DHS HCI RI TMJC 2025 Prelim P1

© DHS 2025 This question paper consists of 6 printed pages and 2 blank pages. [Turn over Name: Centre/Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination 2025 Year 6 FURTHER MATHEMATICS 9649/01 Paper 1 18 September 2025 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Write your name, centre number, index number and class on this question paper. An answer booklet will be provided with this question paper. You should follow the instructions on the front of the answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. Answer all the questions. 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. The number of marks is given in brackets [ ] at the end of each question or part question.

2 DHS 2025 Year 6 H2 Further Mathematics Preliminary Examination Paper 1 1 The terms in the sequence {}nu satisfy the recurrence relation 1 12 3nnuu −=+ (a) Find an expression for nu in terms of n in the case that 0 1u = . [4] (b) Describe the behaviour of the sequence as n tends to infinity. [2] 2 (a) Suppose S1 and S2 are the column spaces of matrix A1 and A2 where 12 1 1 1 1 1 0 0 and 0 3 0 0 0 0 1 a b cc − − −           =  =    AA . Determine the possible values of a, b and c such that 1S is a subspace of 2S . [4] (b) Let  1 2 3,,v v v be a basis for 3  and N be a non-singular 3 3 matrix with real entries. Determine whether the set  231,,Nv Nv Nv is a basis for 3  . [3] 3 The complex number z satisfies i 6 i 5 6izz+ +  − − and 1 3 4i 52 z− +  . (a) Sketch the locus of z. [3] (b) Find the range of values of arg( 4i)z− . [4] 4 The tangent plane to the point (a, b, c) on the surface with equation ( ) ( ) 22 23 5 4x y z− + − + = is a subspace of 3  where 3, 5, 0a b c   . Find the range of values of a. [6]

3 DHS 2025 Year 6 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over 5 A spherical zone is the portion of the curved surface area of a sphere between two parallel planes h units apart as shown in the diagram below. By using a suitable curve and rotation axis, explain if the location of the two planes will affect the curved surface area of the spherical zone. [8] 6 The region enclosed by the curve ln ,y x x= the axisx− and the vertical line 2x= is rotated by 2π radians about the axisy− to form a solid. (a) Show that the volume of the solid is ( ) 2 1 2 g d xx , where ( )g x is a function to be determined. Hence, find the exact volume of the solid. [4] (b) Using Simpson’s Rule with 4n= strips, estimate the value of the integral ( ) 2 1 2 g d xx up to 4 decimal places. You may assume that the intervals are equally spaced. [2] (c) Calculate the percentage error in your answer in part (b). [1] (d) Without performing the calculations to find an approximation using the Trapezium Rule, explain with reasoning whether your answer will be an over -estimate or under - estimate. [2] 7 A differential equation is given by dcos sin 2cos ,d yx y x xx−= where ππ .22 x−   (a) Show that the g eneral solution of the differential equation is 2 tan sec ,y x C x=+ where C is the arbitrary constant. [2] (b) Explain why the solution curves have turning points when Ck− or when ,Ck where k is a constant to be determined. [4] (c) On the same diagram, sketch the family of solution curves where or C k k− . [3] h

4 DHS 2025 Year 6 H2 Further Mathematics Preliminary Examination Paper 1 8 (a) Let cos isin .z =+ By considering 5,z show that 53sin 5 16sin 20sin 5sin .   = − + [3] (b) Using the result in part (a), (i) find the distinct real roots to the equation , giving your answer, where appropriate, in exact trigonometric form 5316 20 5 1 0x x x− + + = [3] (ii) find the exact value of 2 πtan . 5   [4] 9 The polar curves C1 and C2 are given by the following equations in the respective intervals. C1: 2 2sinr =+ , 02  C2: 2sinr =− , 2   (a) On a single diagram, sketch C1 and C2. [4] The region R is enclosed by the polar curves C1 and C2 in the 3rd and 4th quadrants. (b) Show that the perimeter of the region R can be expressed as 24 2 2sin d 3 q p ++ where p and q are exact real numbers to be determined. Evaluate this integral in exact form. [7] [You may use the identity “ 21 sin 2sin 24  + = +  ” without proof.]

5 DHS 2025 Year 6 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over 10 In a population of plants, a particular gene has two versions (called alleles): A and a. A genotype is a combination of two alleles. Hence, for this particular gene, individual plants can have one of three genotypes: AA, Aa or aa. The order of the sequence of genotypes does not matter. To breed a new plant, two plants, termed as its parents, are used. The new plant will inherit one gene from each of its parents’ pairs of genes to form its own particular pair. For example, if one parent is of genotype Aa, it is equally likely that the new plant will inherit an A gene or an a gene from that parent, and if the other parent is of genotype aa, the new plant will definitely inherit an a gene from that parent. Thus, the new plant will either be of genotype Aa or aa. Suppose a farmer has a large population of plants consisting of some distribution of all three possible genotypes AA, Aa and aa. He starts a breeding programme where each plant in the population is always bred with a plant with genotype Aa to produce the next generation of plants. Let the vector i i i p q r    represent the proportion of plants with genotype AA, Aa and aa respectively in generation i, and T be the transformation 33:T →  that maps i i i p q r    to 1 1 1 . i i i p q r + + +    (a) Show that 20 : 2 2 2 02 ii ii ii pp T q q rr                       , where β is a non -zero constant to be determined and find the nullity of T. [4] (b) Find the eigenvalues of T and their corresponding eigenvectors of T. [7] (c) By diagonalising the matrix representing T, find the proportion of the plants in the long run. [4]

6 DHS 2025 Year 6 H2 Further Mathematics Preliminary Examination Paper 1 11 Two tanks are filled with salt water and are interconnected by pipes as shown below. Tank X and Tank Y initially contain 600 litres and 500 litres of salt solution respectively. Salt water with a concentration of 1 gram per litre of salt enters Tank X at a rate of 2 litres per minute. Pure water enters Tank Y at a rate of 13 litres per minute. Through the connecting pipes, the salt solution in Tank X flows into Tank Y at 12 litres per minute. To keep the levels of the tanks the same, salt solution in Tank Y flows into Tank X at 10 litres per minute, and drains out at 15 litres per minute. The amount of salt in Tank X and Y are denoted as x grams and y grams respectively. (a) Show that d 1 12d 50 50 x xyt = − + and d 1 1 .d 50 20 y xyt =− [2] (b) Find a second order differential