ACJC 9649 2023 Prelim P1

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 12 September 2023 3 hours Additional Materials: Cover Sheet Answer Paper List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your index number, class and name on 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. 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 7 printed pages and 1 blank page. [Turn Over

2 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 1 A curve has parametric equations 4 2 4ln , 4 x t t yt   where 0t  . The surface area generated when the arc of the curve from 1t  to tk , where 1k  , is rotated through one complete revolution about the x-axis is 384π square units. Find the value of k. [4] 2 The sequence  nx is given by 1 1x  and 1 1 1 14 1 2 n n nnnx x n n       for 2n . By multiplying the recurrence relation throughout by 1nn  , use a suitable substitution to determine nx as a function of n, simplifying your answer. [7] 3 The variables x and y are related by the differential equation 2 3 2 3d d yx y x x y yx    , (*) and it is given that 0xy . (a) Use the substitution y ux to find the general solution of (*). [5] A particular solution of (*) has 1y when 2x . (b) Use one step of the Euler method to calculate an approximate value of y when 2.5x . [2] (c) Given that the solution curve is concave downwards near the point  2,1 , determine with a sketch whether the approximate value of y found in (b) is an overestimate or underestimate. [1]

3 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 4 (i) A 33 square matrix A is said to be skew symmetric if T AA . Prove that for a given skew symmetric A,  det A is equal to zero. [3] (ii) Let T be the linear transformation such that 33T:  and T( ) x a x where 1 2 3 a a a   a and the multiplication is the usual vector product in 3 . Let M be the matrix representation of T. (a) Find M in terms of 1 2 3 ad, naaa . [2] (b) Determine if M is invertible. [1] (c) State ker(T) and R(T) , the null space and range space of T respectively. Give a geometrical interpretation of your answers. [4] 5 Let T denote the matrix cos 2 sin 2 sin 2 cos 2      . (i) By finding cos sin      T and sin cos      T , show that cos sin     and sin cos     are eigenvectors of T and write down their corresponding eigenvalues. [4] (ii) Express T in the form 1RDR where cos sin sin cos    R . [1] The matrix T represents the reflection of a position vector in 2 about the line through the origin that makes an angle  with the positive x-axis. (iii) Find the reflection of 2 1   about the line 3yx . [3] (iv) The matrix R in (ii) represents the rotation of a position vector in 2 through an angle  about the origin. By finding TT , show that the product of two reflection matrices is a matrix that represents the rotation of a position vector in 2 . [2]

4 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 6 (i) Show that the equation f ( ) 0x  , where 2 1f ( ) 2xx x   , has a root  in the interval  ,1kk  where k is an integer to be determined. [2] (ii) In order to find an approximation  to  , two stages of the linear interpolation process is used on the interval  ,1kk  . Find the value of  , correct to 3 significant figures. [2] (iii) Show how the consideration of f ( )x in the interval  ,1kk  enables you to determine whether  is an under-estimate or an over-estimate of  . [3] (iv) Explain why the use of the Newton -Raphson iteration with initial approximation 1 0 2xk to find  would not work. [1] (v) The root  for the equation f ( ) 0x  satisfies the equation  gxx where 4 1g( ) 2 .x x Use an iterative method based on the form 1 g( )nnxx  with 0 1xk to find an approximation,  to  , correct to 2 decimal places. Justify whether  is a sufficiently accurate approximation. [3] 7 The curve C has parametric equations    sin , 1 cosx a t t y a t    , where 3 20 πt and a is a constant. (i) The region R is bounded by the curve C, the x-axis and the line  3 2 1xa  . Using the shell method, show that V, the volume of solid generated when R is rotated through 2π radians about the y-axis is given by    23 0 2π sin 1 cos d b a t t t t , where b is a constant to be determined. [2] (ii) Find the exact value of V . [6]

5 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 [Turn Over 8 Consider a body moving horizontally in one dimension through a liquid medium. The body experiences a force  F t in the direction of motion and a frictional force in the opposite direction. At lower speeds of motion, the frictional force is directly proporti onal to the velocity  vt of the motion. According to Newton’s second law of motion, for a body with constant mass m, the product of the body’s mass and acceleration is equal to the net force on the body in the direction of motion. This basic model leads to the differential equation    d Fd vm t kv tt  where k is a positive constant. (a) Suppose  F tC where C is a positive constant. Given that the initial velocity of the body is 0v , find an expression for v. [4] (b) Show that, after a long time, the velocity of the body tends to a limiting value that is independent of the initial velocity. [2] (c) Now suppose F( )t C qt where C and q are positive constants. (i) Given that the body is initially at rest, show that 2 1e kt mqm qtCv kk k       . [5] (ii) Show that at the instant when F( ) 0t  , the velocity of the body is positive. [2] (You may use the fact that 1e 1 x x    when 0x .)

6 ANGLO-CHINESE JUNIOR COLLEGE 2023 H2 FURTHER MATHEMATICS 9649/01 9 (a) The terms in the sequence  nu satisfy the recurrence relation 1242r r ru u u  . (i) Find the general solution of this recurrence relation. [3] (ii) Show that 3kkuu   for all 0k  where  is a constant to be determined. Find 32 1 r r u     in terms of 1u . [4] (b) Five different signals are available for transmitting a particular message. Two of these signals require 1 microsecond each for transmission, while the other three signals require 2 microseconds each. It is assumed that the signals in a message are sent immediately one after another. Signals can be repeated within a message. Let na be the number of messages that can be transmitted in exactly n microseconds using these five different signals, where n is a positive integer. (i) Exp