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 that     1 12 20a r g a r g π.zz The complex numbers 1z and 2z are represented by the points 1Z and 2Z in the Argand diagram respectively. (iii) Explain why the locus of all points z such that 1zz z passes through the point 2Z . [1] (iv) The complex number w satisfies the relation    12 1arg argwz z z  . Sketch the locus of the points which represent w in the same Argand diagram. [2] (v) Find the range of values of arg w . [3] 7 CJC/2017/FM/Promo/7 It is given that the complex number 1c i i nos sz   , where ππ  . (i) By considering appropriate tr igonometric identities, or otherwise, show that the argument of z is 2  and find the modulus of z in terms of . [3] (ii) Hence, find the real and imaginary parts of 1c o s i s i n n   , where n  . [3] (iii) By considering the binomial expansion of  isin1c o s n     , show that  1 cos cos 2 cos 2cos cos12 22 n nn n nn n                        , where  ! !! n n r rn r    . [3]

8 RI/2017/FM/Promo/6a On the same Argand diagram, sketch the lo ci of points given by each of the following equations: 1 :2 i 5Lz  , 2 :a r g 3 iLz   , where 1tan 2  . Find, in the form ix y , the complex number which repres ents the point in the Argand diagram which is on both 1L and 2L , giving the exact values of x and y. [5] 9 SAJC/2017/FM/Promo/4 (i) Solve the equation 32 35 9zzz   , giving your answers in exact form. [4] (ii) Show the 3 roots of the equation in (i) on an Argand diagram and find the area of the triangle formed by joining the 3 points. [3] (iii) Write down two roots of the equation 300 200 10035 9zzz   in polar form. [1] 10 SAJC/2017/FM/Promo/9 (ai) Sketch, on a single Argand diagram, the locus of z which satisfies both 2i 2z and 46 i 2 izz  . [3] (ii) It is given that arg( 2i)z    . Find the complex numbers v and w that give the greatest and least values of arg( 2i)z respectively. [4] (bi) The complex number w has modulus 6 and argument 5 6  , and the complex number z has modulus 42 and argument 3 4  . Find the modulus and argument of z w , giving each answer exactly. [3] (ii) Given that the Cartesian forms of w and z are 33 3 i and 44 i respectively, find the exact real part of z w and deduce that 53 1cos 12 22   . [4]

11 TJC/2017/FM/Promo/6 (i) Show that if iez  , then 1 2i sink kzk z  , where k is a positive integer. [1] (ii) Show that 5sin  can be expressed in the form sin sin3 sin5 ,AB C  where the values of A, B and C are to be determined. [4] (iii) Find the particular solution of the differential equation  5d ec o s e cd xy yx  , given that y = 0 when x = 0. [3] 13 TJC/2018/FM/Prelim/I/7 (a) The complex numbers z and w are such that 4i 2 and 2i 1.zw  By considering an Argand diagram, find (i) the least value of zw , (ii) the greatest value of arg( )zw . [4] (b) The points P1 and P2 represent the complex numbers z1 and z2 respectively in an Argand diagram with origin O. Given that 22 11 22 0zz zz  , show that 12 cos isin33zz    . . Hence or otherwise, prove that the triangle 12OPP is equilateral. [7] AJC/2018/FM/Prelim/I/5 12 ACJC/2018/FM/Prelim/I/Q6 Let 1 cos 2 cos 4 cos 6 cos 20C       , sin 2 sin 4 sin 6 sin 20S      . (i) Show that i10sin11ie sinCS   for all π,nn   . [3] (ii) Hence, show that sin 21 1cos 2 cos 4 cos 6 cos 20 2sin 2        . [3] (iii) Deduce that 10 2 1 sin 21 cos 21cos 21sin 2 4sin 4sinr rr      . [3]

14 ๠e complex number z can be expressed as ie where   . (a) Given that z satisfies the equation = 11z , find the possible values of θ by means of a geometrical argument or otherwise. [3] (b) It is given, instead, that z satisfies the equation  21arg 01. . . nzz z    for some positive integer 2n and z ≠ 1. Determine the set of possible values of θ, giving your answer in terms of n. [7] Q Answers 1 23 i ,a  1 or 3 i 2 (ii) least 4n  (iv) 1, 1, 1, 1zi i i i    3 (ii) 52 2 , (iii)(a) b = 2, (