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The real value of alpha for which the ex...

The real value of `alpha` for which the expression `(1-isinalpha)/(1+2isinalpha)` is purely real is

A

`(n + 1)(pi)/(2)`

B

`(2n + 1)(pi)/(2)`

C

`npi`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
Given expression , `z = (1-isin alpha)/(1 +2isinalpha)`
=`((1 - 2isin alpha)(1 - 2isin alpha))/((1 + 2isin alpha)( 1 - 2isin alpha))`
`= (1-isinalpha - 2isinalpha + 2i^(2) sin^(2)alpha)/(1+4i^(2)sin^(2)alpha)`
` = (1-3 isinalpha - 2 sin^(2)alpha)/(1 + 4sin^(2)alpha`
`(1-2sin^(2)alpha)/(1 + 4 sin ^(2)alpha)-(3isinalpha)/(1 +4 sin^(2)alpha`
It is given that z is a purely real.
`:. (-3sinalpha)/(1+ 4 sin^(2)alpha) = 0`
`rarr -3sinalpha = 0 rArr sinalpha = 0 `
`alpha = npi`
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