EJC 2026 Prelim P1

2026 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2026 General Certificate of Education Advanced Level Higher 2 FURTHER MATHEMATICS Paper 1 9649/01 27 August 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (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 5 printed pages and 3 blank pages.

2 2026 JC2 H2 Further Mathematics Preliminary Examination Paper 1 1 The terms in the sequence 0 1 2, , ,u u u  satisfy the recurrence relation 124r r ru u u −−=− for 2r . (a) Find the general solution of this recurrence relation. [2] (b) Find, in an exact form, an expression for ru in terms of r in the case that 0 0u = and 1 2u = . [2] (c) Find the exact value of 0u in the case that 2 41 7 3u =− and ru tends to zero as r tends to infinity. [2] 2 Consider the differential equation 10de d 2 1 xyy xy −= − , where 1y= when 0x= . (a) Use two steps of both the Euler method and the improved Euler method to calculate approximations to the value of y when 1x= , giving each answer correct to 2 decimal places. [5] (b) Explain the large discrepancy between these two answers. [2] 3 The growth of a mold colony on a circular petri dish is the subject of a biology experiment. A student proposes modelling the area of the mold using a recurrence relation: ( )1 1n n nA p A qA+ =− , where p and q are positive constants and nA represents the area of the mold in mm 2, n days after the start of the experiment. (a) It is given that 20p= and 0 100A = . (i) Find the exact value of q for which this model predicts that the area of the mold remains constant. [2] (ii) Determine the range of values of q for which this model predicts that 1A is less than zero. [1] (b) The values of p and q are now chosen so that for 1n , 400, is odd, 100, is even. n nA n= (i) Determine the exact values of p and q . [3] (ii) Investigate what the model, with these values of p and q , predicts about the mold area when 0 99A = and when 0 101A = . Hence comment on the proposed model. [3]

3 2026 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over 4 Let ( ) 5 0.5 1 0.5 sin e d xI x x=+ . (a) Use the trapezium rule with four strips to obtain an approximation to I, correct to 3 decimal places. [3] (b) Use Simpson’s rule with five ordinates to obtain an approximation to I, correct to 3 decimal places. [3] It is given that the approximation to I, correct to 3 decimal places, from the graphing calculator is 6.158. (c) State which of the approximations in (a) or (b) is more accurate and explain, with aid of a diagram, why the less accurate method has such a large discrepancy. [3] 5 The curve C has equation ( ) 2siny x x=+ . The area bounded by the portion of C over the interval 0, 3π , the positive x-axis and the line 3πx= is rotated 2π radians about the y-axis. (a) Find the exact volume of the solid of revolution. [4] (b) Find, correct to 3 decimal places, the area of the surface of revolution that is generated by the portion of C. [3] The same portion of C is now rotated 2π radians about the x-axis instead. (c) Find the area of the new surface of revolution that is generated. [2] 6 The linear transformation T maps points ( , , )x y z to image points ( , , )x y z   such that 3 1 2 1 3 2 446 xx yy zz           =         −−−     . (a) Given that T has an invariant plane, , and an invariant line, l, find vector equations for  and l. [6] (b) (i) Given that a, b and c are constants, state the values of t for which the system of equations (3 ) 2 (3 ) 2 4 4 ( 6 ) t x y z a x t y z b x y t z c − + + = + − + = − − + − − = has no unique solutions. [1] (ii) For each of these values of t, find the relationship(s) between a, b and c such that the system of equations is consistent. [3]

4 2026 JC2 H2 Further Mathematics Preliminary Examination Paper 1 7 Consider the function ( ) ( ) 1.1cos e f x x x= . Solutions are required for the equation ( )f0 x = for 0x . (a) Evaluate ( )f1 and ( )f2 , and hence comment on the possible number of roots of ( )f0 x = in the interval ( )1, 2 . [2] (b) Name the feature that must exist on the graph of ( )fyx= for the number of roots of ( )f0 x = in the interval ( )1, 2 to be non-zero. [1] It is given that  , the 2nd smallest positive root of ( )f0 x = , lies in the interval ( )1, 2 . (c) (i) Use the Newton-Raphson method, with initial approximation 0 1x = , to determine the values of 1x , 2x and 3x . Give each answer correct to 3 decimal places. [3] (ii) Explain why the values of 3x and 4x are the same (correct to 3 decimal places). [2] (iii) Explain why the use of 0 1x = in the Newton-Raphson method does not give convergence to  . [1] (iv) Find the value of  correct to 3 decimal places. [1] (v) By considering the graph of ( )fyx= , for 0x , suggest a reason why the Newton -Raphson method should not be used to find the roots of ( )f0 x = for large values of x. [1] 8 (a) The matrices 2 1 2 1 2 2 1 1 1 −− = − −−−A and 8 4 5 4 1 4 5 4 2 −− = − − −−B have characteristic equations equivalent to ( ) ( ) 2 3 1 0− + = . (i) For each matrix, find a basis for the set of eigenvectors corresponding to the eigenvalue 3. [3] (ii) For each matrix, determine whether it is diagonalisable. [1] (b) The vector space V is 4  . The set S consists of the vectors a, b, c and d in 4  given by ( )1, 0, 1, 2=−a , ( )0, 1, 2, 1=−b , ( )1, 2, 1, 0=−c and ( )2, 1, 0, 1=−d . (i) Show that the set S spans V. [3] (ii) Express the vector 1 (1, 0, 0, 0)=u as a linear combination of a, b, c and d. [2] (iii) Write down similar expressions for each of the vectors 2 (0, 1, 0, 0)=u , 3 (0, 0, 1, 0)=u and 4 (0, 0, 0, 1)=u in terms of a, b, c and d. [3]

5 2026 JC2 H2 Further Mathematics Preliminary Examination Paper 1 9 A curve C has polar equation ( )( )1 sin3 2 cosr = + + for 0 2π . (a) Find, in an exact form, the equation of the tangents at the pole. [2] (b) State the number of loops that C has. [1] (c) For each loop, find the values of  and r for which C is furthest from the pole. [3] (d) Sketch C, labelling clearly the features found in all previous parts above. [3] (e) Find the area enclosed by the smallest loop of C. [2] (f) Find the arc length of the largest loop of C. [2] 10 [In this question, all variables are in standard SI units.] In an experiment to investigate the motion of objects submerged in viscous fluid, an object P of fixed mass m is attached to the end of a metal spring and submerged in a pool of the viscous fluid. The origin O is the equilibrium point for the system , where there will not be any motion unless an external force is applied . P is displaced from O by a displacement of 0x and released from rest. P moves along the x-axis with a displacement from O of x,