JPJC 9649 2023 Prelim P1

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC2 Preliminary Examination 2023 FURTHER MATHEMATICS 9649/01 Higher 2 14 September 2023 Paper 1 3 hours Additional materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the booklets. If you need any additional answer paper ask the invigilator for a continuation booklet. 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 unsupport ed answers from a graphing calculator are not allowed in a question, you are required to pre sent 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. This document consists of 7 printed pages and 1 blank page. [Turn over

2 1 (a) Given that A and B are square matrices with a common eigenvector x corresponding to eigenvalues  and  respectively, show that  is an eigenvalue of the matrix AB with x as a corresponding eigenvector. [2] (b) The matrix 3 1 0 466 5 11 10      A has 1 1 1   as an eigenvecto r. Find the co rresponding eigenvalue. [1] (c) The other two eigenvalues of A are 1 and 2 , with corresponding eigenvectors 1 2 3    and 1 1 2    respectively. A square matrix B has an order 3 and has eigenvalues 2, 3 and 1 with corresponding eigenve ctors 1 1 1   , 1 2 3    and 1 1 2    respectively. Find a matrix P and a diagonal matrix D such that   3 1A B PDP . [3] 2 On the first day of January 2023, a freshwater prawn farmer bought some prawns to start an aquaculture business . It is known that th e population of prawns increase s by 20% every month. O n the last day of every month, the farmer sells 40% of the prawns and bought another 14 000 prawns. Let np  1n denote the number of prawns in the farm n months after he starts his business. (a) Write down an expression for 1np  in terms of np . [2] (b) Given that the farmer bought 220 000 prawns on the fir st da y of January 2023, determine np in terms of n. [3] According to a business analyst, the business will no longer be profitable if the total number of prawns falls below 60 000. If he continues with his present business model without taking any additional measures to rescue his farm , how many mont hs of operation should pass before the farmer considers closing his business? [2]

3 3 (a) Solve the equation 5 16 16 3 i 0z    , giving your answers in the form ier  , where 0r and      . [5] (b) Show these roots on an Argand diagram. Identify th e root for wh ich 2 z is the least. [3] 4 An ellipse has equation 22 22xy . It is given that a point P has coordinates  11,xy and is external to the ellipse. (a) Write down the equation of the line that has gradient m and passing through P. [1] (b) Show that the x-coordinates of the points of intersection of the line and the ellipse are given by the roots of the quadratic equation     2 2 2 2 2 1 1 1 1 1 11 2 4 2 2 4 2 0m x m y mx x y m x mx y        . [2] (c) Given that the line is a tangent to the ellipse, show that   2 2 2 1 1 1 12 2 1 0x m x y m y     . [2] (d) Hence show that the locus of the points P at which the two tangents to the ellipse are perpendicular is a circle, stating its centre and radius. [3] [You may use without proof the formulae: If  and  are the roots of a quadratic equation 2 0, 0ax bx c a    , then ,b a  c a  .] [Turn over

4 5 The linear transformations 44 1 :T  and 44 2 :T  are represented by the matrices 1M and 2M respectively, where 1 1 1 1 2 1 4 7 8 1 7 11 13 1 2 5 5      M and 2 2 0 1 1 5 1 3 3 3 1 1 1 13 1 6 6           M . (a) Find a basis for 1R , the range space of 1T . [2] (b) Find a basis for 2K , the null space of 2T . Hence, show that 2K is a subspace of 1R . [5] The set of vectors which belong to 1R but do not belong to 2K is denoted by W. (c) State whether W is a vector space, justifying your answer. [1] The linear transformation 44 3 :T  is the result of applying 1T and then 2T , in that order. (d) Find the dimension of the null space of 3T . [2] 6 The curve C is defined parametrically by 2,2x at y at , where a is a positive constant and t . (a) Find, in terms of a and t, the distance between   2,2P at at and  ,0Fa . [2] (b) By considering t he distance of P from the line xa , or otherwise, describe the curve C. [2] (c) Find the equation of the tangent to C at P. Show that this tangent meets the x-axis at   2,0Q at . [4] (d) Let D be the foot of perpendicular of P to the line xa . With the aid of a sketch of C and the p oints F, P, Q and D, show that DPQ FPQ  . (Do not quote the Reflection Property of C.) [3]

5 7 (a) It is given that cos isinz . (i) Prove that 1 2cos n nzn z . [2] (ii) Hence, by considering the binomial expansion of 6 1  z z , express 6cos  in the form  1 cos 2 cos 4 cos632     p q r s , where p, q, r and s are integers to be determined. [4] (b) By considering 21 1    N n n z , where e iz , show that   1 sin 2cos 2 1 2sin    N n Nn , where sin 0  . [6] 8 (a) By considering t he images of  1,0 and  0,1 , show that th e matrix as sociated with an anticlockwise rotation of points through an angle  about the origin is cos sin sin cos     . [2] The conic 1C has equation 22 6 4 0x y xy    . 1C is rotated 45 anticlockwise about the origin to obtain the conic 2C . (b) Show that the equation of 2C is 2 2 12 yx  . [5] (c) State whether 2C is a parabola, an ellipse or a hyperbola. [1] (d) For 2C , find the (i) eccentricity, (ii) coordinates of the foci, (iii) equations of the directrices. [4] [Turn over

6 S 9 It is given that the curve C is defined for ln 3 ln 3x   and has equation ee 2 xx y  . (a) Find the exact length of C. [3] A region R bounded by C and the line 5 3y is rotated  radians about the y-axis to form a solid S. (b) Using shell method, show that the volume of the solid S can be expressed as     2 ln 3 ln 33V p q r    , where p, q and r are constants to be determined. [5] The y-ordinate of the c entroid, also known as the centre of gravity, of a solid generated by rotating a region bounded by a curve  fyx , the lines ya and yb , about the y-axis is given by 2 d b a y x y yV   , where V is the volume of the solid generated. (c) Show that the centroid of S is given by   ln3 2 2 2 0 e e d4 xxy x x V    . [2] An object is made such that it has a base in the shape of S as shown in Figure 1 below. Figure 1 (d) Given that th e weight of the materials used to ma ke the part of the object above the base S is negligible, fin d the numeric al value of the height of the centroid of the object