Complex numbers practice · Mixed difficulty

$\cos \left(i \log \frac{a-i b}{a+i b}\right)$ is equal to

$(\cos 2 \theta+i \sin 2 \theta)^{-5}(\cos 3 \theta-i \sin 3 \theta)^6$ $(\sin \theta-i \cos \theta)^3$ is equal to

If for the complex numbers $z_1$ and $z_2$, $\left|1-\bar{z}_1 z_2\right|^2-\left|z_1-z_2\right|^2=k\left(1-\left|z_1\right|^2\right)\left(1-\left|z_2\right|^2\right)$, then $k$ is equal to

The solution set of ${x^2} + 2 \le 3x \le 2{x^2} - 5,$ is

If ${\left( {\sqrt 5 + \sqrt 3 i} \right)^{33}} = {2^{49}}\;z$, then modulus of the complex number $z$ is equal to

The complex number $\mathrm{z}=1+\mathrm{i}$ is rotated through an angle $\frac{3 \pi}{2}$ in anticlockwise direction about the origin and stretched by additional $\sqrt{2}$ unit, then the new complex number is:

If $z=5+t+i \sqrt{25-t^2}$, $(-5 \leq t \leq 5)$, then locus of $z$ is a curve which passes through

The point (4, 1) undergoes the following three transformations successively (i) Refletion about the line $y = x$
(ii) Translation through a distance of 2 unit along the positive direction of $x$-axis
(iii) Rotation through an angle of $\frac{\pi }{4}$ about the origin in the anti-clockwise direction

The final position of the point is

The value of $\frac{{\left[ {(\cos 20^\circ + i\sin 20^\circ )\left( {\cos 75^\circ + i\sin 75^\circ } \right)\left( {\cos 10^\circ + i\sin 10^\circ } \right)} \right]}}{{\sin 15^\circ - i\cos 15^\circ }}$ is

If $\cos \mathrm{\alpha } + \cos \mathrm{\beta } + \cos \mathrm{\gamma } = \sin \mathrm{\alpha } + \sin \mathrm{\beta } + \sin \mathrm{\gamma } = 0,\mathrm{\;then}\cos 3\mathrm{\alpha } + \cos 3\mathrm{\beta } + \cos 3\mathrm{\gamma \;}$ equals

If $\left| {z - i} \right| = 1$ and $\arg \left( z \right) = \mathrm{\theta },$ where $0 < \mathrm{\theta } < \frac{\pi }{2}$, then $\cot \mathrm{\theta } - \frac{2}{z}$ equals

If ${z_1}$ and ${z_2}$ are two complex numbers such that $\left| {\frac{{{z_1} - {z_2}}}{{{z_1} + {z_2}}}} \right| = 1,$ then

The value of $\left(\frac{1+i \sqrt{3}}{1-i \sqrt{3}}\right)^6+\left(\frac{1-i \sqrt{3}}{1+i \sqrt{3}}\right)^6$ is

The complex number $z=1+i$ is rotated through an angle $3 \pi / 2$ in anti-clockwise direction about the origin and stretched by additional $\sqrt{2}$ units, then the new complex number is

If $a = \cos \left( {\frac{{2\pi }}{7}} \right) + i\sin \left( {\frac{{2\pi }}{7}} \right),\;$ then the quadratic equation whose roots are $\mathrm{\alpha } = a + {a^2} + {a^4}\;\mathrm{and}\;\mathrm{\beta } = {a^3} + {a^5} + {a^6}$

Which term of the sequence

$\left( { - 8 + 18i} \right),\;\left( { - 6 + 15i} \right),\left( { - 4 + 12i} \right),\;$ … is purely imaginary?

If $\left|z^2-1\right|=|z|^2+1$, then ${P}(z)$ lies on

Let $\alpha$ and $\beta$ be two fixed non-zero complex numbers and '$z$' a variable complex number. If the lines $\alpha \bar z + \bar \alpha z + 1 = 0$ and $\mathrm{\beta \;\bar z} + \bar \beta \;z - 1 = 0$ are mutually perpendicular, then

Let $p, q \in \mathbb{R}$ and $(1-\sqrt{3} i)^{200}=2^{199}(p+i q), i=\sqrt{-1}$, then $p+q+q^{2}$ and $p-q+q^{2}$ are roots of the equation.

The smallest positive integer $n$ for which ${\left( {1 + i} \right)^{2n}} = {\left( {1 - i} \right)^{2n}}$ is

If $i=\sqrt{-1}$, then $4+5\left(-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{365}$ is equal to

The values of $x$ and $y$ satisfying the equation $\frac{{\left( {1 + i} \right)x - 2i}}{{3 + i}} + \frac{{\left( {2 - 3i} \right)y + i}}{{3 - i}} = i$ are

The values of ${\left( {16} \right)^{1/4}}$ are

If the complex numbers $z_1, z_2$ and $z_3$ represent the vertices of an equilateral triangle such that $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|$, then

If $\alpha ,\beta ,\gamma$ are the cube roots of a negative number $p$, then for any three real numbers $x,\;y,\;z$ the value of $\frac{{x\alpha + y\beta + z\gamma }}{{x\beta + y\gamma + z\alpha }}$ is

The product of cube roots of $- 1$ is equal to

The imaginary part of $\left( {z - 1} \right)(\cos \alpha - i\sin \alpha ) + {\left( {z - 1} \right)^{ - 1}} \times (\cos \alpha + i\sin \alpha )$ zero, if

If $\omega$ is a complex cube root of unity, then the equation ${\left| {z - \omega } \right|^2} + {\left| {z - {\omega ^2}} \right|^2} = \lambda$ will represent a circle, if

The value of $\sum \limits_{r = 1}^8 \left( {\sin \frac{{2\;r\;\pi }}{9} + i\cos \frac{{2\;r\;\pi }}{9}} \right),$ is

If $z=\frac{-1}{2}+i \frac{\sqrt{3}}{2}$, then $8+10 z+7 z^2$ is equal to