1.

The real value of `alpha` for which the expression `(1-isinalpha)/(1+2isinalpha)` is purely real isA. `(n + 1)(pi)/(2)`B. `(2n + 1)(pi)/(2)`C. `npi`D. None of these

Answer» Correct Answer - 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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