If z -log e i i- then the value of complex number z isA- purely imaginaryB- Purely realC- Has both real - imaginary partsD- Can-t say

The correct option is A purely imaginary i^i = [e^i π/2]^i = e^ π/2 log_e i^i = log_e e^ π/2 = π/2 Z = | z | [cos(arg(z)) + i sin(arg(z))] = | z | [0 + i( 1 ...

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Standard XI

Mathematics

Basic Trigonometric Identities

If z =log e i...

Question

If arg(z) = ${log}_{e} i^{i}$ , then the value of complex number z is

Purely real

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purely imaginary

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Has both real & imaginary parts

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Can't say

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Solution

The correct option is B
purely imaginary

$i^{i} = {[e^{i \frac{π}{2}}]}^{i} = e^{- \frac{π}{2}}$

$l o g_{e} i^{i} = l o g_{e} e^{- \frac{π}{2}} = \frac{- π}{2}$

Z = $| z |$ [cos(arg(z)) + i sin(arg(z))]

= $| z |$ [0 + i(-1)]

= $| z |$ .(-i)

$\Rightarrow$ z is purely imaginary number.

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If arg(z) = logeii, then the value of complex number z is

The correct option is B purely imaginary ii=[eiπ2]i=e−π2 logeii=logee−π2=−π2 Z = |z| [cos(arg(z)) + i sin(arg(z))] = |z| [0 + i(-1)] = |z|.(-i) ⇒ z is purely imaginary number.

If arg(z) = ${log}_{e} i^{i}$ , then the value of complex number z is

The correct option is B
purely imaginary

$i^{i} = {[e^{i \frac{π}{2}}]}^{i} = e^{- \frac{π}{2}}$

$l o g_{e} i^{i} = l o g_{e} e^{- \frac{π}{2}} = \frac{- π}{2}$

Z = $| z |$ [cos(arg(z)) + i sin(arg(z))]

= $| z |$ [0 + i(-1)]

= $| z |$ .(-i)

$\Rightarrow$ z is purely imaginary number.