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(RI16/Promo/Q8) (a) The complex number

`z`

is given by

`z=1+cos\alpha _(+)isin\alpha _()`

, where

`0<\alpha <(\pi )/(2)`

. (i) Show that

`z`

can be expressed as

`(2cos((\alpha )/(2)))(cos((\alpha )/(2))+isin((\alpha )/(2)))`

. (ii) Find

`zz^(**)`

, where

`z^(**)`

denotes the conjugate of

`z`

. (iii) Given that

`\alpha =(\pi )/(3)`

, without using a calculator, find the values of

`|z^(6)|`

and

`argz^(6)`

. Deduce the value of

`z^(6)`

.

`3`

(b) The complex numbers

`u`

and

`v`

are such that

`(i(u+v))/((u-v))`

is real. Show that

`|u|=|v|`

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