2021 J2 RVHS H2 Math CT QP

1 ©RIVER VALLEY HIGH SCHOOL 9758/01/2021 RVHS 2021 JC2 H2 MATHS COMMON TEST Section A: Pure Mathematics [60 marks] Sequence & Series not tested in 2023 H2 MA CT 1 A M C a P O b N B As shown in the above parallelogram OACB, OA = a and OB = b. It is given that M is the midpoint of AC and 21::ON NB = .The point P is the intersection of the line OM and AN. (i) By letting OP OM= and AP AN= , find the value of  and  and hence the position vector of P in terms of a and b. [3] (ii) Given that the point Q lies on AN produced and is such that 21::AN NQ = , find the position vector of Q and hence determine if Q, B and C are collinear. [3] 2 Solve the following simultaneous equations for complex numbers z and w. * * 2i 1 4 zw zw −= += [6] 3 Let 1 3iz=+ ( )arg 4w = and 22zw = . (i) Find w in the form ixy+ where ,xy  . [2] (ii) Find the least positive integer value of n such that ( ) n zw is purely imaginary. [3] 4 It is given that 3f1 2!() ()r r=− + , where r is a non-negative integer. Show that 1f f 1 2! ()( ) ( ) () arrr r +− − = + for some constant a to be determined. [1] (i) Using the above result, find 1 1 2!() n r r r= + + in terms of n . [3] (ii) By using the result in part (i), deduce the value for 3 1 ! n r r r= −  , simplifying your answer. [2]

2 ©RIVER VALLEY HIGH SCHOOL 9758/01/2021 Integration not tested in 2023 H2 MA CT 5 The functions f and g are defined by 2 1f 1 for , 1 1x x x x→ −  −−: , 2 4 5 for , 1g x x x x x→ + +  : . (i) Find 1f ()x− and state the domain of 1f− . [3] (ii) State whether the composite functions fg and gf exist, justifying your answer. Hence find the range of the composite function(s) that exist(s). [4] 6 A curve C has parametric equations etx= , sinyt= where 0 t  . (i) Find e sin dt tt . [4] (ii) Sketch the graph of C. [1] (iii) Find the area of the region bounded by C, the x-axis, the lines 1x= and eax= , where a is a positive real constant, in terms of a. [2] 7 The plane  passes through the points with coordinates ( 1, 0, 1),− (2, 1, 1)− and (1, 3, 2)− . The line l passes through the point P (16, 20, 16) and is parallel to the vector 1 2 1    . (i) Find the cartesian equation of  . [3] (ii) Find the coordinates of the point of intersection of l and  . [2]