HCI 2022 FM Prelim P1 - modified

2022 HCI Prelim Paper 1_Modified [96 marks] 1 Consider the function ( ) 2f, x y x ky=+ , where k is a non-zero constant. (a) Find, in terms of k , the linearization of ( )f, xy at the point ( )2, 2 . [3] (b) By using the answer in part (a) and a suitable integer value of k , find the approximate value of 22.1 3.9+ , giving your answer in the form p bq , where p , q and b are integers to be determined. [2] 2 Consider the following system of linear equations, where ,ab   . 0 20 x y az x y az x y z b + + = − + = + + = Let S be the solution set of this system of linear equations. By performing suitable row operations, state the possible values of a and b such that (i)  S = v , where v is a nonzero vector, [2] (ii) S is not a subspace of 3  . Find the possible sets of S in this case. [3] 3 In the diagram, the origin O is the focus of a parabola P with equation 2 4 ( )y h x h=+ , where h is a positive constant. (i) By taking the pole to be at O , find a polar equation of P . [2] (ii) A and B are points on P such that OA is perpendicular to OB . By using differentiation, find the exact minimum area of the triangle OAB in terms of h . [You are not required to prove the area found is a minimum.] [5] B y A x O

4 (a) A fixed point of a function g( )x is a real number  such that g( )= . An approximate value for a fixed point  is to be found by using the fixed point iteration method of the form 1 g( )nnxx+ = , 0, 1, 2, ...n= , with an initial estimate of 0 1.5x = . For each of the following expressions for g( )x , explain with reason(s) or with the aid of suitable diagrams, whether it is suitable for fixed point iteration. If the g( )x is found suitable for fixed point iteration, obtain an estimate for  correct to 4 decimal places. (i) 32g( ) 4 6x x x x= − − + [2] (ii) 1 26g( ) 4xx x =− [1] (iii) 1 26g( ) 4x x = + [3] (b) The function ( ) 21 21f ( ) e 2 x x   − = is the normal density function with a mean value of zero and standard deviation  . The probability of a randomly chosen value x as described by this function lying in the interval [ , ]ab , is given by f ( ) d b a xx . Use Simpson’s rule with 5 ordinates to find an approximation to the probability that a randomly chosen value x will lie in the interval [ , ]− . Give your answer correct to 3 decimal places. [4]

4 5 Do not use a calculator in answering this question. (i) Show algebraically that 3 5 7 9cos cos cos cos cos 010 10 10 10 10              + + + + =                   . [2] (ii) The equation 10 10 (13 1) 0zz + − = has 10 complex roots denoted by iz and * iz , where * iz are the complex conjugates of iz , 1,2,3,4,5i= , respectively. By expressing these roots as i1 13 ez =− , where   −   , or otherwise, evaluate * * * * * 1 1 2 2 3 3 4 4 5 5 1 1 1 1 1 z z z z z z z z z z+ + + + . [8] 6 (i) Let 0 2 1 2(1 ) d t n n x x xI =+ . Show that 2 3 212( 2) (1 ) ( 1)n nnn I t t n I − −+ = + − − , where 2n . [3] (ii) By considering the derivative of 21)ln (x x+ + , find the exact value of 4 3 2 0 1d xx+ in the form ln 3mn+ , where m , n are constants to be determined. [4] (iii) A curve C has parametric equations tanx = , 1 1 cos 2y = + for 22  −   . A part of C bounded by 0x= and 4 3x= is rotated through 2 radians about the x-axis. By converting the equations to cartesian form, or otherwise, f ind the exact value of the surface area of revolution thus formed. [4] 7 Taking the origin as the pole, the equation of the curve G in polar coordinates f ( )r = satisfies the differential equation 2 2 d 4 5sin 3d r r  += for 3 55   . It is given that when 2  = , 1r= and d 2d r  =− . (i) Find the polar equation of G . [7] (ii) Sketch G , indicating clearly the point with polar coordinates (1, )2  on G . [1] (iii) Show that the area enclosed by G is given by 2(sin sin )55ab  +− square units, where a , b are exact rational numbers to be determined. [5]

6 8 In a habitat, the number of rabbits nR (in thousands) and wildcats nW (in thousands) in n years can be modelled by 11 31 22n n nR RW−−=− , 11 31 44n n nW R W −−=+ , where 1n , and 0R and 0W are the initial number of rabbits and wildcats respectively. (i) Find an expression for nR in terms of 1nR − and 2nR − . [2] (ii) Show that the 21 matrix n n R W   can be expressed as 01 0 R W −   PDP , where P is a 2 2 invertible matrix and D is a 2 2 diagonal matrix. [4] (iii) Using the result in part (ii), find nR and nW in terms of n , 0R and 0W . [3] (iv) Given that the population of both rabbits and wildcats decrease and approach a non- zero equilibrium population, find a relationship between 0R and 0W . [3] 9 The sequence of integers  1 2 3, , , ...u u u is defined by the second -order recurrence relation 1ua= , 2u ka= and 21 22n n nu u u++ =− for 1n , where a , k are positive constants. (i) Show that 4 4nnuu+ =− for all 1n . [2] It is given that 42ru − is a multiple of ka for all positive integers r . (iii) Find 4 1 N r r u =  in terms of a , k and N . [3] (iv) Find nu in the form ( cos sin )np A qn B qn+ , where A , B are constants in terms of a , k and p , q are constants to be determined. Hence find 20 1 ( cos sin )r r p A qr B qr = + in terms of a and k . [5]

7 10 A predator-prey model is a pair of first -order non-linear differential equations used to describe the dynamic interactions between two species in eco logical system s, one as predator and the other as prey. A financial analyst adopted the predator-prey model to predict the stock market prices of two companies X and Y . The stock price s $ x and $ y of companies X and Y respectively in t months are modelled as follows. d 0.2 0.01 ,d d 0.05 0.004 .d x x xyt y y xyt =− =− + It is given that the initial stock prices of companies X and Y are $10.00 and $15.00 respectively. (i) A ‘modified’ Euler method with step size h is given by 11 11 21 21 , , d ,d d .d x x y y x x y y xx x h t yy y h t == == =+ =+ Using this ‘modified’ Euler method with step size 0.5 , estimate the stock prices of companies X and Y after 1 month. Give your answers correct to 2 decimal places. [4] (ii) Find an expression for d d y x . Hence, using integration, show that eeg h y m nxy Kx = , where g , h , K , m , n are constants to be determined. [5] (iii) The graph below shows the relationship between x and y . It is given that the tangents of the curve at the points A and C are parallel to the x- axis, while t he tangents of the curve at the points B and D are parallel to the y- axis. By finding the x-coordinate of points A and C , and the y-coordinates of points B and D , determine the coordinates of the points A , B , C and D . [3] (iv) Using the graph and answers found in part (iii), comment on what the predator-prey model indicates about the stock prices of companies X and Y . [1] End of Paper y O x