NYJC 2021 FM CT2 P1 - modified

This document consists of 6 printed pages. NANYANG JUNIOR COLLEGE Internal Examinations © NYJC 2021 [Turn Over NANYANG JUNIOR COLLEGE JC2 COMMON TEST 2 Higher 2 FURTHER MATHEMATICS 9649/01 Paper 1 1st July 2021 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, highlighters, 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 a 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 2021 JC2 Common Test 2 9649/01 [Turn Over 1 Find the general solution of the differential equation 2 2 dd 2 3 cosdd yy yxxx + + = . [5] 2 The linear transformation T: 22→  is represented by the matrix cos sin sin cos   −= A . (i) Find the image of cos sin r r     where r ≥ 0 and 02  under T and give a geometrical description of the transformation T. [2] (ii) Find all values of the angle  for which the matrix A has real eigenvalues and state the corresponding eigenvalue. [5] 3 Let 1e , 2e , 3e be the standard basis for 3  and let 33:T →  be a linear transformation with the properties 1 2 2 1 2 1 2 3 3(( ) , ) 2 ,. )(e e e eT e T e T e ee= = + + + = (i) Find a vector v such that ()T v kv= for some k  . Explain the significance of this result. [2] (ii) Find A , the matrix representation of T with respect to the standard basis. [2] (iii) State the kernel of the linear transformation 1T with matrix representation 2AI− and obtain a basis for the range space, 1R , of 1T . [3] 4 Let A and B be nn matrices. (i) Prove that ker( ) ker( )A BA . [2] (ii) If BA is an invertible matrix, prove that both A and B are invertible. [2] (iii) If +AB is invertible, show that ( ) ( ) 11−− + = +A A B B B A B A . [3]

3 NYJC 2021 JC2 Common Test 2 9649/01 5 A research laboratory is studying a population of hamsters. The hamsters can be grouped into three age classes of equal duration defined as follows: Hamsters in the first age class are for those with 0 age 1 , second age class are for those with 1 age 2 and the third age class are for those with 2 age 3 . The maximum life span of the hamsters is assumed to be 3 years. The population of hamsters in the laboratory has the characteristics listed below: • Half of the population survives its first year. Of those that survive the first year, half survives the second year. • There is no offspring produced by the hamsters in the first age group. The average number of offspring for each hamster of the population is 6 for the second age class and 8 for the third age class. The number of hamsters in the first age class, second age class and third age class after r years are denoted by ,rrab and rc respectively. The information above can be represented by the matrix equation 1nn +=Mx x where M is a 33 matrix, r rr r a b c  = x and r is a positive integer. (i) Write down the matrix M. Given that there are 120 hamsters in each of the three age class initially, show that 1 1680 60 60  = x . Find the number of hamsters in each age class after 2 years. [3] (ii) The laboratory would like to achieve a stable growth pattern, one in which the proportion in each age class remains the same each year. For this stable growth pattern to be achieved, 1n n n +==Mx x x . By finding a suitable eigenvalue and eigenvector, determine the smallest initial population of the hamsters in each age class so that the proportion of hamsters in each class remains the same each year. [4] (iii) Using the initial population values found in (ii), determine the number of years taken for the total population of the hamsters to exceed 5000. [3]

4 NYJC 2021 JC2 Common Test 2 9649/01 [Turn Over 6 The ‘Chinese rings puzzle’ is a disentanglement puzzle featuring a loop which must be disentangled from a sequence of rings on interlinked pillars. The number nx of steps required to solve the ‘Chinese rings puzzle’ with n rings satisfies 1 1x = and 1 2, 2 is odd is e1 ve, n n n n xn nx x + = + (i) Prove that 21 21n n nxx x++ ++= . [2] (ii) Find the maximum number of rings such that the number of steps required to solve the puzzle is less than 2021. [2] (iii) Given that the solution of the recurrence system is of the form (2 ) ( 1)nn nx C A B−= ++ , find the values of A, B and C. [5] 7 Find the coordinates of the stationary points of the surface ( ) 21 πe cos 3 y zx −− =−  in the region ( ) ,| π π, x y x y−     and determine the nature of these stationary points. [8] 8 The cartesian equation of a curve 1 is given by 4 6 3 0x xy− + = . Show that ( ) 242 4 1d1 d4 xy xx ++= . [1] (i) The points A and B on 1 correspond to x = 1 and x = 2 respectively. Show that the length of the arc AB is 17 12 . [2] (ii) The arc AB is rotated through one complete revolution about the x-axis. Find, in exact form, the area of the curved surface generated. [3] Another curve 2 has cartesian equation 4 6 24 3 0x xy x− + + = . State the relationship between the curves 1 and 2. [1] The region bounded by the curves 1 and 2 and the lines x = 1 and x = 2, is denoted by R. (iii) Find the perimeter of R, showing your working clearly. [2] (iv) Obtain the volume of the solid formed when R is rotated completely about the y-axis, giving your answer in an exact form. [2]

5 NYJC 2021 JC2 Common Test 2 9649/01 9 In many practical situations, we are interested to fit a polynomial curve to a set of given points obtained via experiments. Suppose we have n data points given by 11( , )xy , 22( , )xy ,  , ( , )nnxy . We attempt to find a degree 1n− polynomial that will pass through all these points. In another words, we have to find a polynomial 21 0 1 2 1() n np x a a x a x a x − −= + + + +  such that () iip x y= for each 1, 2, ,in= . (i) Show that the values of 0a , 1a ,  , 1na − can be found by solving the system of equations 01 12 23 1nn a ay ay y ay−                =                Q  , where 21 1 1 1 21 2 2 2 21 3 3 3 21 1 1 1 1 n n n n n n n x x x x x x x x x x x x − − − −    =   Q      . [1] (ii) For the case where 3n= , use Gaussian row elimination to show that the determinant of Q is 3 22 1 1 3 )( ( ( ))xx x xxx−−− . [4] It can be proven that the determinant of Q is 1 )( ji i j n xx    − . (iii) If the n data points are such that ijxx if ij , show that the polynomial that passes through all the n data points is unique. [2] (iv) For the case where 4n= , (0) 1p =− , (1) 0p = , (3) 2p = and (4) 5p = , find the equ