EJC 9649 2023 Prelim P1

2023 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2023 General Certificate of Education Advanced Level Higher 2 CANDIDATE NAME CIVICS GROUP INDEX NO. FURTHER MATHEMATICS Paper 1 [100 marks] 9649/01 12 September 2023 3 hours Additional Materials: Answer Booklet List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST Write your name, civics group and index number on the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all 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 6 printed pages.

2 2023 JC2 H2 Further Mathematics Preliminary Examination Paper 1 1 A curve has parametric equations 42 4ln , 4 , 0.x t t y t t    The arc of the curve from 1t to tk , where 1k  , is rotated through one complete revolution about the x-axis. Express the surface area generated as a polynomial in k. Given that this surface area is 384 , find the value of k. [4] 2 The sequence  nx is given by 1 1x  and 1 1 1 14 1 2 for 2. n n nnnx x n n n        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 the differential equation. [5] A particular solution of the differential equation (*) is such that 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 over-estimate or under-estimate. [1] 4 The curve C has parametric equations ( sin ), (1 cos )x a t t y a t    where 3π0 2t and a is a positive constant. (i) The region R is bounded by the curve C, the x-axis and the line 3 12xa   . The volume of the solid generated when R is rotated through 2 radians about the y-axis is given by V. Using the shell method, show that    23 0 2π sin 1 cos d b V a t t t t   , where b is a constant to be determined. [2] (ii) Find the exact value of V . [6]

3 2023 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over 5 (i) A 33 square matrix A is said to be skew symmetric if T AA . Given that A is skew symmetric, show that  det A = 0. [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] 6 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 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 2023 JC2 H2 Further Mathematics Preliminary Examination Paper 1 7 It is given that 2 1f ( ) 2xx x   . (i) Show that the equation f ( ) 0x  has a root  in the interval  ,1kk  where k is an integer to be determined. [2] In order to find an approximation  to  , two stages of the linear interpolation process is used on the interval  ,1kk  . (ii) Find the value of  , correct to 3 significant figures. [2] (iii) By considering f ( )x in the interval  ,1kk  , determine whether  is an under -estimate or an over - estimate of  . [3] The Newton-Raphson iteration can be used to estimate  . (iv) Explain why an initial approximation 0 21 2 kx  would not work. [1] The root  for the equation f ( ) 0x  satisfies the equation g( )xx where 4 1g( ) 2 .x x (v) Use an iterative method based on the form 1 g( )nnxx  with 0 1xk to find an approximation,  to  , correct to 2 decimal places. Verify that your answer is accurate up to 2 decimal places. [3] 8 Consider a body moving along a straight line 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 proportional 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 F( ) ( )d 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) (i) Now suppose F( )t C qt where C and q are positive constants. 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. (You may use the fact that 1e 1 x x    when 0x .) [2]

5 2023 JC2 H2 Further Mathematics Preliminary Examination Paper 1 [Turn over 9 (a) The terms in a sequence 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) A message is sent by transmitting a sequence of signals with each signal being one of the five different signals. Two of these signals require 1 microsecond each to transmit, while the other three signals require 2 microseconds each. It is assumed that the signals in a message are transmitted without additional time between signals. Let na be the number of messages that can be transmitted within n microseconds using these five different signals, where n is a positive integer. (i) Explain why the sequence 1 2 3, , ,a a a satisfies the rec