Complex Numbers Questions - JEE

If arg$\left(\frac{z}{|z|} - \frac{z_1}{|z|}\right) = \frac{\pi}{2}$ and z - z1 = 3, then |z1| equals to

Let $z_1 = a + ib$, $z_2 = a - ib$, $z_3 = c + id$, $z_4 = c - id$. Then $\arg\left(\dfrac{z_1}{z_4}\right) + \arg\left(\dfrac{z_2}{z_3}\right)$ equals:

For Problems 23–25: Consider the equation $az + b\bar{z} + c = 0$, where $a, b, c \in \mathbb{Z}$.If $|a| = |b|$ and $\bar{a}c \neq b\bar{c}$, then $z$ has

If $|z - 2i| = \sqrt{2}$, where $i = \sqrt{-1}$, then the maximum value of $|3 + i(z - 1)|$ is

What is the digit in the unit's place of $(2419)^{111213}$?

If $z_1$, $z_2$ are two distinct complex numbers such that $\left|\dfrac{z_1-2z_2}{\frac{1}{2}-z_1\bar{z}_2}\right|=2$, then:

Let z1, z2, z3 are represented by points A, B and C on Argand diagram. The minimum value of \[\frac{1}{|z_1 - z_2||z_1 - z_3|} + \frac{1}{|z_2 - z_1||z_2 - z_3|} + \frac{1}{|z_3 - z_1||z_3 - z_2|}\] is equal to ___.

$z \neq 1$ and $\dfrac{z^2}{z - 1}$ is real, then the point represented by the complex number $z$ lies

Let $\omega$ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt{-3}$. If $\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^2 - 1 & \omega^2 \\ 1 & \omega^2 & \omega^7 \end{vmatrix} = 3k$, then $k$ is equal to

If $\left|\dfrac{z-2}{z-3}\right| = 2$ represents a circle, then find its centre and radius.

If $\dfrac{3}{2+e^{i\theta}} = ax + iby$, then the locus of $P(x, y)$ will represent

How many solutions does the system of equations $\||z + 4| - |z - 3i|\| = 5$ and $|z| = 4$ have?

Let z be a non-real complex number which satisfies the equation $z^{23} = 1$. Then the value of $\displaystyle\sum_{k=1}^{22} \dfrac{1}{1+z^{8k}+z^{16k}}$ is ___.

If complex number $z = x + iy$ satisfies the equation $\text{Re}(z + 1) = |z - 1|$, then $z$ lies on:

How many solutions does the system of equations $\arg(z + 2 - 3i) = -\pi/4$ and $|z + 4| - |z - 3i| = 5$ have?

If $z_1, z_2, z_3$ are distinct nonzero complex numbers and $a, b, c \in \mathbb{R}^+$ such that $\dfrac{a}{|z_1 - z_2|} = \dfrac{b}{|z_2 - z_3|} = \dfrac{c}{|z_3 - z_1|}$, then find the value of $\dfrac{a^2}{z_1 - z_2} + \dfrac{b^2}{z_2 - z_3} + \dfrac{c^2}{z_3 - z_1}$.

If α and β are imaginary cube roots of unity, then find the value of α4 + β4 + 1/(αβ).

Let $z = x + iy$ be a complex number, where $x$ and $y$ are real numbers. Let $A$ and $B$ be the sets defined by $A = \{z : |z| \le 2\}$ and $B = \{z : (1-i)z + (1+i)\bar{z} \ge 4\}$. Find the area of region $A \cap B$.

For Problems 14–16: Consider a quadratic equation $az^2 + bz + c = 0$, where $a, b, c$ are complex numbers.If equation has two purely imaginary roots, then which of the following is not true.

For every large value of $\lambda$, the roots are approximately

For Problems 17–19: Suppose $z$ and $\omega$ are two complex numbers such that $|z| \leq 1$, $|\omega| \leq 1$, and $|z + i\omega| = |z - i\bar{\omega}| = 2$.The complex number $\omega$ can be

For Problems 26–28: Complex numbers $z$ satisfy the equation $|z - (4/z)| = 2$.The difference between the least and the greatest moduli of complex numbers is

For Problems 23–25: Consider the equation $az + b\bar{z} + c = 0$, where $a, b, c \in \mathbb{Z}$.If $|a| = |b| \neq 0$ and $\bar{a}c = b\bar{c}$, then $az + b\bar{z} + c = 0$ represents

If a complex number $z$ satisfies $|2z + 10 + 10i| \leq 5(\sqrt{3} - 5)$, then the least principal argument of $z$ is

