NYJC 2023 JC1 Promos 9649

This document consists of 6 printed pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2023 [Turn Over NANYANG JUNIOR COLLEGE JC1 End of Year Examination Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 29th September 2023 3 Hours Additional Materials: Answer Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name and class on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. 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. 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.

2 NYJC 2023 JC1 End-of-Year Examination 9649/01 [Turn Over 1 A nn matrix A is said to be idempotent if 2 AA . (a) Show that det( )A is 0 or 1. [2] (b) Show that if det( ) 1A , then AI . [1] Let wx yz  A be a 22 idempotent matrix where 0w , 0z . (c) If det( ) 0A , find the value of wz . [2] 2 Let  fs i n 2x x . Prove by mathematical induction that   2 2 d f4 s i n 2d n n n x xx  for .n   [5] Hence, explain why the Maclaurin expansion of f(x) has only odd powers of x. [2] 3 (a) The curve C is defined parametrically by cos ln tan 2 txa t   , sinya t where 34 t  . and a is a positive constant. Find the exact length of the arc of C, in terms of a. [4] (b) Let 2 1 0 d 4 n nxIx x     , where n is a non-negative integer. Prove that for 2n ,   241 3nnnI n I   . [3] Let R be the region bounded by the curve 3 24 xy x   , the line 1 3 y and the y-axis. Find (i) the exact area of R, [3] (ii) the exact volume of the solid obtained when R is rotated 2 radians about the y-axis. [4]

3 NYJC 2023 JC1 End-of-Year Examination 9649/01 [Turn Over 4 (a) The sequence of positive numbers nu for 1, 2, 3,n Κ satisfy the recurrence relation 1 n n n ku au uk    , where a and k are positive real constants. (i) Given that the sequence converges, find its limit in terms of a. [2] (ii) If nua , by considering 1nnuu  , show that 1nnuu  . [2] (b) Two sequences     and nnx y are related by 11 110.8 0.1 and 0.2 0.9nn n nn nx xy y xy   For 1.n (i) Given that 1nnxy , show that 10.7 0.1nnxx  . [2] (ii) Given that 1 0.9x  , solve the recurrence relation in (i) to find nx and ny in terms of n. [4] 5 Let 44:T  be a linear transformation such that 14 01 3 19 01 0 T                 , 11 15 14 16 T                , 16 11 3 07 37 T                 and 11 1 02 7 21 6 41 7 T                 . (a) Show that 111 1 01 1 0,, ,110 2 01 3 4                            is a basis for 4 . [3] Let M be the matrix representation of T. (b) Write down the matrix B such that 111 1 01 1 0 110 2 01 3 4       MB . Hence, with the aid of the graphing calculator, find M. [2] (c) Find a basis for W, the range space of T. Hence, state the dimension of the kernel of T. [3] (d) Given that 0 b c d       belongs to the range space of T, find the conditions on b, c and d. [2]

4 NYJC 2023 JC1 End-of-Year Examination 9649/01 [Turn Over 6 (a) Use the substitution tan 2 xt   to find 1 d32 s i n c o s xxx  . [4] (b) It is given that fc o s 3 s i n 3xm x n x , where m and n are integers and 0mn . (i) Find cos3 sin 3 dmx nx x in terms of m and n. [2] (ii) Find  2 0 fd xx   where m is even and n is odd, giving your answer in non- trigonometric form, in terms of m, n and  . [3] 7 When a frame consisting of two parallel rings with the same radius is dipped into soap solution, a soap film is formed between the rings. Coordinate axes have been superimposed so that one could model the soap film as the surface of revolution of a curve C around the y-axis (see figure). The equation of curve C is given by ee2 yy kkkx    , yaa  , where k is a positive constant. (a) Let S be the curved surface area of the soap film. Show that 22 e4e2 aa kkkSk a k    . [5] (b) Let A be the curved surface area of a cylinder with height 2a and the same radius as the rings. Determine, to 3 decimal places, the value of a k such that 2SA . [5] (c) Let V be the volume of the region bounded by the soap film and the planes of the rings. Show that 1 2Vk S . [3] Parallel Rings Soap film Curve C a –a O x y

5 NYJC 2023 JC1 End-of-Year Examination 9649/01 [Turn Over 8 The matrices A and B are such that 01 0 1 20 a n d 2 0 . 12 12 kk kk kk   AB (a) Find the exact values of k such that ABx = 0 has non trivial solution only. [5] (b) (i) Given that the linear transformation  T xA x has an invariant point at (1, 0, –1), determine the value of k. [1] (ii) Using the value of k found in (b)(i), find a matrix P and a diagonal matrix D such that A = P– 1DP. [6] 9 (a) Two vectors in 3 are said to be orthogonal if their dot product is zero. Let V be a subspace of 3 . The orthogonal complement of V, is defined as   3 : 0 for every WV   ww v v . Show that W is a subspace of 3 . [3] (b) It is of interest during war to devise a plan for military assets to get as close as possible to the enemy command centre without be ing detected by the enemy’ s detection system. A new detection system under testing follows the model Au b , where 124 51 0 2 0 248   A is the detection matrix constructed with refe rence to the terrain and weather, and x y z   u is the position vector of the location of the military asset relative to the enemy command centre. (i) Find a basis for the null space of A . [2] (ii) The military asset is at a blind spot and is not detected by the detection system if b0 . Give a geometrical interpretation of the blind spots of the detection system. [1] (iii) Give a geometrical interpretation of the orthogonal complement of the blind spots of the detection system. [1]

6 NYJC 2023 JC1 End-of-Year Examination 9649/01 10 In the country of Everland, the production of Imperial carrots, in hundreds, is denoted by An and is defined by 12 1 15, 8 and 3 2 2f o r . 8 nn nAA AA A n     (a) Determine the solution for nA , giving nA in the form of   2 nn pk qm where p, k ,q and m are exact constants to be determined. [7] However in the midst of farming, these carrots are devoured by wild rabbits. Peter hypothesized the number of wild rabbits is growing with a recurrence relation, nB , in hundreds, in year n, taking year 2023 as year 1. The model proposed is given by 32 1 ,nnBA for 1.n (b) (i) Calculate the values of for 1, 2, 3, 4 and 5.nBn  [2] (ii) Peter discovers that the sequence nB follows a non-homogeneous second order recurrence relation. Find this recurrence relation. [2] (c) Find the year when there are more than 2000 wild ra