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Polar form of complex numbers - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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14 Questions around this concept.

Solve by difficulty

Correct result for $\left ( 4+4\sqrt{3}\:\: i \right )^{\frac{4}{5}}$ is :

A

$\left ( 8 \right )^{\frac{4}{5}}\left ( \cos \frac{\pi }{15} +i\sin\frac{\pi }{15} \right )$

B

$\left ( 4 \right )^{\frac{4}{5}}\left ( \cos \frac{7\pi }{15} +i\sin\frac{7\pi }{15} \right )$

C

$\left ( 8 \right )^{\frac{4}{5}}\left ( \cos \frac{4\pi }{15} +i\sin\frac{4\pi }{15} \right )$

D

None of these

What is the polar term of $z=-4+4 i$ ?

A

$4\left(\operatorname{Cos} \frac{3 \pi}{4}+i \operatorname{Sin} \frac{3 \pi}{4}\right)$

B

$4 \sqrt{2}\left(\operatorname{Cos} \frac{3 \pi}{4}+i \operatorname{Sin} \frac{3 \pi}{4}\right)$

C

$4 \sqrt{2}\left(\operatorname{Cos} \frac{\pi}{4}-i \operatorname{Sin} \frac{\pi}{4}\right)$

D

$4\left(\operatorname{Cos} \frac{3 \pi}{4}-i \operatorname{Sin} \frac{3 \pi}{4}\right)$

Concepts Covered - 1

Polar form of complex numbers

In polar form, we represent the complex number through the argument and modulus value of complex numbers.

Let $z=x+i y$ be a complex number,

And we know that

$|z|=\sqrt{x^2+y^2}=r$

And let arg(z) = θ

From the figure, $x=|z| \cos (\theta)=r \cos (\theta)$
and $y=|z| \sin (\theta)=r \sin (\theta)$
So, $z=x+i y=r \cos (\theta)+i . r \sin (\theta)=r(\cos (\theta)+i . \sin (\theta))$
This form is called polar form with $r=$ principal value of $\arg (z)$ and $r=|z| . \mid$

For general values of the argument

$\mathrm{z}=\mathrm{r}[\cos (2 \mathrm{n} \pi+\theta)+i \sin (2 \mathrm{n} \pi+\theta)]$, where $n \in$ Integer

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Polar form of complex numbers

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