If the complex number associated with the vertices $A$, $B$, $C$ of $\triangle ABC$ are $e^{i\theta}$, $\omega$, $\bar{\omega}$, respectively [where $\omega$, $\bar{\omega}$ are the complex cube roots of unity and $\cos\theta > \text{Re}(\omega)$], then the complex number of the point where the angle bisector of $A$ meets the circumcircle of the triangle is

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

If the equation $z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$, where $a_1, a_2, a_3, a_4$ are real coefficients different from zero, has a purely imaginary root, then the expression $\dfrac{a_3}{(a_1 a_2)} + \dfrac{a_1 a_4}{(a_2 a_3)}$ has the value equal to

Let $z = 1 - t + i\sqrt{t^2 + t + 2}$, where $t$ is a real parameter. The locus of $z$ in the Argand plane is

If z1, z2, z3, ..., zn are n nth roots of unity, then for k = 1, 2, 3, ..., n

The value of \sum_{k=1}^{6} \left(\sin\frac{2\pi k}{7} + i\cos\frac{2\pi k}{7}\right), where i = \sqrt{-1}, 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 $z$ be a complex number with nonzero imaginary part and $a=z^2+z+1$ is real. Then '$a$' cannot take the value

Let $\alpha_1,\alpha_2,\ldots,\alpha_8$ be roots of $1+z+z^2+\cdots+z^8=0$. Three dice thrown; sum is $n$. Probability that $\sum_{i=1}^8(\alpha_i)^n=8$ is $a/b$ (coprime). Then $|7a-b|$

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 any complex number such that $|z + 4| \leq 3$, then find the greatest value of $|z + 1|$.

If $z=x+iy$ and $|z-1|=|z+1|$, then

Area bounded by $-8z^2-8\bar{z}^2+20z\bar{z}=144$ (where $z$ is a complex number)

If a, b, c are three complex numbers on the unit circle . Then find the value of .

The number of points in the complex plane satisfying both $|z-2|=2$ and $z(1-i)+\bar{z}(1+i)=4$ (where $i=\sqrt{-1}$) is

A value of $\theta$ for which $\dfrac{2+3i\sin\theta}{1-2i\sin\theta}$ is purely imaginary is

Let $(1)^{1/n} = z \Rightarrow z^n - 1 = 0$. For $n = 11$, $z^{11} - 1 = 0$. If $\alpha = \cos 0 + i\sin 0 = 1$ and $\alpha_k = \cos\left(\dfrac{-2\pi k}{11}\right) + i\sin\left(\dfrac{-2\pi k}{11}\right) - 1$, then $(\alpha_k)^{1/11}$ equals:

Let z = x + iy. If z satisfies $(2iy)^2 = 12(x^2 + y^2) - 4$, find the maximum value of $3\sqrt{3}\,\text{Re}(z) + 8\,\text{Im}(z)$.

If $|z+\bar{z}|+|z-\bar{z}|=4$ and $|z+i|+|z-i|=4$, then the area of region in which $z$ lies is

Match List-I with List-II (for $z=x+iy$, $x,y\in\mathbb{R}$): P) Area of triangle formed by $z,\omega z,z+\bar{\omega}z$ is $16\sqrt{3}$ then $|z|$; Q) Area of locus of $z$ if $|\text{Re}\,z|+|\text{Im}\,z|=3$; R) If $|z+6/z|=5$ then maximum of $|z|$; S) Number of complex numbers $z$ satisfying $z^2=\bar{z}$

Write the complex number in $a + ib$ form using cube roots of unity:(a) $\left(-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i\right)^{1000}$(b) If $z = \dfrac{(\sqrt{3}+i)^{17}}{(1-i)^{50}}$(c) $(i + \sqrt{3})^{100} + (i - \sqrt{3})^{100} + 2^{100}$

If $\arg\!\left(\dfrac{z-(10+6i)}{z-(4+2i)}\right)=\dfrac{\pi}{4}$, then the perimeter of locus of $z$ is

If center of a regular hexagon is at the origin and one of the vertices on the Argand diagram is $1 + 2i$, then its perimeter is

Let $z_1=3+3i$, $z_2=9-6\sqrt2+(6\sqrt2-3)i$, $\triangle ABC$ right-angled at $C$. Min and max possible values of $\arg(z_3)$ are $a,b$ ($\arg z_3\in(-\pi,\pi]$). Then $b-a=\dfrac{\pi}{k}$. Find $k$.

If $z$ is a complex number lying in the fourth quadrant of the Argand plane and $|[kz/(k+1)] + 2i| > \sqrt{2}$ for all real values of $k$ ($k \neq -1$), then range of $\arg(z)$ is

If $\omega = \dfrac{2}{z - \dfrac{1}{3}i}$ and $|\omega| = 1$, then $z$ lies on

Complex Numbers Questions (654)