RVHS 9758 2023 Prelim P2

1 ©RIVER VALLEY HIGH SCHOOL 9758/02/2023 Section A: Pure Mathematics [40 marks] 1 The complex numbers 12 and zz are such that i 3 1 4ze π = and i3 2 2ze π− = . (i) On an Argand diagram with origin O, mark out the points which represent 12 and zz respectively. Deduce the exact value of 12 zz− . [3] (ii) Write down a complex number a such that 12z az= . Hence or otherwise, describe geometrically the relation between 1z and 2z . [2] 2 A sequence 0u , 1u , 2u , … is such that 1 41 2 33 3 nn nnuu Q+  −= − +    , where Q is a constant and 0n≥ . (i) Given that 0 3u = , 1 4 3u = , find Q and 2u . [2] (ii) Show algebraically that for any real positive constant ε , there exists an integer 0n such that 1nnuu ε+ −≤ for 0nn≥ . Find an integer 0n when 0.001ε = . [4] 3 (a) A curve has equation ( )fyx= , where ( ) 2 for 0 1, 13f for 1 3,22 0 otherwise. xx xx x  ≤≤ =− + <≤   (i) Sketch the curve for 14 x−≤ ≤ . [3] (ii) On a separate diagram, sketch the curve with equation 1f1 2yx = − , for 14 x−≤ ≤ . Write down the coordinates of the end points of this curve. [2] (b) State a sequence of transformations that will transform the curve with equ ation lnyx= onto the curve with equation ( ) 3 ln 2yx= − . [2] 4 The line 1l has equation ( )3 λ= ++r k ij where λ is a parameter. Referred to the origin O , the points A and P have position vectors 4+ik and 345−+ +i jk respectively. (i) Find the cartesian equation of the plane 1π containing the line 1l and the point A. [2] (ii) The point Q is the foot of perpendicular of P on 1l . Find the position vector of Q . [3] (iii) The line 2l passes through P and is parallel to the vector 3 m++ij k where m is a constant. Find the value of m given that 2l is parallel to 1π . h denotes the distance between 2l and 1π . Find the value of h . [3]

2 ©RIVER VALLEY HIGH SCHOOL 9758/02/2023 Section B: Statistics [60 marks] (iv) The plane 2π is parallel to 1π . The 3l passing through P and Q , intersects 2π at the point R . The points P , Q and R are such that : 2:5PQ PR = . Without finding the equation of 2π , find the possible value(s) of the distance between 1π and 2π in terms of h . [2] 5 (i) Using the substitution 2 1ux= + , evaluate 1 1 1d a xx x − − +∫ , giving your answer in terms of a. [5] (ii) The curve C has equation 1y xx= + . The region R is bounded by C, the lines 2y= and 1x=− . Find the exact volume of the solid obtained when R is rotated through 2π radians about the line 2y= . [7] 6 Two fair six-sided dice are thrown and the highest score X is recorded. (i) State the probability distribution for X. [1] (ii) Find the expected value and variance for X. [3] 7 A group of 8 people consisting of 2 children, 4 women and 2 men goes for lunch, photo - shoot and a game.