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Let $z= \frac{\left ( -1+\sqrt{3}i \right )\left ( \sqrt{3}-i \right )}{\left ( 1+i \right )}$ then in Euler's form z will be represented as

Option 1)
$z= 2\sqrt{2}e^{i\left ( \frac{3\pi }{4} \right )}$

Option 2)
$z= 2\sqrt{2}e^{i\left ( \frac{\pi }{4} \right )}$

Option 3)
$z= 2e^{i\left ( \frac{\pi }{3} \right )}$

Option 4)
$z= 2e^{i\left ( \frac{\pi }{4} \right )}$

Answers (1)

best_answer

$r= \left | z \right |= \left | \frac{\left ( -1+\sqrt{3i} \right ) \left ( \sqrt{3-i} \right )}{1+i} \right |$

$r= \frac{\left | -1+\sqrt3i \right | \left | \sqrt3-i \right |}{\left | 1+i \right |}= \frac{\sqrt{4}\sqrt{4}}{\sqrt2}= 2\sqrt2$

$r= 2\sqrt2\rightarrow \left ( 1 \right )$

$arg(z)= arg (-1+\sqrt3i) + arg (\sqrt3-i)- arg(1+i)+2n\pi$

$arg(z) =\pi -\tan ^{-1}\left | \frac{\sqrt3}{-1} \right | -\tan ^{-1}\left | \frac{-1}{\sqrt3} \right |- \tan ^{-1}\left | \frac{1}{1} \right | + 2n\pi$

$arg(z)= \pi - \frac{\pi }{3} - \frac{\pi }{6} -\frac{\pi }{4}+2n\pi$

$arg(z)= \frac{\pi }{4} + 2n\pi$

for n= 0 , we get arg(z) $=\pi /4$

$z= 2\sqrt2e^{i\frac{\pi }{4}}$

Properties of Argument of a Complex Number -

$Arg\left(\frac{z}{w}\right)=Arg(z)-Arg(w)+2n\pi$

- wherein

$n\epsilon Integer$ and it is chosen such that Arg(z/w) lies in principal value range of Argument.

Option 1)

$z= 2\sqrt{2}e^{i\left ( \frac{3\pi }{4} \right )}$

This is incorrect

Option 2)

$z= 2\sqrt{2}e^{i\left ( \frac{\pi }{4} \right )}$

This is correct

Option 3)

$z= 2e^{i\left ( \frac{\pi }{3} \right )}$

This is incorrect

Option 4)

$z= 2e^{i\left ( \frac{\pi }{4} \right )}$

This is incorrect

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