Complex Numbers Questions - JEE

Let $S = \{z\in\mathbb{C}: z^2+\bar{z}=0\}$. The number of elements in $S$ is:

If z, z, z$\i_n C are the vertices of a_n equilateral triangle, whose centroid is z, then$$\sum 1 2 3 0 3$k=1$($zk -$z_{0}$$) 2 is equal to$

Let$A = 2$cos$\$theta+i$sin$$\theta {$$\theta$$\i_n$[0, 2$\pi$]:$1 + 10$Re( ) = 0}. cos$\$theta-3i$sin$$\theta Then$$\sum$$\theta$$\inA$$\theta 2 is equal to$

If $ \left| z - \dfrac{4}{z} \right| = 2 $, then the maximum value of $ |z| $ is:

If $ z + z^2 = 0 $, then which of the following must be true on the complex plane?(A) $ \text{Re}(z) = 0 $(B) $ \text{Im}(z) = 0 $(C) Both $ \text{Re}(z) = 0 $ and $ \text{Im}(z) = 0 $(D) $ z^2 = 1 $

On the complex plane, the locus of $ z $ satisfying $ \arg(z - 3) = \dfrac{\pi}{3} $ is:

If $ \alpha, \beta \in \mathbb{R} $ with $ \alpha^2 + \beta^2 = 1 $, then the locus of $ z = \dfrac{1 + \alpha + i\beta}{1 - \alpha - i\beta} $ is:

If $ |z - z_1| = |z - z_2| $ and $ \arg\left(\dfrac{z - z_1}{z + z_1}\right) = \dfrac{\pi}{2} $, then $ z $ lies on:

On the complex plane, the locus of $ z $ satisfying $ 2 < |z - 3| \leq 3 $ denotes:

Let $ z_1 $ and $ z_2 $ be complex roots of $ z^2 + az + b = 0 $ where $ a^2 < 4b $. If the roots satisfy certain symmetry, the value of $ a + b $ is:

If $|z-1|=1$, then $\dfrac{z-2}{z}$ is:

The minimum value of $|z+1+i|+|z-1-i|+|z-1+i|+|z+1-i|$ for $z\in\mathbb{C}$ is:

The line joining $z_1=1+i$ and $z_2=-1-i$ divides the complex plane into two half-planes. The condition on $z=x+iy$ for it to lie in the same half as $z=0$ and $z=2i$ is:

Let $ z_1 = 2+i $ and $ z_2 = 1+3i $. The value of $ \left|\text{Re}\left(\dfrac{\bar{z}_1 \cdot z_2}{|z_1|^2}\right)\right| $ is:

On the complex plane, the locus of a point $ z $ satisfying $ 2 \leq |z - 3| < 3 $ denotes:

If $ z = \dfrac{\sqrt{3} - i}{2} $, then the principal argument of $ z^{99} $ is:

Given that $\alpha\bar{z}+\bar{\alpha}z+\gamma=0$ is equation of a line where $\gamma$ is purely imaginary. If $\alpha^2-\beta^2=2$, where $\text{Re}(\alpha)$ and $\text{Im}(\alpha)$ are whole numbers, then slope of the line is

Let $ z = \left(\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}\right)^5 + \left(\dfrac{\sqrt{3}}{2}-\dfrac{i}{2}\right)^5 $. If $\text{Re}(z)=a$ and $\text{Im}(z)=b$, then $(a,b)$ equals:

Number of complex numbers satisfying the equation $z^2 = \bar{z}\cdot 2^{1-k}$ is

If $Z$ is a complex number such that $\arg\left(z(1+\bar{z})\right) + \arg\left(\dfrac{|z|^2}{z - |z|^2 i}\right) = 0$, then

Let $z$ be a complex number with nonzero imaginary part and $a=z^2+z+1$ is real. Then '$a$' cannot take the value

Let $P(z)$, $R(z^4)$ and $Q(z^2)$ be three points in the Argand plane such that $PR + RQ = PQ$; ($z\neq 0,1$). Then $z$ lies on

If all roots of $z^3+az^2+bz+c=0$ are of unit modulus, then

If complex number $z_1$ satisfies $\arg z = \dfrac{\pi}{4}$ and $|z-2-i|=1$, and $z_2$ lies on the curve $|z-25i|=15$ with the least positive argument, then the minimum value of $|z_1-z_2|$ is:

