EJC 9649 2025 Prelim P1

2025 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2025 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. FURTHER MATHEMATICS Paper 1 9649/01 02 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all 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. This document consists of 6 printed pages and 2 blank pages.

2 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 1 1 The matrices A and B are given by 1 1, 1 α βγ β γα γ αβ  = A 1 1 33 3 β γα α βγ γ αβ  = B . (a) Show that det ( )( )( )αββγγα= − −−A . [3] (b) Hence, or otherwise, find det B . [2] 2 (a) Let cos 4z xy π= +  where etx= and ln(1 )yt= + . By finding z x ∂ ∂ and z y ∂ ∂ , evaluate the exact value of d d z t when 0t = . [4] (b) Suppose the function f( , )xy satisfies f (, ) f (, ) 0xy yxxy xy= = for all x and y. What does this imply about how fx and fy depend on x and y? Hence deduce a general form of the function f( , )xy . [2] 3 The diagram shows a shaded region bounded by an arc PQ having cartesian equation 2 ln 24 xxy= − , vertical lines 2x= , 4x= and the x-axis. (a) Show that the exact length of the arc PQ is 16 ln 24+ . [4] (b) Both the arc and the shaded region are rotated completely about the y -axis. (i) Find the area of the surface of revolution formed by arc PQ , correct to 2 decimal places. [3] (ii) Find the volume of the solid of revolution formed by the shaded region, correct to 2 decimal places. [2]

3 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over 4 Do not use a calculator in answering this question. Use de Moivre’s theorem to show that 42cos5 cos (16cos 20cos 5)θθ θ θ= −+ . [5] Hence, or otherwise, evaluate 2cos 10 π and deduce that 15cos54 π += . [5] 5 The complex number z satisfies the three relations below: • 13 5z−≤ • 8 18 10izz−≥−− • 1111tan arg( ) tan22 z−−−≤≤ (a) (i) Illustrate all three relations clearly on a single Argand diagram. [6] (ii) One of the relations is implied by another. Identify the two relations involved, justifying your answer. [2] (b) Determine the maximum and minimum possible distances of z from the origin. [2] 6 Let ab bc = A be a real symmetric matrix. The matrix A is said to be positive definite if T 0>x Ax for all non-zero vectors x y = x . (a) Show that if A is symmetric and positive definite, then both the eigenvalues of A are positive. [2] (b) Show that 2 2 T2 b ac bax y yaa  −= ++   x Ax if a is non-zero. Hence deduce the conditions for A to be positive definite. [5] (c) Let 22f( , ) 2 2 3x y x xy y=++ defined for all 2(,)xy ∈R . By expressing f( , )xy in the form of Tx Ax for a suitable symmetric matrix A, explain why the origin is the global minimum. [3]

4 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 1 7 Let nu be a sequence defined by the recurrence relation 1 32nnuu+ = + for 0n≥ with 0uk= where k is a constant. (a) Find nu in terms of n. [4] Let nv be another sequence defined by the recurrence relation 1 3 2 () n nnvv+ = +∗ for 0n≥ with 0 1v = . Let 0 f( ) n n n x vx ∞ = =∑ , 1 3x < . (b) By multiplying 1nx + to both sides of ()∗ , show that f( ) 13 12 ABx xx= +−− where A and B are constants to be determined. [4] (c) Hence find an expression for nv in terms of n. Determine also the limiting behaviour of 3 n n v as n→∞ . [3] 8 In polar coordinates, curve C is defined by 22 2 cos sin 0rr θθ− += for ππ 44 θ−≤≤ , 0r≥ . (a) By solving for r in terms of θ , sketch the curve C , indicating clearly the behaviour of the curve near 0r = and near π 4θ =± . [5] (b) Show that area of the region enclosed by curve C is given by the definite integral π 4 2 0 4 1 2sincos dθθθ −∫ , and find the exact area, using the substitution sin 2sinu θ= . [7]

5 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over 9 Consider the equation f( ) 0x = , where 53f( ) 10 45 24 3xx x x= − +− . It is given that there is a positive root near 2x= . (a) Use Newton-Raphson method with 1 2x = to obtain a sequence of iterates until, correct to 3 decimal places, 1nnxx += . Write down the least value of n for which this is so and state the corresponding value of nx . [3] (b) Verify, by substitution, that 3 is a root for f( ) 0x = . Supporting working must be shown. [2] Equations can have repeated roots. So, for example, the equation 2( 1)( 3) 0xx+ −= is said to have three roots: 1x=− (of multiplicity 1) and 3x= (a repeated root of multiplicity 2). If the root has multiplicity 1, it is said to be a simple root. The rate of convergence of the Newton -Raphson method is slowed if the root is not simple. One method of handling the problem of repeated roots is to define f( )h( ) f( ) xx x= ′ . Let R be a root of f( ) 0x = with multiplicity 2m≥ . Assume f( )x can be writt en as f () ( )q () mx xR x= − , where q( ) 0R ≠ . (c) Find h( )lim xR x xR→ − and hence explain why R is a simple root of h( ) 0x = . [3] (d) By applying the Newton-Raphson method to h( )x , show that ( ) 1 2 f () f () f () f () f () nn nn n nn xxxx x xx + ′= − ′ ′′− . [2] (e) Use the result in (d) with 1 2x = to obtain another sequence of iterates until, correct to 3 decimal places, 1nnxx += . Write down the least value of n for which this is so, state the corresponding value of nx and comment on the efficiency. [3]

6 2025 JC2 H2 Further Mathematics Preliminary Examination Paper 1 10 A pain‑killing drug is injected into the bloodstream. It then diffuses into the brain, where it is absorbed. The quantities at time t of the drug in the blood and the brain are ()yt and ()zt respectively, satisfying d 2( )d y yzt = −− , dd 3dd zy ztt= −− . (a) Obtain a second‑order differential equation for y. [2] (b) Hence, derive the general solution 6eettyA B −−= + , 61 e 2e2 ttzA B −−= − , where A and B are arbitrary constants. [4] (c) When a single dose of the drug is injected into the bloodstream at initial time 0t = such that (0) 5y = , given that no drug is initially in the brain (0) 0z = , find the particular solution for the resulting quantity of the drug in the brain, denoted by 1()zt . [2] (d) In another regimen, identical doses of the drug are periodically injected at each unit time, and the system has reached a steady state: (0) (1)z zc = = , where c is a given constant. Find, in terms of c, the particular solution for quantity of the drug in the brain in this regimen, denoted by 2 ()zt . [3] (e) In steady state, each preceding injection at times ... 3, 2, 1,0t =−−− contributes its own 1z profile. Show that the superposition of all earlier dose responses 12 0 ( ) () n zt n z t ∞ = +=∑ when c takes a particular constant value, and find it. [3]

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