Complex Numbers

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 :

If \(z\) is a complex number, then the number of common roots of the equations \(z^{1985}+z^{100}+1=0\) and \(z^3+2 z^2+2 z+1=0\), is equal to

Let \(\alpha, \beta\) be the roots of the equation \(x^2-x+2=0\) with \(\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)\). Then \(\alpha^6+\alpha^4+\beta^4-5 \alpha^2\) is equal to ___________.

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

Let \(\alpha, \beta\) be the roots of the equation \(x^2-\sqrt{6} x+3=0\) such that \(\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)\). Let \(a, b\) be integers not divisible by 3 and \(n\) be a natural number such that $$\frac{\alpha^{...

Let \(\mathrm{r}\) and \(\theta\) respectively be the modulus and amplitude of the complex number \(z=2-i\left(2 \tan \frac{5 \pi}{8}\right)\), then \((\mathrm{r}, \theta)\) is equal to

If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7=A+B \alpha+C \alpha^2, A, B, C \geqslant 0$, then $5(3 A-2 B-C)$ is equal to ____________.

If $S=\{z \in C:|z-i|=|z+i|=|z-1|\}$, then, $n(S)$ is :

Let the complex numbers \(\alpha\) and \(\frac{1}{\bar{\alpha}}\) lie on the circles \(\left|z-z_0\right|^2=4\) and \(\left|z-z_0\right|^2=16\) respectively, where \(z_0=1+i\). Then, the value of \(100|\alpha|^2\) is __________.

Let $\mathrm{P}=\{\mathrm{z} \in \mathbb{C}:|z+2-3 i| \leq 1\}$ and $\mathrm{Q}=\{\mathrm{z} \in \mathbb{C}: z(1+i)+\bar{z}(1-i) \leq-8\}$. Let in $\mathrm{P} \cap \mathrm{Q}$, $|z-3+2 i|$ be maximum and minimum at $z_1$ and $z_2$ respectiv...

Let $\mathrm{S}=|\mathrm{z} \in \mathrm{C}:| z-1 \mid=1$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2} \mid$. Let $z_1, z_2 \in \mathrm{S}$ be such that $\left|z_1\right|=\max\limits_{z \in s}|z|$ and $\left|z_2\right|=\min\limits _{...

If $z$ is a complex number such that $|z| \leqslant 1$, then the minimum value of $\left|z+\frac{1}{2}(3+4 i)\right|$ is :

If for \(z=\alpha+i \beta,|z+2|=z+4(1+i)\), then \(\alpha+\beta\) and \(\alpha \beta\) are the roots of the equation :

Let \(A=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.\) is purely imaginary \(\}\). Then the sum of the elements in \(\mathrm{A}\) is :

For \(\alpha, \beta, z \in \mathbb{C}\) and \(\lambda > 1\), if \(\sqrt{\lambda-1}\) is the radius of the circle \(|z-\alpha|^{2}+|z-\beta|^{2}=2 \lambda\), then \(|\alpha-\beta|\) is equal to __________.

Let \(a \neq b\) be two non-zero real numbers. Then the number of elements in the set \(X=\left\{z \in \mathbb{C}: \operatorname{Re}\left(a z^{2}+b z\right)=a\right.\) and \(\left.\operatorname{Re}\left(b z^{2}+a z\right)=b\right\}\) is equ...

For all \(z \in C\) on the curve \(C_{1}:|z|=4\), let the locus of the point \(z+\frac{1}{z}\) be the curve \(\mathrm{C}_{2}\). Then :

The complex number $z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ is equal to :

Let \(z=1+i\) and \(z_{1}=\frac{1+i \bar{z}}{\bar{z}(1-z)+\frac{1}{z}}\). Then \(\frac{12}{\pi} \arg \left(z_{1}\right)\) 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}\ri...

Let \(\alpha = 8 - 14i,A = \left\{ {z \in c:{{\alpha z - \overline \alpha \overline z } \over {{z^2} - {{\left( {\overline z } \right)}^2} - 112i}}=1} \right\}\) and \(B = \left\{ {z \in c:\left| {z + 3i} \right| = 4} \right\}\). Then $$\...

Let \(\mathrm{z_1=2+3i}\) and \(\mathrm{z_2=3+4i}\). The set \(\mathrm{S = \left\{ {z \in \mathbb{C}:{{\left| {z - {z_1}} \right|}^2} - {{\left| {z - {z_2}} \right|}^2} = {{\left| {{z_1} - {z_2}} \right|}^2}} \right\}}\) represents a

Let \(z\) be a complex number such that \(\left| {{{z - 2i} \over {z + i}}} \right| = 2,z \ne - i\). Then \(z\) lies on the circle of radius 2 and centre :

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

The value of \({\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}\) is

If the center and radius of the circle \(\left| {{{z - 2} \over {z - 3}}} \right| = 2\) are respectively \((\alpha,\beta)\) and \(\gamma\), then \(3(\alpha+\beta+\gamma)\) is equal to :

Let \(a,b\) be two real numbers such that \(ab < 0\). IF the complex number \(\frac{1+ai}{b+i}\) is of unit modulus and \(a+ib\) lies on the circle \(|z-1|=|2z|\), then a possible value of \(\frac{1+[a]}{4b}\), where \([t]\) is greatest int...

