . [6] Another service reservoir is identified to be a nearer water source. Let nx denote the volumes of water (in million litres) in this reservoir at the end of month n. The volume of water in the reservoir is modelled by ( )21 1.4 0.45n n nx x x F Gn++ = − + − , where 0n , 0 100x = and 1 200x = . (b) State the significance of the positive constants F and G. [1] To remain sustainable, the reservoir must not be depleted. To determine whether the reservoir can sustain the new town over a 30 -month horizon ( 0 30n ), the value of F is initially fixed at 100 while G is varied, where 49 G . (c) For the case when 4G= , use your graphing calculator to investigate the behaviour of nx over the 30- month horizon and comment on its sustainability. [1] (d) By considering the behaviour of nx over the 30 -month horizon when 6G= and 9G= , comment on the proposed model and how the value of F should be adjusted, so that the reservoir remains sustainable over the 30-month horizon for all values of G satisfying 49 G . [2] 7 Consider the system of equations 1 2 3 (sin ) (cos ) (1 , sin ) ( cos ) xy yz x y z b b b      +=  +=  ++ + + = where 1 2 3, , , b b b are real constants and  )0,2π . (a) Show that if 0,  the system is always consistent for any values of 1 2 3, , b b b and the solution set is a point that does not pass through the origin when 1 2 30, 0, 0.b b b   [3] (b) Explain the geometrical relationship of the solution set when π0, 2== . Discuss how this relationship depends on the values of 1 2 3, , b b b . [2] (c) Write down the coefficient matrix of the above system of linear equations and find its determinant in terms of  and  . Explain how the determinant confirms the condition in part (a). [2] (d) Given 1 = and π = , determine if the coefficient matrix is diagonalisable over  . [4]

5 NYJC 2026 JC2 Preliminary Examination 9649/01 [Turn Over 8 A public health researcher studies the spread of an infectious disease in a closed population. Individuals are classified into three groups: • Healthy: individuals who are not infected but remain susceptible to the disease. • Infected: individuals who carry the disease but do not require medical treatment. • Hospitalised: individuals whose symptoms are severe enough to require hospital care. Individuals who recover from hospital treatment become healthy again but may later become infected. The movement between the three groups is illustrated below: The model assumes that each week: • 10% of the healthy i