If $z = \cos \frac{\pi }{6} + i\sin \frac{\pi }{6}$ then
If $z = \cos \frac{\pi }{6} + i\sin \frac{\pi }{6}$ then
- A
$|z|\, = 1,\,\,\,\,arg\,z = \frac{\pi }{4}$
- B
$|z|\, = 1,arg\,z = \frac{\pi }{6}$
- C
$|z|\, = \frac{{\sqrt 3 }}{2},\,arg\,z = \frac{{5\pi }}{{24}}$
- D
$|z|\, = \frac{{\sqrt 3 }}{2},\,\,arg\,z = {\tan ^{ - 1}}\frac{1}{{\sqrt 2 }}$
Similar Questions
Let $a = lm\left( {\frac{{1 + {z^2}}}{{2iz}}} \right)$, where $z$ is any non-zero complex number. The set $A = \{ a:\left| z \right| = 1\,and\,z \ne \pm 1\} $ is equal to
Let $a = lm\left( {\frac{{1 + {z^2}}}{{2iz}}} \right)$, where $z$ is any non-zero complex number. The set $A = \{ a:\left| z \right| = 1\,and\,z \ne \pm 1\} $ is equal to
- [JEE MAIN 2013]
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- [IIT 1988]
Let $z$ be complex number satisfying $|z|^3+2 z^2+4 z-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-$I$ to the correct entries in List-$II$.
List-$I$
List-$II$
($P$) $|z|^2$ is equal to
($1$) $12$
($Q$) $|z-\bar{z}|^2$ is equal to
($2$) $4$
($R$) $|z|^2+|z+\bar{z}|^2$ is equal to
($3$) $8$
($S$) $|z+1|^2$ is equal to
($4$) $10$
($5$) $7$
The correct option is:
Let $z$ be complex number satisfying $|z|^3+2 z^2+4 z-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-$I$ to the correct entries in List-$II$.
| List-$I$ | List-$II$ |
| ($P$) $|z|^2$ is equal to | ($1$) $12$ |
| ($Q$) $|z-\bar{z}|^2$ is equal to | ($2$) $4$ |
| ($R$) $|z|^2+|z+\bar{z}|^2$ is equal to | ($3$) $8$ |
| ($S$) $|z+1|^2$ is equal to | ($4$) $10$ |
| ($5$) $7$ |
The correct option is:
- [IIT 2023]
The inequality $|z - 4|\, < \,|\,z - 2|$represents the region given by
The inequality $|z - 4|\, < \,|\,z - 2|$represents the region given by
- [IIT 1982]
If $z_1, z_2, z_3$ $\in$ $C$ such that $|z_1| = |z_2| = |z_3| = 2$, then greatest value of expression $|z_1 - z_2|.|z_2 - z_3| + |z_3 - z_1|.|z_1 - z_2| + |z_2 - z_3||z_3 - z_1|$ is
If $z_1, z_2, z_3$ $\in$ $C$ such that $|z_1| = |z_2| = |z_3| = 2$, then greatest value of expression $|z_1 - z_2|.|z_2 - z_3| + |z_3 - z_1|.|z_1 - z_2| + |z_2 - z_3||z_3 - z_1|$ is