paper 1 dhs hci ri tmjc 2022 QP

2 A y x F O B E D G 1 In the diagram, a part of a hyperbola with eccentricity e is shown with a chord AB passing through the focal point F . The distance from the nearest directrix to F , denoted by EF, is p . (i) Let D, G be the points on the directrix nearest to A and B respectively. Show that ep AFAF AB BF AF −= − . [3] (ii) Hence, or otherwise, find 11 AF BF+ in terms of e and p . [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 [Turn over 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] 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] B y A x O

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 roo