ACJC, NJC, SAJC 2026 Prelim P1

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 15 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 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. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. __________________________________________________________________________________ This document consists of 9 printed pages and 1 blank page. [Turn Over

2 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/01 1 Consider ( ) ( )( ) 7 f sin π 0.1xx=+ for 02 x . (a) Use your calculator to evaluate ( ) 2 0 fd xx to 6 decimal places. [1] (b) Estimate ( ) 2 0 fd xx using Simpson’s Rule with four strips. Give your answer to 6 decimal places. [2] A student estimates ( ) 2 0 fd xx using the trapezium rule with four strips, and finds that the percentage error of the estimate is 58.5%. (c) Calculate the percentage error when Simpson’s Rule is used in part (b). [1] (d) Sketch the graph of ( )fyx= and explain why the percentage error is greater when using Simpson’s Rule than the trapezium rule. [3] 2 A family of polar curves has equations ( ) 1: 1 cos , for 0 πnnC r ak −= +   where n is a positive integer, 0a and 01 k . (a) Without using a calculator, find the arc length of 1C , leaving your answer in terms of a. [4] The area of the region bounded by nC and the horizontal axis is denoted by nA . (b) Show that 1 n n A  =  is finite. [4]

3 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 3 It is given that the complex number iieez =+ , where 11 22ππ −   and 11 22ππ −   . (a) Show that ( ) ( )12arg z =+ and write down z as a single trigonometric expression in terms of  and  . [2] (b) By considering ( ) 2iiee+ , prove that the identity ( ) ( )( )sin 2 2sin sin 2tan cos 2 2cos cos 2       + + ++= + + + . holds if 1 2 π+  . [2] The complex numbers p, q and r have modulus 1 and positive real part. (c) Deduce that ( )( )( )p q q r r p pqr + + + is a real number. [3] 4 Consider the differential equation ( )134sin πd d2 xy xy= , with 1y= when 1x= . (a) Find the value of y when 5 4x= , giving your answer correct to 4 decimal places. [3] (b) Use one step of both the Euler method and the improved Euler method to calculate approximations to the value of y when 5 4x= , giving each answer correct to 4 decimal places. [4] (c) Determine which method is better and explain why the method with the worse approximation did not perform as well. [3]

4 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/01 5 Let A be a 33 real matrix with eigenvalues 1, 2, 3 and corresponding eigenvectors 1 2 3 1 1 0 0 , 1 , 1 1 0 1            = = =               v v v . (a) Explain why A is diagonalisable. Hence find A. [3] (b) Find all vectors 3y  satisfying 100 101=A y A y . [3] (c) Find all vectors 𝐱 ∈ ℝ3 such that 1lim 3 nnn→ A x = 0 . [3] 6 In an Argand diagram, the point P represents the complex number z satisfying 9 4i 3 1 4izz− + = − − . Let the locus of P be the curve C. (a) By letting iz x y=+ and finding the Cartesian equation of C, show that C is a circle, and state the centre and radius of C. [3] (b) Find the exact minimum value of ( )arg 9 4iz−+ . [2] A given complex number a with ( ) 120 arg πa has the property that there is exactly one complex number w that satisfies 9 4i 3 1 4iww− + = − − and * * 1wawa− =+ . (c) Find all possible value(s) of a with this property, and state the corresponding value(s) of w in each case. [5]

5 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 7 A surface S is defined implicitly by the equation 3 3 212 3 15 12 4z z x xy x y+ = + − − − for all real values of x, y and z. (a) Show that the x-coordinates of the stationary points of the surface S satisfy the equation 42 5 4 0xx− + = . Hence, find the coordinates of all four stationary points of S. [5] (b) Using the second partial derivative test, determine the nature of each of these four stationary points. [5] (c) Let 0 0 0( , , )x y z be the coordinates of the stationary point where z has a minimum value. Find the quadratic approximation of z in the region of 00( , )xy . [3] Show that this quadratic approximation can be expressed in the form Tzc+X MX where 0 0 xx yy −= −X , pq qr =M for real constants c, p, q and r, and TX MX is a scalar and solve the matrix M. [2]

6 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/01 8 (a) Suppose 1S and 2S are the column spaces of matrices 1A and 2A where 1 12 00 00 a b c   = A and 2 1 0 1 0 2 0 01 k −  = A . Determine the possible values of a, b, c and k such that 1S is a subspace of 2S . [5] (b) The matrix M is defined as 1 0 1 0 1 1 1 1 2  =M . The matrix M defines a linear transformation from T: 3 →  3  . Find a basis for (i) the row space of M; [2] (ii) the null space of M. [2]

7 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 9 A curve has equation 1siny x x −= for 1 203 x . The length of the curve is denoted by L. (a) Explain, with the aid of a diagram, why L must satisfy the inequality 23 π 4 12L+ . [2] (b) Write down the integral that represents L and find its numerical value. [2] The region R is bounded by the curve 1siny x x −= , the y-axis and the line 3 π6y= . The region R is rotated completely about the y-axis to form a solid of revolution S. (c) Show that the volume of S can be expressed as ( ) 2 0 3π 2π f d24 3 k xx+  , where ( )f x is to be determined. State the exact value of the constant k. Hence, use the substitution 21tx=− to find the exact volume of S. [8]

8 ANGLO-CHINESE JUNIOR COLLEGE 2026 H2 FURTHER MATHEMATICS 9649/01 10 A physicist is studying steady -state heat conduction on thin rectangular plates with non - uniform heating. In one particular experiment, the physicist uses a thin rectangular plate occupying the region 01 x , 1 20 πy . The steady-state temperature ( ),u x y , in C , at any point ( ),xy on the plate is governed by Poisson’s equation ( ) 22 2 22 3 2 cosuu xyxy + = − . The physicist uses an ice -water bath to maintain the temperature of the top edge at 0C , and uses programmable heating strips to control the temperature profiles of the remaining edges of the plate such that • ( )0, cosu y y = , • ( )1, 4cosu y y = , and • ( ) ( ),0u x v x = , where v is a function dependent only on x, as shown in Figure 1 below. The physicist models the steady -state temperature of the plate using a separable solution of the form cosu v y= . (a) Show that Poisson’s equati