If the set $\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in \mathbb{C}, \operatorname{Re}(z)=3\right\}$ is equal to the interval $(\alpha, \beta]$, then $24(\beta-\alpha)$ is equal to :

Let \(w=z \bar{z}+k_{1} z+k_{2} i z+\lambda(1+i), k_{1}, k_{2} \in \mathbb{R}\). Let \(\operatorname{Re}(w)=0\) be the circle \(\mathrm{C}\) of radius 1 in the first quadrant touching the line \(y=1\) and the \(y\)-axis. If the curve $$\ope...

Let \(S=\left\{z \in \mathbb{C}: \bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}\). Then \(\sum_\limits{z \in \mathrm{S}}|z|^{2}\) is equal to :

Let \(\mathrm{C}\) be the circle in the complex plane with centre \(\mathrm{z}_{0}=\frac{1}{2}(1+3 i)\) and radius \(r=1\). Let \(\mathrm{z}_{1}=1+\mathrm{i}\) and the complex number \(z_{2}\) be outside the circle \(C\) such that $$\left|z...

Let \(w_{1}\) be the point obtained by the rotation of \(z_{1}=5+4 i\) about the origin through a right angle in the anticlockwise direction, and \(w_{2}\) be the point obtained by the rotation of \(z_{2}=3+5 i\) about the origin through a ...

Let \(\mathrm{S}=\left\{z \in \mathbb{C}-\{i, 2 i\}: \frac{z^{2}+8 i z-15}{z^{2}-3 i z-2} \in \mathbb{R}\right\}\). If \(\alpha-\frac{13}{11} i \in \mathrm{S}, \alpha \in \mathbb{R}-\{0\}\), then

\(242 \alpha^{2}\) is equal to _________.

For \(a \in \mathbb{C}\), let \(\mathrm{A}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z}) > \operatorname{Im}(\bar{a}+z)\}\) and \(\mathrm{B}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z})<\operatorname{Im}(\bar{a}+z)\}\). Then among th...

Let the complex number \(z = x + iy\) be such that \({{2z - 3i} \over {2z + i}}\) is purely imaginary. If \({x} + {y^2} = 0\), then \({y^4} + {y^2} - y\) is equal to :

Let \(S = \left\{ {z = x + iy:{{2z - 3i} \over {4z + 2i}}\,\mathrm{is\,a\,real\,number}} \right\}\). Then which of the following is NOT correct?

The real part of the complex number \({{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }}\) is equal to :

Let \(S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\}\). Let \(|z - 4i|\) attains minimum and maximum values, respectively, at z1 \(\in\) S and z2 \(\in\) S. If $$5(|{z_1}{|^2} + |{z_2}{|^2}) = \alpha + \beta \sqrt 5...

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 arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z \(-\) 1) \(-\) arg(z + 1) = \({\pi \over 4}\) intersect :

If \(z=2+3 i\), then \(z^{5}+(\bar{z})^{5}\) is equal to :

Let \(\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|<2,|z+i|=|z-1|\}\). Then the set of all values of \(x\), for which \(w=2 x+i y \in \mathrm{S}\) for some \(y \in \mathbb{R}\), is :

If \(z \neq 0\) be a complex number such that \(\left|z-\frac{1}{z}\right|=2\), then the maximum value of \(|z|\) is :

The number of elements in the set {z = a + ib \(\in\) C : a, b \(\in\) Z and 1 < | z \(-\) 3 + 2i | < 4} is __________.

Sum of squares of modulus of all the complex numbers z satisfying \(\overline z = i{z^2} + {z^2} - z\) is equal to ___________.

Let \(S_{1}=\left\{z_{1} \in \mathbf{C}:\left|z_{1}-3\right|=\frac{1}{2}\right\}\) and \(S_{2}=\left\{z_{2} \in \mathbf{C}:\left|z_{2}-\right| z_{2}+1||=\left|z_{2}+\right| z_{2}-1||\right\}\). Then, for \(z_{1} \in S_{1}\) and $$z_{2} \in ...

Let \(\mathrm{z}=a+i b, b \neq 0\) be complex numbers satisfying \(z^{2}=\bar{z} \cdot 2^{1-z}\). Then the least value of \(n \in N\), such that \(z^{n}=(z+1)^{n}\), is equal to __________.

The area of the polygon, whose vertices are the non-real roots of the equation \(\overline z = i{z^2}\) is :

The number of points of intersection of \(|z - (4 + 3i)| = 2\) and \(|z| + |z - 4| = 6\), z \(\in\) C, is :

Let \(S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}\). Then \(\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))\) is equal to ______________.