PJC P2

3 Section A : Pure Mathematics [40 marks] 1 (i) Given that    1 2 1 cos3h 2 sin     , and that  is a sufficiently small angle, show that   13h 2     . [2] (ii) Given that n is a positive integer, by expanding binomially, show that   h n ab  , where a and b are constants to be found in terms of n. [2] (iii) State the range of values of  for which the expansion is valid. [1] (iv) Verify your answers in (ii) by finding a suitable Maclaurin’s series. [2] 2 The complex number z satisfies the relations 4i 12 z z   , 2i 2z ,   1 3arg 2 4z  . (i) Illustrate these loci on a single Argand diagram, showing clearly the locus of z. [5] (ii) Find the complex number z such that z is the smallest, giving your answer in exact form. [3] [Turn over Name : Class :

4 3 The diagram shows the parts of the graph of 1 1y x  . The four rectangles each of equal width, as shown in diagram below, are rotated through  radians about the x-axis. Prove that the volume generated may be expressed as   4 2 1 4 8r r    . [2] n rectangles, each of equal width under the curve between 1x and 2x , are used to estimate the volume of solid obtained when the region bounded by the graph, the lines 1x and 2x , is rotated through  radians about the x-axis. Find the estimated volume in the form   1 f n r r   , where f(r) is to be determined in terms of r and n. [2] (i) Deduce that   21 11 62r n nnr   . [3] (ii) State the value of   21 2 n r n nr   as n . [1] x 1 1y x  O y x=1 x=2

5 4 (a) A, B and C are constants such that   2 22 4 1 xB A x Cx    for all values of x. Find the values of A, B and C. [1] State precisely a sequence of transformations by which the graph of 2 2 4 xy x   may be obtained from the graph of 2 1 1 y x   . [3] (b) The sketches above show the graphs of 2 f( )yx and f ( )'yx for a certain function f. (i) Sketch the graph of f ( )yx . [2] (ii) Sketch the graph of  f1yx . [2] [Turn over x y 0 f '( )yx 1 1 x y 0 2 f ( ) yx  1 ,3  1 , 3 2 2y 2y

6 5 The functions f and g are defined as follows f : , , , and xx x x x         , 2g : 3 6 2,x x x x  