Complex Numbers

The number of values of $z \in \mathbb{C}$, satisfying the equations $|z-(4+8 i)|=\sqrt{10}$ and $|z-(3+5 i)|+|z-(5+11 i)|=4 \sqrt{5}$, is $:$

Let the set of all values of $k \in \mathbb{R}$ such that the equation $z(\bar{z}+2+i)+k(2+3 i)=0, z \in \mathrm{C}$, has at least one solution, be the interval $[\alpha, \beta]$. Then $9(\alpha+\beta)$ is equal to:

Let $S=\left\{z \in \mathbb{C}: z^2+\sqrt{6} i z-3=0\right\}$. Then $\sum\limits_{z \in S} z^8$ is equal to :

Let $a, b \in \mathbb{C}$. Let $\alpha, \beta$ be the roots of the equation $x^2+a x+b=0$. If $\beta-\alpha=\sqrt{11}$ and $\beta^2-\alpha^2=3 i \sqrt{11}$, then $\left(\beta^3-\alpha^3\right)^2$ is equal to:

Let $z_1, z_2 \in \mathbb{C}$ be the distinct solutions of the equation $z^2+4 z-(1+12 i)=0$.
Then $\left|z_1\right|^2+\left|z_2\right|^2$ is equal to :

Let $z$ be a complex number such that $|z+2|=|z-2|$ and arg $\left(\frac{z+3}{z-i}\right)=\frac{\pi}{4}$. Then $|z|^2$ is equal to:

Let $\mathrm{S}=\left\{z \in \mathrm{C}: z^2+4 z+16=0\right\}$. Then $\sum\limits_{z \in \mathrm{~S}}|z+\sqrt{3} \mathrm{i}|^2$ is equal to :

Let $x$ and $y$ be real numbers such that $50\left(\frac{2 x}{1+3 i}-\frac{y}{1-2 i}\right)=31+17 i, i=\sqrt{-1}$. Then the value of $10(x-3 y)$ is :

Let the circles $C_1:|z| = r$ and $C_2:|z - 3 - 4i| = 5$, $z \in \mathbb{C}$, be such that $C_2$ lies within $C_1$.If $z_1$ moves on $C_1$, $z_2$ moves on $C_2$ and $\min |z_1 - z_2| = 2$, then $\max |z_1 - z_2|$ is equal to :

Let $z$ be a complex number such that $|z-6|=5$ and $|z+2-6 i|=5$. Then the value of $z^3+3 z^2-15 z+141$ is equal to :

Let$A = \{ z \in \mathbb{C} : |z - 2| \leq 4 \}$ and$B = \{ z \in \mathbb{C} : |z - 2| + |z + 2| = 5 \}$.Then the max $\{|z_1 - z_2| : z_1 \in A \text{ and } z_2 \in B \}$ is :

Let $\mathrm{S}=\left\{z \in \mathbb{C}:\left|\frac{z-6 i}{z-2 i}\right|=1\right.$ and $\left.\left|\frac{z-8+2 i}{z+2 i}\right|=\frac{3}{5}\right\}$.
Then $\sum\limits_{z \in \mathrm{~s}}|z|^2$ is equal to :

Let $z=(1+i)(1+2 i)(1+3 i) \ldots .(1+n i)$, where $i=\sqrt{-1}$. If $|z|^2=44200$, then $n$ is equal to $\_\_\_\_$

Let $\mathrm{S}=\{z: 3 \leqslant|2 z-3(1+\mathrm{i})| \leqslant 7\}$ be a set of complex numbers.
Then $\operatorname{Min}_{z \in S}\left|\left(z+\frac{1}{2}(5+3 i)\right)\right|$ is equal to :

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}, i=\sqrt{-1}$, then $\left(z^{201}-i\right)^8$ is equal to

Let $\alpha=\frac{-1+i \sqrt{3}}{2}$ and $\beta=\frac{-1-i \sqrt{3}}{2}, i=\sqrt{-1}$. If
\((7-7 \alpha+9 \beta)^{20}+(9+7 \alpha-7 \beta)^{20}+(-7+9 \alpha+7 \beta)^{20}+(14+7 \alpha+7 \beta)^{20}=m^{10},\)
then $m$ is $\_\_\_\_$

