If $z$ is a complex number, then $z.\,\overline z = 0$ if and only if
If $z$ is a complex number, then $z.\,\overline z = 0$ if and only if
- A
$z = 0$
- B
${\mathop{\rm Re}\nolimits} (z) = 0$
- C
${\mathop{\rm Im}\nolimits} \,(z) = 0$
- D
None of these
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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
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Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
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Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
$(\bar{z})^2+\frac{1}{z^2}$
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- [IIT 2022]
The amplitude of $\frac{{1 + \sqrt 3 i}}{{\sqrt 3 + 1}}$ is
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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]
If $\frac{\pi }{2} < \alpha < \frac{3}{2}\pi $ , then the modulus and argument of $(1 + cos\, 2\alpha ) + i\, sin\, 2\alpha $ is respectively
If $\frac{\pi }{2} < \alpha < \frac{3}{2}\pi $ , then the modulus and argument of $(1 + cos\, 2\alpha ) + i\, sin\, 2\alpha $ is respectively