The real and imaginary part of the complex number 1 - -i where i - -1 are

The correct options are A 1 1√(2) and 1√(2) respectively C 1 + 1√(2) and 1√(2) respectively √( i ) = cis( π #160; 2 #160; ) ...

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Standard XII

Mathematics

Complex Numbers

The real and ...

Question

The real and imaginary part of the complex number $1 + \sqrt{i}$ where $i = \sqrt{- 1}$ are

1 - \frac{1}{\sqrt{2}}

and

- \frac{1}{\sqrt{2}}

respectively

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1 - \frac{1}{\sqrt{2}}

and

\frac{1}{\sqrt{2}}

respectively

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1 + \frac{1}{\sqrt{2}}

and

\frac{1}{\sqrt{2}}

respectively

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1 + \frac{1}{\sqrt{2}}

and

- \frac{1}{\sqrt{2}}

respectively

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Solution

The correct options are
A $1 - \frac{1}{\sqrt{2}}$ and $- \frac{1}{\sqrt{2}}$ respectively
C $1 + \frac{1}{\sqrt{2}}$ and $\frac{1}{\sqrt{2}}$ respectively
$\sqrt{i} = {c i s (\frac{π}{2})}^{0.5}$
From De moivres theorem,
$\sqrt{i} = {c i s (\frac{π}{2})}^{0.5} = \pm c i s (0.5 \frac{π}{2}) = \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}$ ,

$∴$ the real and imaginary parts are $1 + \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$ and $1 - \frac{1}{\sqrt{2}}, - \frac{1}{\sqrt{2}}$

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The real and imaginary part of the complex number 1+√i where i=√−1 are

The correct options are A 1−1√2 and −1√2 respectively C 1+1√2 and 1√2 respectively√i=cis(π2)0.5From De moivres theorem,√i=cis(π2)0.5=±cis(0.5π2)=1√2+i1√2,∴ the real and imaginary parts are 1+1√2,1√2 and 1−1√2,−1√2

The real and imaginary part of the complex number $1 + \sqrt{i}$ where $i = \sqrt{- 1}$ are