Complex number z=i-1 ( / 3)+i ( / 3) polar form is - Vidyadip

MCQ

Complex number $z=\frac{i-1}{\cos (\pi / 3)+i \sin (\pi / 3)}$ polar form is

✓
$\sqrt{2}\left(\cos \frac{5 \pi}{12}+ i \sin \frac{5 \pi}{12}\right)$
B
$\sqrt{2}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$
C
$\sqrt{2}\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)$
D
None of these

✓

Answer

Correct option: A.

$\sqrt{2}\left(\cos \frac{5 \pi}{12}+ i \sin \frac{5 \pi}{12}\right)$

(A)
$Z=\frac{i-1}{\cos (\pi / 3)+i \sin (\pi / 3)}$
$=\frac{i-1}{\frac{1}{2}+\frac{i \sqrt{3}}{2}}=\frac{2(i-1)}{1+i \sqrt{3}}$
$=\frac{2( i -1)}{1+ i \sqrt{3}} \times \frac{1- i \sqrt{3}}{1- i \sqrt{3}}=\frac{2 i +2 \sqrt{3}-2+2 i \sqrt{3}}{1+3}$
$=\frac{2(-1+i+\sqrt{3}+i \sqrt{3})}{4}$
$=\frac{1}{2}[(\sqrt{3}-1)+ i (\sqrt{3}+1)]$
$\therefore|z|=\sqrt{\frac{1}{4}(3+1-2 \sqrt{3}+3+1+2 \sqrt{3})}=\sqrt{\frac{8}{4}}=\sqrt{2}$
$\theta=\tan ^{-1}\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)=\tan ^{-1}\left(\frac{1+\frac{1}{\sqrt{3}}}{1-\frac{1}{\sqrt{3}}}\right)$
$=\tan ^{-1} 1+\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi}{4}+\frac{\pi}{6}=\frac{5 \pi}{12}$
∴ the polar form of $z=\sqrt{2}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Complex Numbers→Complex Numbers — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

The circle $x^2 + y^2 + 2gx + 2fy + c = 0$ does not intersect $x-$axis, if:

If $\tan\theta_1\tan\theta_2=\text{k},$ then $\frac{\cos(\theta_1-\theta_2)}{\cos(\theta_1+\theta_2)}=$

The real values of $x$ and $y$ for which the equation $(x+i y)(2-3 i)=4+i$ is satisfied, are

If $\cos\text{x}=\frac{1}{2}\Big(\text{a}+\frac{1}{\text{a}}\Big),$ and $3\text{x}=\lambda\Big(\text{a}^3+\frac{1}{\text{a}^3}\Big),$ then $\lambda=$

Four persons are selected at random out of $3$ men, $2$ women and $4$ children. The probability that there are exactly $2$ children in the selection is:

Equations of the two straight lines passing through the point (3, 2) and making an angle of \[45^{\circ}\] with the line x - 2y = 3 are

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is :

If $\omega$ is a non real cube root of unity, then $(a+b)(a+b \omega)\left(a+b \omega^2\right)$ is

If $f(x)=\frac{x^2-10 x+25}{x^2-7 x+10}$ for $x \neq 5$ and $f$ is continuous at $x=5$, then $f (5)=$

Letters in the word HULULULU are rearranged. The probability of all three $L$ being together is