Complex Numbers

If $\,\left| {z + 4} \right|\,\, \le \,\,3\,$, then the maximum value of $\left| {z + 1} \right|$ is :

Let $\alpha \,,\beta$ be real and z be a complex number. If ${z^2} + \alpha z + \beta = 0$ has two distinct roots on the line Re z = 1, then it is necessary that :

If $\left| {z - 4} \right| < \left| {z - 2} \right|$, its solution is given by :

If ${z^2} + z + 1 = 0$, where z is complex number, then value of ${\left( {z + {1 \over z}} \right)^2} + {\left( {{z^2} + {1 \over {{z^2}}}} \right)^2} + {\left( {{z^3} + {1 \over {{z^3}}}} \right)^2} + .......... + {\left( {{z^6} + {1 \over {{z^6}}}} \right)^2}$ is :

If $\,\left| {z - {4 \over z}} \right| = 2,$ then the maximum value of $\,\left| z \right|$ is equal to :

The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $2\sqrt 2$ units in the south-westwardsdirection. Then its new position in the Argand plane is at the point represented by :

A value of $\theta \,$ for which ${{2 + 3i\sin \theta \,} \over {1 - 2i\,\,\sin \,\theta \,}}$ is purely imaginary, is :

Let $\omega$ be a complex number such that 2$\omega$ + 1 = z where z = $\sqrt {-3}$. If \left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k, then k is equal to :

For two non-zero complex numbers $z_{1}$ and $z_{2}$, if $\operatorname{Re}\left(z_{1} z_{2}\right)=0$ and $\operatorname{Re}\left(z_{1}+z_{2}\right)=0$, then which of the following are possible? A. $\operatorname{Im}\left(z_{1}\right)>0$ and $\operatorname{Im}\left(z_{2}\right) > 0$ B. $\operatorname{Im}\left(z_{1}\right) < 0$ and $\operatorname{Im}\left(z_{2}\right) > 0$ C. $\operatorname{Im}\left(z_{1}\right) > 0$ and $\operatorname{Im}\left(z_{2}\right) < 0$ D. $\operatorname{Im}\left(z_{1}\right) < 0$ and $\operatorname{Im}\left(z_{2}\right) < 0$ Choose the correct answer from the options given below :

The set of all $\alpha$ $\in$ R, for which w = ${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$ is purely imaginary number, for all z $\in$ C satisfying |z| = 1 and Re z $\ne$ 1, is :

Let A = $\left\{ {\theta \in \left( { - {\pi \over 2},\pi } \right):{{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}is\,purely\,imaginary} \right\}$ . Then the sum of the elements in A is :

If ${{z - \alpha } \over {z + \alpha }}\left( {\alpha \in R} \right)$ is a purely imaginary number and | z | = 2, then a value of $\alpha$ is :

Let z $\in$ C be such that |z| < 1. If $\omega = {{5 + 3z} \over {5(1 - z)}}$z, then :

Let z $\in$ C with Im(z) = 10 and it satisfies ${{2z - n} \over {2z + n}}$ = 2i - 1 for some natural number n. Then :

The region represented by {z = x + iy $\in$ C : |z| – Re(z) $\le$ 1} is also given by the inequality : {z = x + iy $\in$ C : |z| – Re(z) $\le$ 1}

Let z = x + iy be a non-zero complex number such that ${z^2} = i{\left| z \right|^2}$, where i = $\sqrt { - 1}$ , then z lies on the :

If $S = \left\{ {z \in C:{{z - i} \over {z + 2i}} \in R} \right\}$, then :

Let $\alpha$ and $\beta$ be the roots of the equation x² + (2i $-$ 1) = 0. Then, the value of |$\alpha$⁸ + $\beta$⁸| is equal to :

Let S be the set of all $(\alpha, \beta), \pi<\alpha, \beta<2 \pi$, for which the complex number $\frac{1-i \sin \alpha}{1+2 i \sin \alpha}$ is purely imaginary and $\frac{1+i \cos \beta}{1-2 i \cos \beta}$ is purely real. Let $Z_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in S$. Then $\sum\limits_{(\alpha, \beta) \in S}\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right)$ is equal to :