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Standard XII

Mathematics

Evaluation of a Determinant

For any non-z...

Question

For any non-zero complex number z, arg (z) + arg $(\bar{z})$ = ____________.

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Solution

For complex number z
Say z = x + iy = re^iθ where r = modulus of z, θ = argument of $\bar{z} = x - i y$
$\Rightarrow \bar{z} = r e^{- i θ}$
Let us arg z = θ
Since arg $\bar{z}$ = – arg z
⇒ arg z + arg $\bar{z}$ = θ + (–θ)
i.e arg z + arg $(\bar{z})$ = 0

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For any non-zero complex number z, arg (z) + arg z¯ = ____________.

For complex number z Say z = x + iy = reiθ where r = modulus of z, θ = argument of z¯=x-iy ⇒z¯=re-iθ Let us arg z = θ Since arg z¯ = – arg z ⇒ arg z + arg z¯ = θ + (–θ) i.e arg z + arg z¯ = 0

For any non-zero complex number z, arg (z) + arg $(\bar{z})$ = ____________.

For complex number z
Say z = x + iy = re^iθ where r = modulus of z, θ = argument of $\bar{z} = x - i y$
$\Rightarrow \bar{z} = r e^{- i θ}$
Let us arg z = θ
Since arg $\bar{z}$ = – arg z
⇒ arg z + arg $\bar{z}$ = θ + (–θ)
i.e arg z + arg $(\bar{z})$ = 0