Let $\mathrm{S}=\left\{z \in \mathbb{C}: 4 z^2+\bar{z}=0\right\}$. Then $\sum\limits_{z \in \mathrm{~S}}|z|^2$ is equal to:

If $x^2+x+1=0$, then the value of $\left(x+\frac{1}{x}\right)^4+\left(x^2+\frac{1}{x^2}\right)^4+\left(x^3+\frac{1}{x^3}\right)^4+\ldots+\left(x^{25}+\frac{1}{x^{25}}\right)^4$ is:

Let $z$ be the complex number satisfying $|z-5| \leq 3$ and having maximum positive principal argument.Then $34 \left| \frac{5z - 12}{5iz + 16} \right|^2$ is equal to:

Let $ A = \left\{ \theta \in [0, 2\pi] : 1 + 10\operatorname{Re}\left( \frac{2\cos\theta + i\sin\theta}{\cos\theta - 3i\sin\theta} \right) = 0 \right\} $. Then $ \sum\limits_{\theta \in A} \theta^2 $ is equal to

Among the statements
(S1) : The set $\left\{z \in \mathbb{C}-\{-i\}:|z|=1\right.$ and $\frac{z-i}{z+i}$ is purely real $\}$ contains exactly two elements, and
(S2) : The set $\left\{z \in \mathbb{C}-\{-1\}:|z|=1\right.$ and $\frac{z-1}{z+1}...

If the locus of z ∈ ℂ, such that Re$ \left( \frac{z - 1}{2z + i} \right) + \text{Re} \left( \frac{\overline{z} - 1}{2\overline{z} - i} \right) = 2 $, is a circle of radius r and center $(a, b)$, then $ \frac{15ab}{r^2} $ is equal to :

Let $\mathrm{A}=\{z \in \mathrm{C}:|z-2-i|=3\}, \mathrm{B}=\{z \in \mathrm{C}: \operatorname{Re}(z-i z)=2\}$ and $\mathrm{S}=\mathrm{A} \cap \mathrm{B}$. Then $\sum_{z \in S}|z|^2$ is equal to _________.

If $\alpha$ is a root of the equation $x^2+x+1=0$ and $\sum_\limits{\mathrm{k}=1}^{\mathrm{n}}\left(\alpha^{\mathrm{k}}+\frac{1}{\alpha^{\mathrm{k}}}\right)^2=20$, then n is equal to _________.

Let the product of $\omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta$ and $\omega_2=(1+8 i) \sin \theta+(4+7 i) \cos \theta$ be $\alpha+i \beta$, $i=\sqrt{-1}$. Let p and q be the maximum and the minimum values of $\alpha+\beta$ respectively....

Let $z \in C$ be such that $\frac{z^2+3 i}{z-2+i}=2+3 i$. Then the sum of all possible values of $z^2$ is :

\(If\,\,{z_1},{z_2},{z_3} \in \,\,are\,\,the\,\,vertices\,\,of\,\,an\,\,equilateral\,\,triangle,\,\,whose\,\,centroid\,\,is\,\,{z_0},\,\,then\,\,\sum\limits_{k = 1}^3 {{{\left( {{z_k} - {z_0}} \right)}^2}\,is\,\,equal\,\,to}\)

Let $z$ be a complex number such that $|z|=1$. If $\frac{2+\mathrm{k}^2 z}{\mathrm{k}+\bar{z}}=\mathrm{k} z, \mathrm{k} \in \mathbf{R}$, then the maximum distance of $\mathrm{k}+i \mathrm{k}^2$ from the circle $|z-(1+2 i)|=1$ is :

Let $ |z_1 − 8−2i| \leq 1 $ and $ |z_2−2+6i| \leq 2 $, $ z_1, z_2 \in \mathbb{C} $. Then the minimum value of $ |z_1 − z_2| $ is :

Let integers $\mathrm{a}, \mathrm{b} \in[-3,3]$ be such that $\mathrm{a}+\mathrm{b} \neq 0$. Then the number of all possible ordered pairs (a, b), for which $\left|\frac{z-\mathrm{a}}{z+\mathrm{b}}\right|=1$ and $\left|\begin{array}{ccc}z+1...

