ACJC 2025 Prelim P1

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 16 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 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 6 printed pages. [Turn Over

2 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/01 1 The curve 1C and 2C have equations 3 2cosr =− and 2r= respectively for 02 π . (i) Sketch 1C . [1] (ii) Find the exact area of the region that is inside both 1C and 2C . [4] 2 The function f is defined by ( ) 2 1 2fe xx −=− . The equation ( )f0 x = has only one real root  in the interval ( )0.75,1 . (a) Carry out linear interpolation once on the interval ( )0.75,1 to find an approximation x1 for  , giving your answer to 4 decimal places. By considering ( )f x and  f x  , explain with an aid of a diagram whether x1 is an overestimate or underestimate of the root  . [4] (b) Using the value of x1 found in part (i) as the initial value, apply the Newton-Raphson method to find  correct to 6 decimal places. Justify the accuracy of your answer. [3] 3 (i) Use de Moivre’s theorem to prove that ( ) 2 11 0 sin 1 sinsin sin 1 2 cos     +− = −+= − − + nn r n r r x nxx x n xx where x is a real number not equal to 1 , and  is a real number not equal to πk for any integer k. [4] (ii) By substituting 1 3 π = in part (i), find the sum of the first 4m terms of the series 124 815 07 1 ...− − + − − + ++x xxxxxxx , expressing your answer in the form ( ) ( ) p q x x , where ( )p x and ( )q x are polynomials in x with degrees 61 +m and 2 respectively. [3]

3 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 4 The variables x and y are related by the differential equation ( ) ( )dsin cos 1d yx x yx =+ , with 2y= when 1 2 πx= . (a) Use two steps of the Euler method to find an approximation to the value of y when 1 2 π1x=+ . Give your answer to 3 decimal places. [3] (b) Find the solution of the given differential equation and obtain the value of y when 1 2 π1x=+ . Give your answer to 3 decimal places [5] (c) Find the percentage error in the approximation. [1] (d) Explain why the answer in part (a) is not a good approximation to the value of y when 1 2 π1x=+ . [1] 5 The space-transformation T is given by the matrix 3 1 1 1 3 1 0 0 4 − =− A . The eigenvalues of the matrix A are denoted by 1 , 2 and 3 where 1 2 3   , and their corresponding eigenvectors are denoted by 1e , 2e and 3e . (i) Find the eigenvectors 1e , 2e and 3e of A . [5] (ii) Give a geometrical interpretation of each eigenvalue’s set of eigenvectors in relation to T. [2] (iii) Find matrices Q and D such that ( ) 12 3 4 5 1 − −+ + + + + =I A A A A A QDQ where D is a diagonal matrix. [3]

4 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/01 6 Let  nx be a sequence of positive terms defined by 1 1 1 nn n nn xxx xx − + − = + , for 2,n with 12 1 and 1.xx== (a) By letting 1 n n y x= , show that  ny satisfies a second order linear homogeneous recurrence relation and find this recurrence relation. [2] (b) Hence, find nx in terms of n. [6] (c) Using the expression derived in part ( b), describe the long -term behaviour of the sequence  nx as n→ . Justify your answer. [2] 7 (a) The surface 1S has equation ( )f,z x y= where ( ) 23f, 35x xy yx y x + − + −= . Find the exact coordinates of the stationary points of 1S and determine their nature. [7] (b) The surface 2S has equation ( )g,z x y= where ( ) ( ) 1g , tan 2x y x y −=+ . The quadratic approximation for 2S in the region of the point ( )1 41,0, π is given by ( )Q,z x y= . Find ( )Q, xy . [5]

5 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 8 The linear transformation 33T: →   is represented by the matrix 1 17 5 11 17    −−   + . (a) Find the set of values of  such that the nullity of T is zero. [4] It is now given that 2 = . (b) Find a basis for the range space of T. State a geometrical interpretation of the range space of T. [3] (c) State a basis for the null space of T and its nullity. Hence, find the subset P of 3  whose image under T is the line 2 , 5 11 6 2 1      =+ r  . [5] 9 (a) The curve C has equation 2, for 0 2.y x x=   Show that the length of the curve C is given by 2 2 0 1117 d 2 14 x x + +   . Use Simpson’s rule with 5 ordinates to find an approximation to the value of 2 2 0 1 d 14 x x+   and hence, find an approximation to the length of the curve C, giving your answer to three decimal places. [8] (b) The ellipse with equation 2 2 14 x y+= is rotated by π radians about the x-axis to form a solid. Find the exact surface area of the solid by integration using the substitution 4 sin 3 x = . [7]

6 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 FURTHER MATHEMATICS 9649/01 10 The population of a new breed of fish in a farm is being studied. In an initial survey, the number, P (in thousands), of the fish at time t months is modelled by the differential equation ( )d 4d P P P nt = − − , where n is a positive real constant. (a) Find the condition for n so that the population of fish in the farm is sustainable in the long run. [2] After a new survey, it is found that P is better modelled by the differential equation ( )d 4d4 P kPPPtP= − − + , where k is a positive constant. (b) Given that H is the non -zero equilibrium point, find H in terms of k and give the necessary condition on k for H to exist. [3] (c) Suppose that HP . Show that ( ) 22 d d4 P H PP tP − = + . Show that the general solution of this differential equation is ( ) ( ) e CA tB P H P H P D− + = , where D is a positive real constant and A, B and C are to be determined in terms of H. [7]