Let $w$ $(Im\, w \neq 0)$ be a complex number. Then the set of all complex number $z$ satisfying the equation $w - \overline {w}z = k\left( {1 - z} \right)$ , for some real number $k$, is
Let $w$ $(Im\, w \neq 0)$ be a complex number. Then the set of all complex number $z$ satisfying the equation $w - \overline {w}z = k\left( {1 - z} \right)$ , for some real number $k$, is
- [JEE MAIN 2014]
- A
$\left\{ {z:\left| z \right| = 1} \right\}$
- B
$\left\{ {z:z = \overline z } \right\}$
- C
$\left\{ {z:z \ne 1} \right\}$
- D
$\left\{ {z:\left| z \right| = 1,z \ne 1} \right\}$
Similar Questions
The number of solutions of the equation ${z^2} + \bar z = 0$ is
The number of solutions of the equation ${z^2} + \bar z = 0$ is
Let $z_k=\cos \left(\frac{2 k \pi}{10}\right)+ i \sin \left(\frac{2 k \pi}{10}\right) ; k =1,2, \ldots 9$.
List $I$
List $II$
$P.$ For each $z_k$ there exists a $z_j$ such that $z_k \cdot z_j=1$
$1.$ True
$Q.$ There exists a $k \in\{1,2, \ldots ., 9\}$ such that $z_{1 .} . z=z_k$ has no solution $z$ in the set of complex numbers.
$2.$ False
$R.$ $\frac{\left|1-z_1\right|\left|1-z_2\right| \ldots . .\left|1-z_9\right|}{10}$ equals
$3.$ $1$
$S.$ $1-\sum_{k=1}^9 \cos \left(\frac{2 k \pi}{10}\right)$ equals
$4.$ $2$
Codes: $ \quad P \quad Q \quad R \quad S$
Let $z_k=\cos \left(\frac{2 k \pi}{10}\right)+ i \sin \left(\frac{2 k \pi}{10}\right) ; k =1,2, \ldots 9$.
| List $I$ | List $II$ |
| $P.$ For each $z_k$ there exists a $z_j$ such that $z_k \cdot z_j=1$ | $1.$ True |
| $Q.$ There exists a $k \in\{1,2, \ldots ., 9\}$ such that $z_{1 .} . z=z_k$ has no solution $z$ in the set of complex numbers. | $2.$ False |
| $R.$ $\frac{\left|1-z_1\right|\left|1-z_2\right| \ldots . .\left|1-z_9\right|}{10}$ equals | $3.$ $1$ |
| $S.$ $1-\sum_{k=1}^9 \cos \left(\frac{2 k \pi}{10}\right)$ equals | $4.$ $2$ |
Codes: $ \quad P \quad Q \quad R \quad S$
- [IIT 2014]
Find the modulus and the argument of the complex number $z=-\sqrt{3}+i$
Find the modulus and the argument of the complex number $z=-\sqrt{3}+i$
Let ${z_1}$ and ${z_2}$ be two complex numbers with $\alpha $ and $\beta $ as their principal arguments such that $\alpha + \beta > \pi ,$ then principal $arg\,({z_1}\,{z_2})$ is given by
Let ${z_1}$ and ${z_2}$ be two complex numbers with $\alpha $ and $\beta $ as their principal arguments such that $\alpha + \beta > \pi ,$ then principal $arg\,({z_1}\,{z_2})$ is given by
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
$(\bar{z})^2+\frac{1}{z^2}$
are integers, then which of the following is/are possible value($s$) of $|z|$ ?
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
$(\bar{z})^2+\frac{1}{z^2}$
are integers, then which of the following is/are possible value($s$) of $|z|$ ?
- [IIT 2022]