EJC_H2_2021_Prelim_P2

EUNOIA JUNIOR COLLEGE JC2 Preliminary Examination 2021 General Certificate of Education Advanced Level Higher 2 Section A: Pure Mathematics [40 marks] 1 Do not use a calculator in answering this question. The function f is defined by ( ) 32f 4 12 13 10z z z z= − + − . Given that 1f i 02  += , find all the roots of ( )f z . [4] 2 It is given that cos isinz =+ , where 0 2  . (i) Show that i 2e sin icos    − =− . [1] (ii) Hence, or otherwise, show that ( ) 2arg 1 2z − = − and find the modulus of 21 z− . [3] (iii) Hence, represent the complex number 21 z− on an Argand diagram. [2] (iv) Given that ( ) * 32 1 z zz − is real, find the possible values of  . [3] 3 (a) The function f is given by 11 , 0 .26f : cos , x x kxx   +    (i) State the largest exact value of k for which the function 1f− exists. [1] For the rest of the question, the domain of f is 4 3 , 0 .xx    (ii) Write down the equation of the line in which the graph of f ( )yx= must be reflected in order to obtain the graph of 1f ( ).yx −= Hence, sketch on the same diagram, the graphs of f ( )yx= and 1f ( )yx −= , indicating the exact coordinates of the endpoints of both graphs. [3] (iii) State the value(s) of x for which ( ) ( ) 11ff f f xx−− = . [1] (b) The functions g and h are defined by 2h : 2 3, , 0, hg : 2 3 2 , , where x x x x x x a x a a + +   + − Find g( )x and state the domain of g. [3]

4 The curve C has equation 2 103. 24y xx=− −+ (i) Without using a calculator, determine the exact coordinates of the stationary point of C. [2] (ii) Sketch C, stating clearly the equations of any asymptotes. [2] The region R is bounded by C, the line 3 16 15yx+= and the y-axis. (iii) Find the exact area of the region R. [3] (iv) Find the volume generated when R is rotated through 2 radians about the y-axis. [3] 5 Referred to the origin O, points A and B have position vectors a and b respectively. Point P is on the line AB such that ::AP PB m n= where m and n are positive integers. Point C is on OP extended such that : 1: 2OP PC = . (i) Show that 23n m mAC m n m n −   =+   ++    ab . [3] (ii) If =ab and the angle between vectors a and b is 3  , find the area of the triangle ABC in terms of a . [4] (iii) Find the ratio :AP PB such that AC is parallel to OB. [2] Section B: Probability and Statistics [60 marks] 6 For events A and B it is given that ( ) 3P, 5A = ( ) 7P 60AB= and ( ) 13P '| . 20AB = (i) Show that ( ) 1P. 3B = [3] For a third event C, it is given that ( ) 3P, 10C = and that ( ) 1P. 10BC= (ii) Determine if events B and C are independent. [1] (iii) Given also that ( ) 1P, 15A B C  = find the greatest and least possible values of ( )P ' .A B C [3] 7 John has 4 different pairs of socks in his drawer. Without looking at his drawer, he randomly draws one sock at a time from his drawer without replac