Complex Numbers

Let S₁, S₂ and S₃ be three sets defined as S₁ = {z$$\in$$C : |z $$-$$ 1| $$\le$$ $$\sqrt 2$$} S₂ = {z$$\in$$C : Re((1 $$-$$ i)z) $$\ge$$ 1} S₃ = {z$$\in$$C : Im(z) $$\le$$ 1} Then the set S₁ $$\cap$$ S₂ $$\cap$$ S₃ :

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 :

If z is a complex number of unit modulus and argument $$\theta$$, then arg $$\left( {{{1 + z} \over {1 + \overline z }}} \right)$$ equals :

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 :

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 :