- (i) (a) Show that
$$\frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } = k ( 1 + \mathrm { i } )$$
where \(k\) is a constant to be determined.
(Solutions relying on calculator technology are not acceptable.)
Given that
- \(n\) is a positive integer
- \(\left( \frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } \right) ^ { n }\) is a real number
(b) use the answer to part (a) to write down the smallest possible value of \(n\).
(ii) The complex number \(z = a + b \mathrm { i }\) where \(a\) and \(b\) are real constants.
Given that
- \(\left| z ^ { 10 } \right| = 59049\)
- \(\arg \left( z ^ { 10 } \right) = - \frac { 5 \pi } { 3 }\)
determine the value of \(a\) and the value of \(b\).