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