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)