Let $z_1,z_2,z_3$ be three unimodular complex numbers which are also roots of the equation $z^3+az^2+bz+1=0$ (where $a,b$ are complex numbers). Tangents are drawn to $|z|=1$ at points $z_1,z_2,z_3$ which intersect pairwise at points $\omega_1,\omega_2,\omega_3$. Then $\omega_1,\omega_2,\omega_3$ are roots of the equation:

Let $A(z_1)$, $B(z_2)$, $C(z_3)$ be vertices of $\triangle ABC$ such that $|z_1-z_2|=3$, $z_1^2=z_2z_3$, $z_2^2=z_1z_3$ ($z_1z_2z_3\neq 0$). The area of $\triangle ABC$ equals:

Match the complex loci: P) $|z-(5+3i)|-|z-1-i\sqrt{3}|=\sin(\pi/3-\arg(z-1-i\sqrt{3}))$ → locus type Q) $5|z-3|=(4+3i)z+(4-3i)\bar{z}+2|z|^2-3$ → locus type R) $\big||z-1|-|z-4+4i|\big|=5$ → locus type S) $\arg\!\left(\frac{z-2026i}{z-25i}\right)=0$ → locus type List-II: 1)Line segment; 2)Pair of rays; 3)Circle; 4)Parabola; 5)Hyperbola

Let $z_1,z_2,z_3$ be three unimodular complex numbers which are also roots of the equation $z^3+az^2+bz+1=0$ (where $a,b$ are complex numbers). Tangents are drawn to $|z|=1$ at points $z_1,z_2,z_3$ which intersect pairwise at points $\omega_1,\omega_2,\omega_3$. Then $\omega_1,\omega_2,\omega_3$ are roots of the equation:

Let $A(z_1)$, $B(z_2)$, $C(z_3)$ be vertices of $\triangle ABC$ such that $|z_1-z_2|=3$, $z_1^2=z_2z_3$, $z_2^2=z_1z_3$ ($z_1z_2z_3\neq 0$). The area of $\triangle ABC$ equals:

Find the principal argument of (a) $-1 - i\sqrt{3}$

Modulus of nonzero complex number z satisfying $\bar{z} + z = 0$ and $|z|^2 - 4zi = z^2$ is ___.

The diagram shows several numbers in the complex plane; the circle is the unit circle centred at the origin. One of the labeled numbers is the reciprocal of $ F $. Which one?

The value of $\lambda + \mu$ is equal to:

If $z_1, z_2, z_3$ be three complex numbers such that $|z_1+1|\leq 1, |z_2+2|\leq 2$ and $|z_3+4|\leq 4$, then the maximum value of $|z_1|+|z_2|+|z_3|$ is :

If $n$ is positive integer and a complex number with unit modulus is a solution of the equation $Z^n + z^{-1} = 1$, then the value of $n$ can be

Number of complex numbers satisfying the equation $z^2 = \bar{z}\cdot 2^{1-k}$ is

If $|z| = 2$ and $\frac{z_1 - z_3}{z_2 - z_3} = \frac{z - 2}{z + 2}$, then $z_1, z_2, z_3$ will be vertices of a/an:

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

Consider a square $OABC$ in the Argand plane, where 'O' is origin and $A=A\left(z_0\right)$. Then the equation of the circle that can be inscribed in this square is : (vertices of square are given in anti-clockwise order)

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

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

If $\frac{z-z_1}{z-z_2}=3$, where $z_1$ and $z_2$ are fixed complex numbers and $z$ is a variable complex number, then $z$ lies on a :

If $z$ is a complex number such that $\left|z + \frac{1}{z}\right| = 2$ then minimum value of $|z|$ is ____.

If $A(2 + 3i), B(3i)$ and $D(4i)$ are two vertices of a square $ABCD$ (taken in anticlockwise order) in a complex plane, then the value of $|Z_1|^2 - |Z_2|^2$ (where $C$ is $Z_1$ and $D$ is $Z_1$) is equal to

$z_1,z_2\in\mathbb{C}$ with $|z_1|=|z_2|=1$, $z_1^2+z_2^2=1$. Then locus of $z_1+z_2$ is

Number of points of intersection of $\arg(z-2-7i)=\cot^{-1}2$ and $\arg\!\left(\dfrac{z-5i}{z+2-i}\right)=\pm\dfrac{\pi}{2}$

Find the number of complex numbers $z$ satisfying both $|z + \bar{z}| + |z - \bar{z}| = 2$ and $|z + i| + |z - i| = 2$.

If z is a complex number of unit modulus and argument θ, then arg (1 + z)/(1 + z̄) equals to

Complex Numbers Questions (654)