DHS Complex Numbers (9649) Topical Revision

H2 Double Math COMPLEX NUMBERS [part FM] 1 HCI/2014/I/3 One of the roots of the equation 32 21 3 i 0zz a z    is iz  . Find the complex number a and the other roots. [5] 2 MI/2014/I/10 (i) By using de Moivre’s theorem, or otherwise, show that  2 ππ12 c o s s i n . 44 n n nnii    [2] (ii) Using the result in (i) or otherwise, find the least positive integer n for which 1 n i is real and negative. Solution by trial and error will not be accepted. [3] (iii) For the equation 432 0za zb zc z d   where a, b, c and d are real, give a brief explanation and determine the possible number of complex roots the equation can have. [2] (iv) Solve the equation 4 40 ,z  expressing the solutions in the form xi y where x and y are real. [4] 3 ACJC/2014/II/2 The complex number z satisfies the relations  1 4arg 3 3i πz   and 33 i ,zb  where b is a constant and 13 .b (i) Illustrate each of the above relations on a single Argand diagram. [2] (ii) Find the exact least possible value of 5i .z [1] (iii) Given that the least possible value of z is 18 2 , (a) find the value of b, [1] (b) hence find an exact expression for z, in the form ixy . [2] (c) State the cartesian equation of the locus of the point representing complex variable w such that 1ww z , where 1z is the complex number found in part (b). [1]

4 HCI/2014/II/3 (a) (i) Find the fifth roots of –32, expressing the roots in the form ier  , where 0r  and ππ  . [2] (ii) The roots representing 1z and 2z are such that    120a r g a r g πzz . State the complex number w in the form ier  where 21zw z . [1] (b) The complex number z satisfies 33 i 1izz   and  11 63π arg πz . (i) On an Argand diagram, sketch the re gion in which the point representing z can lie. [3] (ii) Find the area of the region in part (b)(i). [3] (iii) Find the range of values of  arg 5 iz . [2] 5 RI/2014/II/4 Do not use a calculator in answering this question. The complex number z satisfies both the relations 23 i 4z  and 5 6 π arg iz   . (i) On an Argand diagram, shade the region in which the point representing z can lie. [4] (ii) Find the least possible value of .z [2] (iii) State the cartesian form of the complex number z when iz is greatest. [1] (iv) Find the range of values of  arg 4 3 iz . [2]

6 RVHS/2014/II/2 (i) Solve the equation 6 64 0z  , giving the roots in the form ier  , where 0r and ππ   . [3] (ii) Show the roots on an Argand diagram. [2] The roots denoted by 1z and 2z are such tha