TMJC DHS HCI RI FM 2024 Prelim P1

__________________________________________________________________________________ H2 FURTHER MATHEMATICS 9649/01 Paper 1 12 SEPTEMBER 2024 3 hours Additional material: Answer Booklet List of Formulae (MF26) __________________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Write your name and civics group on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB 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. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 100. _________________________________________________________________________________ This document consists of 7 printed pages and 1 blank page. TAMPINES MERIDIAN JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION

2 1 The matrix A is given by 11 0 0.5 = A . (a) Form a conjecture for nA in the form 11 , for , 0 n n ab n b − + −   where a and b are real constants to be determined. [2] (b) Using mathematical induction, prove the conjecture formed in part (a). [4] 2 Let tanx = . (a) Use De Moivre’s Theorem to express tan 5 in terms of x, showing your working clearly. [4] (b) Hence, without using a calculator, find the exact value of 2 πtan 5   . [3] 3 The positive numbers nx satisfy the relation ( ) 2 1 3 1,nnxx+ =− for .n + (a) Find the values of  and , where , such that when 1x = or 1x = , the sequence remains constant. [2] (b) By considering ( ) ( ) 22 1 ,nnxx+ − or otherwise, show algebraically that if 1 ,x then the sequence increases and converges to . You may assume that the sequence converges. [5] 4 (a) Express 1 1 x− as a series, where 1.x  [1] The Fibonacci sequence is given as 1 nn+2 n+u u u=+ for , 0nn with initial conditions 0 0u = and 1 1u = . Let ( ) 0 f. r r r x u x  = = You may assume that ( )f x converges. (b) By considering ( ) ( ) ( ) 2f f f ,x x x x x−− express ( )f x in partial fractions. Hence find nu in terms of n, giving your answer in exact form. [8]

3 [Turn Over Tampines Meridian Junior College 2024 JC2 Preliminary Examination H2 Further Mathematics 5 The curve C has an equation ( )f,yx= where ( ) 1f 2 sin . 3x x x= − − (a) Show algebraically that C cuts the x-axis exactly once in the interval 0 1.x [2] Let ( )f 0. = (b) Use Newton-Raphson method with initial approximation 0 0x = to find an estimate for , correct to 4 decimal places. [3] The region 1R is bounded by C and both axes. The region 2R is bounded by C, the x-axis and the line 1.x= Let A be the total area of 1R and 2.R (c) Explain why Simpson’s Rule with four strips might not give a good estimation of the value of A. [1] (d) Use Simpson’s Rule with six strips to estimate the value of A, giving your answer correct to 5 decimal places. [3] 6 (a) The p oints ( ) ( ) ( )120,0 , ,0 , 0,O P k Q k and ( )12,R k k have images ', ', 'O P Q and 'R respectively under the transformation 22:T → represented by the matrix .ab cd = A (i) Find the coordinates of ', ', ' and 'O P Q R in terms of a, b, c, d, 1k and 2.k [2] (ii) Show that the area of the geometrical shape ' ' ' 'O P R Q is equivalent to ( )det A area of rectangle OPRQ. [2] (b) An nn square matrix M is said to be an orthogonal matrix if TT==MM M M I , where I denotes the identity matrix. (i) Explain why the product of two orthogonal matrices is still orthogonal. [2] (ii) Find the possible value(s) of the determinant of an orthogonal matrix. [2] (iii) Show that there exists a 22 non-orthogonal matrix with determinant of a value found in part (ii). [2] (iv) Let cos 0 sin 0 1 0 sin 0 cos    = − B . Verify that B is orthogonal and hence write down 1−B , showing clearly how you obtain your answer. [3]

4 7 (a) It is given that 12 3 1  − is an eigenvector of the matrix 3 1 3 1 1 , 13 k k  = − − M where k is a real constant to be determined. Find the eigenvalues and corresponding eigenvectors of M. Show your working clearly. [7] (b) The transformation T is the linear mapping from 3 to 3 given by the matrix M. The plane P is invariant under T, such that if any point in P undergoes the transformation T, the image of the point will still lie in P. Given that P intersects the line 11 01 32         =+       −    r ,  , at a unique point, find a cartesian equation of P. [4] 8 (a) The line L has cartesian equation ( ) ( )cos sin ,x y r += where 0.r  Find, in terms of r, the shortest distance from the origin to the line L and explain the significance of  . [4] The curve C has cartesian equation 22 22 1,xy ab+= where ,0ab  . (b) Determine, in terms of a and b, the equation of the tangent to C at the point ( )00,,P x y in the form ( )f , 1,xy = where f is a function of x and y. [3] The line l passes through the origin and is perpendicular to the tangent to C at P. The line l and the tangent line intersect at point X. (c) The locus of X as P varies is known as the pedal curve of C. Use the results of parts (a) and (b) to show that the polar equation of the pedal curve of C is 2 2 2 2 2cos sin .a b r += [3] (d) Given than 3a= and 2,b= find the arc length of the pedal curve of C. [3]

5 [Turn Over Tampines Meridian Junior College 2024 JC2 Preliminary Examination H2 Further Mathematics 9 For an object falling through the atmosphere, the ‘terminal velocity’ is the value approached by the velocity after a long time. A skydiver of mass m kg falls vertically after jumping off the aircraft at a high altitude. He opens his parachute when he reaches terminal velocity 1 1 ms ,v − which is when his velocity is constant. At time t seconds after the skydiver opens his parachute, his vertical displacement is s meters from the initial point where he opened his parachute. It is given that 0s= and 1 d d s vt = at 0.t = The net force acting on the skydiver is given by the following differential equation 2 2 dd ,dd ssm mg Ktt =− where g 2ms− is the gravitational acceleration constant and 1 kgsK − is the air resistance proportionality constant associated with an open parachute. (a) Find s in terms of t, 1v , m, g and K. [7] (b) Write down, in terms of m, g and K, the second terminal velocity, 2v , reached by the skydiver after opening the parachute. [1] It is given that 75m= , 9.8g = , 110K = and 1 54v = . (c) If the skydiver opens the parachute at an altitude of 1000 m etres from the ground, find the altitude of the skydiver when he is within 2% of 2v . [4] 10 (a) A solid ceramic display is to be built at the centre of an observatory deck, as part of an art exhibition. The display can be modelled by rotating the region bounded by the curve 2 e, xy −= the x-axis and the lines xr= and xr=− through π radians about the y-axis, where r is a positive constant. Determine the theoretical maximum volume of the solid ceramic display. [3]

6 (b) A tourist uses a Cassegrain telescope on the