F08 - Further Complex Numbers - Assignment 1 (Solutions)

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Assignment 1 Solutions) Page 1 of 5 National Junior College 2016 – 2017 H2 Further Mathematics Topic F7: Further Complex Numbers (Assignment 1 Solutions) 1.   4 2i 3 2i 2 3 8 8 3 i 16e 16e , k z k                    i 6 22e , 2, 1,0,1. k z k         2 5i i6 2 6When 2, 2e 2e .k z                i i6 2 3When 1, 2e 2e .k z                0i i6 2 6When 0, 2e 2e .k z               2i i6 2 3When 2, 2e 2e .k z                   8 4 8 4 24 4 16 256 0 16 64 192 8 192 8 192 8 3 i z z z z z z                         *4 4 5 2i i i i4 6 3 6 3 5 2* i i i i4 6 3 6 3 8 8 3 i or 8 8 3 i 8 8 3 i 8 8 3 i 2e , 2e , 2e , 2e or 8 8 3 i 2e , 2e , 2e , 2e z z z z                                                                      www.KiasuExamPaper.com 797

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Assignment 1 Solutions) Page 2 of 5 2. (i)        12 12 12 12 12 i i43 1 11 3i 2e 2 e 2 cos 4 i sin 4 2 4096                   . (ii)         /6 3 2 2 3 3 3 3 12 126 6 i0 i 26 i i i i i 1 11 3i 1 1 3i e4096 4096 1 e , 0, 1, 2,34096 1 e , 0, 1, 2,34 1 1 1 1 1 1e , e , e , e , ,4 4 4 4 4 4 n n z z z n z n z                              (iii)    2 2 2 2 ( ) (2cos ) z re z re z r e e z r z r z r                i i i i (iv)       2 2 3 3 3 3 2 2 2 2 2 2 126 i i i i 1 3i 1 1 1 1 1 1 14096 e e e e4 4 4 4 4 4 1 1 1 2 1 14096 cos cos2 3 16 2 3 16 16 16 4 1 16 4 1 16 1 z z z z z z z z z z z z z z z z z                                                           1 4 1 4 2 3-i1 e4  3i1 e4  3-i1 e4  2 3i1 e4  π/3 www.KiasuExamPaper.com 798

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Assignment 1 Solutions) Page 3 of 5 3. (i)       2 1 1 1 1 2 2 1( ) 1 11 1 1 1 1 (1 )1 (1 ) kn n k k k n k k n k k n n z zz z z z z zz z n zz z z znz z n z z zz z                                    (ii)   i i i i i i2 2 2 e 1 1 e e e e e cos i sin2 2 cos i sin cos isin2 2 2 2 cos isin i2 2 i2isin 2 i cos isin2 22sin 2 z z                                          www.KiasuExamPaper.com 799

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Complex Numbers (Assignment 1 Solutions) Page 4 of 5 (iii)    1 2 1 2 2 2 (