SAJC P2

2 [Turn Over Section A: Pure Mathematics [40 Marks] 1 (i) By expressing 11 ! ( 1)!kk  as a single fraction, find the sum of 2 3 4 2 1 .3! 4! 5! (2 2)! n n      (ii) Hence show that 2 2 21 ( 2)! 6 n r r  . [4] [2] 2 Each time a ball falls vertically on to a horizontal floor, it rebounds to three-quarters of the height from which it fell. It is initially dropped from a point 10m above the floor. (i) Find, and simplify, an expression for the total distance the ball travels when it touches the floor for the (n+1)th time. (ii) Find the number of times the ball has bounced when it has travelled 55m. (iii) When the ball is at the maximum height after bouncing for the 4th time, the floor rises to 2m above the ground. Calculate the total distance it travels before coming to rest. [2] [2] [3] 3(a) The graphs of f '( )yx and )(f2 xy  are shown below: Sketch the graphs of y = f(x), labelling the asymptotes, point(s) of intersection with the axes and turning points if any. [2] (b) Find the values of the constants A and B such that 2 22 4B A( 2) ( 2) xx xx - =+-- for all values of x except 2x = . Hence state a sequence of transformations by which the graph of 2 2 4 ( 2) xxy x -= - may be obtained from the graph of 2 1y x= . [4] 2 f '( )yx x -2 )(f2 xy  x -2 2 -2 2 -3 y y O O

3 [Turn Over 4 A virus is found to be present in Country A, containing a population of 10,000 people. The rate at which the number of infected people, x, in thousands, is increasing at any time t is proportional to the product of the infected people and the number that have yet to be infected. Initially, there are 2,000 people infected. Show that the number of infected people at time t is 10 10 10 kt kt ex Ae  where 0A is a constant to be determined. Sketch, on the same diagram, three solution curves for 0, 0, 0.k k k State the range of values of k such that the virus will be eliminated. Give a possible reason for that to happen. [4] [2] [2] 5(a) The complex numbers z and w each have modulus 1, and have argument  and  respectively. Prove that () 22 sin( ) 2 i z w i e    . Hence find k such that ( zw )( **zw )= k ( 1 cos (  )), where kZ . [5] (b) Solve the equation 2 3 1 i 320 1 i 3 z z     , giving the roots in the form