Complex numbers practice · Mixed difficulty

If $(1+z)^n=a_0+a_1 z+a_2 z^2+\cdots+a_n z^n$, where $a_0, a_1$, $a_2, \ldots, a_n$, are real, then $\left(a_0-a_2+a_4-a_6+\cdots\right)^2+\left(a_1-a_3+a_5-a_7+\cdots\right)^2=$

If ${\log _{10}}7 = 0.8451$, then the position of the first significant figure of ${7^{ - 20}}$ is

Reflection of the line $\bar{a} z+a \bar{z}=0$ in the real axis is

The complex number $i+\sqrt{3}$ in polar form can be written as

The maximum distance from the origin of coordinates to the point $z$ satisfying the equation $\left| {z + \frac{1}{z}} \right| = a$ is

If the real part of $\frac{{\bar z + 2}}{{\bar z - 1}}$ is 4, then the locus of the point representing $z$ in the complex plane is

Let $z_1, z_2$ be two complex numbers such that $z_1 \neq 0$ and $z_2 / z_1$ is purely real, then $\left|\frac{2 i z_1+5 z_2}{2 i z_1-5 z_2}\right|$ is equal to

If $z=x+i y, x y \neq 0$, satisfies the equation $z^2+i \bar{z}=0$, then $\left|z^2\right|$ is equal to :

A point $P$ which represents a complex number $z$ moves such that $\left| {z - {z_1}} \right| = \left| {z - {z_2}} \right|$, then its locus is

For $n \in N$, let $S_n=\left\{z \in C:|z-3+2 i|=\frac{n}{4}\right\}$ and $T_n=\left\{z \in C:|z-2+3 i|=\frac{1}{n}\right\}$. Then the number of elements in the set $\left\{n \in N: S_n \cap T_n=\phi\right\}$ is

The modulus of $\sqrt{2{i}}-\sqrt{-2 {i}}$ is:

The value of $\frac{(\cos 2 \theta-i \sin 2 \theta)^7(\cos 3 \theta+i \sin 3 \theta)^{-5}}{(\cos 4 \theta+i \sin 4 \theta)^{12}(\cos 5 \theta+i \sin 5 \theta)^{-6}}$ is

Let $Z=\left[\begin{array}{ll}7+5 i & 3+3 i \\ 8+5 i & 5+2 i\end{array}\right]$ then $\operatorname{det}(Z \bar{Z})$, (where $\bar{Z}$ is the matrix obtained by taking conjugate of each element of $Z$ ) is equal to

If $x + iy = \frac{3}{{2 + \cos \mathrm{\theta } + i\sin \mathrm{\theta }}},\;\,\mathrm{then}\,\;{x^2} + {y^2}$ is equal to

If $x_1 \leq x_2$ and $y_1 \leq y_2$ for any two complex numbers $z_1=x_1+i y_1$ and $z_2=x_2+i y_2$, then we say $z_1 \cap z_2.$

Let $z$ be a complex number with $1 \cap z$. Then

If $x+i y=\frac{3}{2+\cos \theta+i \sin \theta}$ then $x^2+y^2=$

If $p,q,r$ are positive and are in AP, then roots of the quadratic equation $p{x^2} + qx + r = 0$ are complex for

${i^2} + {i^4} + {i^6} + ...\mathrm{upto}\;\left( {2k + 1} \right)\mathrm{terms},\;k \in N$ is

The modulus and amplitude of ${\left( {1 + i\sqrt 3 } \right)^8}$ are respectively

If $x = {\log _b}a,\;y = {\log _c}b,\;z = {\log _a}c$, then $xyz$ is

Let ${z_1}{\text{ }}\,\,{\text{and}}\,\,{\text{ }}{z_2}$ be two complex numbers such that $\overline {{z_1}} = i\,\overline {{z_2}} {\text{ }}\,\,{\text{and }}\,\,\arg \left( {\frac{{{z_1}}}{{\overline {{z_2}} }}} \right) = \pi .$ then

Consider the following statements.

I. Additive inverse of $(1-i)$ is equal to $-1+i$.

II. If ${z}_1$ and ${z}_2$ are two complex numbers, then ${z}_1-{z}_2$ represents a complex number which is sum of $z_1$ and additive inverse of $z_2$.
III. Simplest form of $\frac{5+\sqrt{2} i}{1-\sqrt{2} i}$ is $1+2 \sqrt{2}i.$

Choose the correct option.

The inequality $a+i b > c+i d$ is true when

If $\omega$ is a complex cube root of unity, then $\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}\pi - \frac{\pi }{4}} \right)} \right\}$ is equal to

The locus of point $z$ satisfying ${\mathop\mathrm{Re}\nolimits} \left( {\frac{1}{z}} \right) = k$, where $k$ is a non-zero real number, is

If $\frac{3}{{2 + \cos \theta + i\sin \theta }} = a + ib$, then $\left[ {{{\left( {a - 2} \right)}^2} + {b^2}} \right]$ is equal to

If $1,\;\omega ,{\omega ^2}$ are the cube roots of unity, then $\Delta = \left| {\begin{array}{cccccccccccccccccccc} 1&{{\omega ^n}}&{{\omega ^{2n}}}\\ {{\omega ^n}}&{{\omega ^{2n}}}&1\\ {{\omega ^{2n}}}&1&{{\omega ^n}} \end{array}} \right|$ is equal to

If $z=5 i\left(\frac{-3}{5} i\right)$, then $z$ is equal to $3+bi.$ The value of '${b}$' is

$e^{2 m i \cot ^{-1} p} \cdot\left(\frac{p i+1}{p i-1}\right)^m$ is equal to

If $z=\frac{\sqrt{ } 3-i}{2}$, where $i=\sqrt{-1}$, then $\left(i^{101}+z^{101}\right)^{103}$ equals to