Ch 9 Complex Numbers Tut 9A APQ Solutions (RVHS)

2023 JC2 H2 Mathematics 9758 RVHS Mathematics Department 1 Chapter 9 Complex Numbers I Tutorial (A): Algebra of Complex Numbers Solutions Additional Practice Questions 1. Since the coefficients are all real, another root of the equation is x = –2 – i. [ x – (–2 + i) ] [ x – (–2 – i) ] = (x + 2 – i)(x + 2 + i) = x2 + 2x – ix + 2x + ix + 22 – i2 = x2 + 4x + 5 By comparing coefficients: x4 + 4x3 + x2 + ax + b = (x2 + 4x + 5)(x2 – 4) = x4 + 4x3 + x2 – 16x – 20 So a = –16, b = –20. The other three roots of the equation are –2 – i, 2, –2. Alternative solution (–2 + i)4 + 4(–2 + i)3 + (–2 + i)2 + a(–2 + i) + b = 0 – 12 + 16i – 2a + ai + b = 0 (b – 2a) + ai = 12 – 16i Comparing imaginary parts, a = –16 Comparing real parts, b – 2a = 12 b = –20 The other three roots of the equation are –2 – i, 2, –2 from GC. 2. 4 3 2 4 3 2 4 2 3 2 2 2 2 ( i) ( i) ( i) ( i) 0 i i 0 0, i i = 0 , 0 0. a k b k c k d k e ak bk ck dk e dak ck e bk dk k b d dThus a c e ad cdb ebb b                                  2 2 2 1, 3, 13, 27, 36 27 9(36) 1053 1053 a b c d e ad b e bcd           2 27 9 33 Thus. two roots are 3i. dk k b     

2023 JC2 H2 Mathematics 9758 RVHS Mathematics Department 2 3 z = 1 + ip , w = 1 + iq zw = (1 + ip)(1 + iq) 3 – 4i = 1 – pq + i(p + q)  3 = 1 − pq … … (1) &  4 = p + q  q = – 4 – p Substitute into (1)  3 = 1 − p(– 4 – p) p2 + 4p – 2 = 0 p = 62)1(2 )2)(1(4164  Since p > 0  p = 62 // q = 4 – ( 62 ) = 62 // 4 iw z  --- (1) 2i 2w z   2 2iw z  --- (2) Substitute (2) in (1):  2 2i iz z   22i 2 i 0z z   2 4 4(2i)( i) 2 4 2 2i 1 i 4i 4i 4i 2iz               Thus,  1 i i 1 1 i2i i 2z      or  1 i i 1 1 i2i i 2z        2i 1 i 1 i1+i 1 iw     2i 1 i 1 i1+i 1 iw        Alternative solution iw z  --- (1) 2i 2w z  2 2i wz  --- (2) Substitute (2) in (1): 2 i2i ww      2 2 2 0w w   2 4 4(2) 2 4 2 2i 1 i2 2 2w         

2023 JC2 H2 Mathematics 9758 RVHS Mathematics Department 3 i.e., 1 iw   or 1 iw   1 i 2 2iz   1 i 2 2iz    1 i i 1 1 i2i i 2       1 i i 1 1 i2i i 2      

2023 JC2 H2 Mathematics 9758 RVHS Mathematics Department 4