13_2016_-_2017_H2_Maths_Complex_Numbers_Tutorial_Solutions_updated

National Junior College Mathematics Department 2017 Complex Numbers (Tutorial Solutions) Page 1 of 11 National Junior College 2016 – 2017 H2 Mathematics Complex Numbers Tutorial Solutions Basic Mastery Questions 1 (a)   3 8i 5 7i 15 21i-40i 56     71 19i (b) 7 5i 7 5i 4 3i 4 3i 4 3i 4 3i                  28 21i 20i 15 16 9 13 41 i25 25      2 (a)  2i 2 , arg 2i 2     i 22i 2e 2 cos isin 22                      (b) 1 3i 1 3 2       2i 3 2arg 1 3i 33 1 3i 2e 222 cos isin33             (c)   1 i 1 3i     1 i 1 3i    22   arg 1 i 1 3i       arg 1 i arg 1 3i 32 43 12           =    i12 1 i 1 3i 2 2e 2 2 cos isin 12 12                         (d) 1 3i1 3i 2 21 i 1 i 2       

National Junior College Mathematics Department 2017 Complex Numbers (Tutorial Solutions) Page 2 of 11 1 3iarg 1i      arg 1 3i arg 1 i 23 34 17 12           17Principalarg( ) 2 12 7 12 z     7i 121 3i 2e1i 772 cos i sin12 12                    (e) i 3 1 3 3 31 e 1 cos isin 1 i i3 3 2 2 2 2           3 3 9 3 12i32 2 4 4 4     13 3 3arg i tan2 2 3 6     ii661 e 3e 3 cos isin66      Alternative Method for 2(e):   ii i i 63 6 6 i 6 i i i6 6 6 1 e e e e e cos isin cos isin6 6 6 6 3e 2cos 2 e 3 e62                                          (f) 33e 3   i , 3arg 3e 3     i , 33e 3 cos isin 33   i (g) 100 100 ,  arg 100 π ,   i100 100 cos isin 100e     (h)   115 5 5 11 i 2 ii i i i i 666 6 6 63e 3 1 e 3e e 3e 3e 3e 3 cos isin66                                      

National Junior College Mathematics Department 2017 Complex Numbers (Tutorial Solutions) Page 3 of 11 Alternatively,       5π 5πii 66 5π 5πii 66 3e 3 e 3 5π 11π πarg 3e arg 3 arg e π 6 6 6             5πi 6 ππ3e 3 cos isin 66                  3 (a) 5 5 3 12 cos isin 2 i 3 i6 6 2 2                       (b) i 3 132e 2 cos + isin 2 cos isin 2 i 1 i 33 3 3 3 2 2                                   4 1 1 1   Practice Questions 1 3 i 4 4izw    2 4 2 5arg( ) arg( )64 zw zw    (i) 1 1 1 2zz   15arg arg( 1) arg( ) 66zz        11 cos isin2 6 6z    (ii) * * 1 1 1 1 2zz z    * * 15arg arg(1) arg( ) 0 arg( ) 6zzz       * 1 1 5 5 cos isin2 6