# The real value of α for which the expression 1-isinα1+2isinα is purely real is ______. - Mathematics

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The real value of α for which the expression (1 - i sin alpha)/(1 + 2i sin alpha) is purely real is ______.

#### Options

• (n + 1)  pi/2

• (2n + 1)  pi/2

• None of these, where n ∈N

#### Solution

The real value of α for which the expression (1 - i sin alpha)/(1 + 2i sin alpha) is purely real is .

Explanation:

Let z = (1 - i sin alpha)/(1 + 2i sin alpha)

= ((1 - i sin alpha)(1 - 2i sin alpha))/((1 + 2i sin alpha)(1 - 2i sin alpha))

= (1 - 2i sin alpha - i sin alpha + 2i^2 sin^2 alpha)/((1)^2 - (2i sin alpha)^2

= (1 - 3i sin alpha - 2 sin^2 alpha)/(1 - 4i^2 sin^2 alpha)

= ((1 - 2 sin^2 alpha) - 3i sin alpha)/(1 + 4 sin^2 alpha)

= (1 - 2 sin^2 alpha)/(1 + 4 sin^2 alpha) - (3sin alpha)/(1 + 4 sin^2 alpha) .i

Since, z is purely real.

Then (-3 sin alpha)/(1 + 4 sin^2 alpha) = 0

⇒ sinα = 0

So, α = nπ, n ∈ N.

Concept: Algebraic Operations of Complex Numbers
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 5 Complex Numbers and Quadratic Equations
Exercise | Q 36 | Page 95

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