Let $O$ be the origin, the point $A$ be $z_1=\sqrt{3}+2 \sqrt{2} i$, the point $B\left(z_2\right)$ be such that $\sqrt{3}\left|z_2\right|=\left|z_1\right|$ and $\arg \left(z_2\right)=\arg \left(z_1\right)+\frac{\pi}{6}$. Then

If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha \gamma + \beta \delta$ is equal to:

If $\alpha$ and $\beta$ are the roots of the equation $2 z^2-3 z-2 i=0$, where $i=\sqrt{-1}$, then $16 \cdot \operatorname{Re}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \cdot \operatorname{lm}\...

Let $\left|\frac{\bar{z}-i}{2 \bar{z}+i}\right|=\frac{1}{3}, z \in C$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $(0,0), C$ and $(\alpha, 0)$ is 11 square units, then $\alp...

Let $\alpha, \beta$ be the roots of the equation $x^2-\mathrm{ax}-\mathrm{b}=0$ with $\operatorname{Im}(\alpha)<\operatorname{Im}(\beta)$. Let $\mathrm{P}_{\mathrm{n}}=\alpha^{\mathrm{n}}-\beta^{\mathrm{n}}$. If $\mathrm{P}_3=-5 \sqrt{7} i,...

The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is :

Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg \left(z_1\right)=\frac{-\pi}{4}, \arg \left(z_2\right)=0$ and $\arg \left(z_3\right)=\frac{\pi}{4}$. If $\left|z_1 \bar{z}_2+z_2 \bar{z}_3+z_3 \bar{z}_1\righ...

Let the curve $z(1+i)+\bar{z}(1-i)=4, z \in C$, divide the region $|z-3| \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha-\beta|$ equals :

The sum of the square of the modulus of the elements in the set \(\{z=\mathrm{a}+\mathrm{ib}: \mathrm{a}, \mathrm{b} \in \mathbf{Z}, z \in \mathbf{C},|z-1| \leq 1,|z-5| \leq|z-5 \mathrm{i}|\}\) is __________.

Let \(z\) be a complex number such that the real part of \(\frac{z-2 i}{z+2 i}\) is zero. Then, the maximum value of \(|z-(6+8 i)|\) is equal to

If the set \(R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\}\) has \(m\) elements and \(\sum_\limits{n=1}^m\left(1-i^{n !}\right)=x+i y\), where \(i=\sqrt{-1}\), then the value of \(m+x+y\) is

Let \(z\) be a complex number such that \(|z+2|=1\) and \(\operatorname{lm}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}\). Then the value of \(|\operatorname{Re}(\overline{z+2})|\) is

The sum of all possible values of \(\theta \in[-\pi, 2 \pi]\), for which \(\frac{1+i \cos \theta}{1-2 i \cos \theta}\) is purely imaginary, is equal to :

If \(z_1, z_2\) are two distinct complex number such that \(\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2\), then

Consider the following two statements :
Statement I: For any two non-zero complex numbers $$z_1, z_2,(|z_1|+|z_2|)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right) \t...

Let \(S_1=\{z \in \mathbf{C}:|z| \leq 5\}, S_2=\left\{z \in \mathbf{C}: \operatorname{Im}\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right\}\) and \(S_3=\{z \in \mathbf{C}: \operatorname{Re}(z) \geq 0\}\). Then the area of the r...

Let \(\alpha\) and \(\beta\) be the sum and the product of all the non-zero solutions of the equation \((\bar{z})^2+|z|=0, z \in C\). Then \(4(\alpha^2+\beta^2)\) is equal to :

The area (in sq. units) of the region \(S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{lm}(z) \geq 0\}\) is

If \(\alpha\) denotes the number of solutions of \(|1-i|^x=2^x\) and \(\beta=\left(\frac{|z|}{\arg (z)}\right)\), where $$z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}\right], i=\sqrt{-1...

Let \(z_1\) and \(z_2\) be two complex numbers such that \(z_1+z_2=5\) and \(z_1^3+z_2^3=20+15 i\) Then, \(\left|z_1^4+z_2^4\right|\) equals -