Complex numbers practice · Easy

The value of $\left( {\cos \frac{\pi }{2} + i\sin \frac{\pi }{2}} \right)\left( {\cos \frac{\pi }{4} + i\sin \frac{\pi }{4}} \right)\left( {\cos \frac{\pi }{8} + \;i\sin \frac{\pi }{8}} \right) \ldots \infty$ is

If ${m_1},{m_2},{m_3}$ and ${m_4}$ respectively denote the moduli of the complex numbers $1 + 4i,\;3 + i,\;1 - i$ and $2 - 3i$ then the correct one, among the following is

If $z$ is a complex number, then $z^2+\bar{z}^2=2$ represents

The argument of $\frac{{1 + i\sqrt 3 }}{{1 - i\sqrt 3 }}$ is

If the number $\frac{(1-i)^n}{(1+i)^{n-2}}$ is real and positive, then $n$ is

If $z \in \mathbf{C}$ and $2 z=|z|+i$, then $z$ equals

If $(1-i)^n=2^n$, then the value of $n$ is

If $\omega$ is a complex cube root of unity, then the value of ${\omega ^{99}} + {\omega ^{100}} + {\omega ^{101}}$ is

If $n$ is a positive integer, then ${\left( {1 + i\sqrt 3 } \right)^n} + {\left( {1 - i\sqrt 3 } \right)^n}$ is equal to

If $z = r(\cos \theta + i\sin \theta )$, then the value of $\frac{z}{{\bar z}} + \frac{{\bar z}}{z}$ is

If $|z+4|=3$, then greatest possible value of $|z-7| / 7$ is

A particle $P$ starts from the point $z_0=1+2 i, i=\sqrt{-1}$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_1$. From $z_1$, the particle moves $\sqrt{2}$ units in the direction of the vector $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and then it moves through an angle $\frac{\pi}{2}$ in anti-clockwise direction on a circle with centre at origin, to reach a point $z_2$, then the point $z_2$ is given by

If $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$, then one of the values of $\arg \left(z_2 / z_1\right)$ is

If $z$ is a non-real ${7^{th}}$ root of $- 1$, then ${z^{86}} + {z^{175}} + {z^{289}}$ is equal to

If $\left| {\frac{{z + i}}{{z - i}}} \right| = \sqrt 3$, then radius of the circle is

If $\alpha, \beta$ are distinct complex numbers with $|\beta|=1$, then value of $\left|\frac{\beta-\alpha}{1-\bar{\alpha} \beta}\right|$ equals

If $\omega(\neq 1)$ is a cube root of unity and $(1+\omega)^7=A+B \omega$, then $(A, B)$ equals to

If the cube roots of unity are $1, \omega, \omega^2$, then the roots of the equation $x^3-3 x^2+3 x+63=0$ are

If $\frac{{3\left( {x - 2} \right)}}{5} \ge \frac{{5\left( {2 - x} \right)}}{3},$ then $x$ belongs to the interval

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

The area of the triangle whose vertices are $i,\omega$ and ${\omega ^2},$, where $i = \sqrt { - 1}$ and $\omega ,{\omega ^2}$ are complex cube roots of unity, is

If $|z|=1$ and $z \neq \pm 1$, then all the values of $\frac{z}{1-z^2}$ lie on

If $(1+i)(1+2 i)(1+3 i) \ldots(1+n i)=\alpha+i \beta$, then $2 \cdot 5 \cdot 10 \ldots\left(1+n^2\right)$ is equal to

If $z^2+z+1=0$, where $z$ is a complex number, then values of $S=\left(z+\frac{1}{z}\right)^2+\left(z^2+\frac{1}{z^2}\right)^2+\left(z^3+\frac{1}{z^3}\right)^2+\cdots+\left(z^6+\frac{1}{z^6}\right)^2 \text { is }$

The value of ${\log _2}{\log _3} \ldots {\log _{100}}{100^{{{99}^{{{98}^{{.^{{.^{{2^1}}}}}}}}}}}$ is equal to

Which of the following is a fourth root of $\frac{1}{2} + i\frac{{\sqrt 3 }}{2}$ ?

If $z=\frac{1}{2}-2 i$ is such that $|z+1|=\alpha z+\beta(1+i)$, $i=\sqrt{-1}$ and $\alpha, \beta \in {R}$, then $\alpha+\beta$ is equal to

The points representing complex number $z$ for which $\left| {z - 3} \right| = \left| {z - 5} \right|$ lie on the locus given by

If $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$ and $z_1+z_2+z_3=0$, then area of the triangle whose vertices are $z_1, z_2, z_3$ is

If ${z_1},\;{z_2},\;{z_3},\;{z_4}$ are four complex numbers represented by the vertices of a quadrilateral taken in order such that ${z_1} - {z_4} = {z_2} - {z_3}\;\;\mathrm{and}\;\arg \left( {\frac{{{z_4} - {z_1}}}{{{z_2} - {z_1}}}} \right) = \frac{\pi }{2},\;$ then the quadrilateral is