NJ Topical - Complex Numbers

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 81 of 103 F08: Further Complex Numbers 1 (a) Show that if iez  , then 1 2i sin ,n nz n z   where n is a positive integer. [1] Hence, or otherwise, show that 5sin  can be expressed in the form sin sin 3 sin 5a b c     , where the numbers a, b and c are to be determined. [3] Deduce that 5cos  can be expressed in the form cos cos3 cos5p q r    , where the numbers p, q and r are to be determined. [2] (b) On the same Argand diagram sketch the loci of points given by each of the following equations:   1 2 : 3 3i 3 2, 5: arg 3 2 3 3i . 6 L z L z         Find, in the form ix y , the exact complex number which represents the point on both 1L and 2L in the Argand diagram. [5] (2016 RI / JC1 / MYE / Q6) 2 Do not use a calculator in answering this question. The complex number z is given by  1 3 i  . (a) Let   3 2P 2 10w w aw w b    , where a and b are real. If  P 0z  , find the values of a and b and determine all the roots of the equation  P 0w  . [4] (b) Find the smallest positive integer n such that  2 * n z z is a positive real number. [3] (2016 ACJC / JC1 / Promo / Q2) 3 Consider the polynomial   4 3 2P 1z z z z z     . (i) By considering    1 Pz z , find the solutions to the equation  P 0z  , expressing the solutions in the form ier  , where 0r and π π   . [3] (ii) Show that   2 2 P 1z w wz    , where 1w z z  . Determine the exact values of w such that  P 0z  . [3] (iii) Using the results from parts (i) and (ii), find the exact value of 2πcos 5      in surd form. [3] (2016 ACJC / JC1 / Promo / Q7)

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 82 of 103 4 Two complex numbers 1z and 2z , where    1 20 arg a πrg 2z z   , are roots to the equation   6 2 232 32 i 0z    . (i) Show that π 1 i 242ez  and 3i 8 2 π 2ez  . [3] (ii) Show all the roots to the equation on an Argand diagram. [2] (iii) Given that 1z and 2z satisfy the equation z w r  , state, in exact forms, the cartesian equation of the line that the point corresponding to w lies on and the minimum value of .r [2] (2016 CJC / JC1 / Promo / Q4) 5 It is given that the complex number 1 c i inos sz    , where π π   . (i) By considering appropriate trigonometric 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  1 cos isin n   , where n  . [3] (iii) By considering the binomial expansion of  isin1 cos n       , show that    1 cos cos 2 cos 2cos cos1 2 2 2 n n n n nn n                                        , where   ! ! ! n n r r n r       . [3] (2016 CJC / JC1 / Promo / Q7) 6 (i) Express 3 i 3 i     in the form ier  , where 0r and 0 2   . [2] (ii) Hence find the smallest positive value of n for which 3 i 3 i n        is real and positive. [3] (2016 IJC / JC1 / Promo / Q3) 7 Write down, in the form ier  , the five roots of the equation 5 1 0w   . Hence show that the roots of the equation     5 5 1 1 0z z    are i tan , where 0, 1, 2.5 k k       [6] (2016 IJC / JC1 / Promo / Q4) 8 The complex numbers 1z and 2z are given by 1 3i and 1 i respectively. (i) Find 1 2 z z in the form ix y , giving x and y in the exact form. [3] (ii) By considering the exponential forms of 1z and 2z , show that tan 2 312       . [4] (2016 IJC / JC1 / Promo / Q8)

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 83 of 103 9 (a) (i) Solve 5 iz  , giving your answers in the form ier  , where 0, π πr     . [3] (ii) Hence, solve the equation 5 i 2 2i z      , giving your answers in a similar form. [3] (b) If z is a non-zero complex number, we define  K z by the equation    ln i argK z z z  ,  π arg πz   . Show that      1 2 1 2K z z K z K z  . [2] (2016 JJC / JC1 / Promo / Q6) 10 (a) On the same Argand diagram, sketch the loci of points give n by each of the following equations: 1 : 2 i 5L z    ,  2 : arg 3 iL z    , where 1tan 2  . Find, in the form ix y , the complex number which rep resents the point in the Argand diagram which is on both 1L and 2L , giving the exact values of x and y. [5] (b) (i) Show that i 1 1 i cot1 e cos    , where 0 2   . [2] (ii) Given that 1z  and 0 2   , state the sum of the infinite series 2 3 2cos cos .z z z      [1] (iii) By putting iez  in (ii) and using the result in (i), find 2 3sin sin 2 cos sin 3 cos sin 4 cos          , simplifying your answer. [3] (2016 RI / JC1 / Promo / Q6) 11 (i) Show that if iez  , then 1 2i sink kz k z   , where k is a positive integer. [1] (ii) Show that 5sin  can be expressed in the form sin sin 3 sin 5 ,A B C    where the values of A, B and C are to be determined. [4] (iii) Find the particular solution of the differential equation   5d e cos ecd xy yx  , given that 0y when 0x . [3] (2016 TJC / JC1 / Promo / Q6)

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 84 of 103 12 (i) Sketch the locus of z that satisfies 2 i 2z   and  Im 1z  . [3] (ii) Find the maximum and minimum values of 2 iz  . [3] (iii) Find z in the form i where ,x y x y   , such that  arg 2 iz  is a maximum. [6] (2016 TJC / JC1 / Promo / Q10) 13 (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z which satisfy both the inequalities 2 1 4i 5z   and 2 2 12iz z    . [5] (ii) Determine exactly the greatest and least possible values of 3z for points in this region. [5] (2017 ACJC / JC2 / BT / Q1) 14 The complex number z satisfies the inequalities 2 2iz z   and  5 3 arg 2 2i4 2 z      . (i) Sketch the locus of z on an Argand diagram. [5] (ii) Find the exact range of  arg 3iz . [3] (2017 TJC / JC2 / BT / Q6) 15 (a) Solve  3 4 2 1 iz   , giving your answers in the form ier  , where 0r  and π π   . [3] The complex number 1z is a root of  3 4 2 1 iz   with  1 π arg 02 z   . Given that 1w z w  for some complex number w, find w, showing your working clearly. [2] (b) On the same Argand diagram, sketch the loci of the points given by each of the following equations: 1 : 1 i 3 2L z    ,   2 π: arg 1 i 2 3 4L z     . [3] The complex numbers 1z and 2z represent the two points in the Argand diagram which are on both 1L and 2L . Given that    2 1Re Rez z  , find  2arg z without using a calculator, leaving your answer in terms of π . [3] (2017 ACJC / JC2 / MYE / Q2)

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 85 of 103 16 Show that * * ( ) 0,zz c z z d    where ,c d are real numbers and 2c d , represents a circle in the Argand diagram, stating its centre and radius. [4] Shade in an Argand diagram the region for which * * 3( ) 5 0.zz z z    [2] (2017 CJC / JC2 / MYE / Q1) 17 The variable complex number z satisfies the following inequalities : i 3i 1 2z   and i 3i 1z z   . (i) On an Argand diagram, sketch the region R satisfied by the point P which represents z. [4] (ii) Find the range of 3 i .z  [2] (iii) Find the exact value of z where arg( 3 i)z  is least. [2] (2017 DHS / JC2 / MYE P1 / Q5) 18 (a) (i) Find the roots of the equation 8 1 0z   in the form ie , 0, .r r       [2] (ii) On the Arga