Complex Numbers Questions - JEE

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

If $\arg\left(\dfrac{z-2}{z-2i}\right)=\dfrac{\pi}{4}$, then which of the following is correct?

In $\triangle ABC$: $A=2025\omega+2024i$, $B=2024i\omega^2+2025$, $C=2024\omega^2-2025i$ where $\omega$ is non-real cube root of unity with positive imaginary part. The internal bisector of $\angle ACB$ meets $AB$ at $D$ and $\angle BDC=\frac{k\pi}{24}$. Then $k$ is

The points $z_1 = 3 + \sqrt{3}\,i$ and $z_2 = 2\sqrt{3} + 6i$ are given on a complex plane. The complex number lying on the bisector of the angle formed by the vectors $z_1$ and $z_2$ is

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

Let $S_1=\{z\in\mathbb{C}:|z-2|\leq|\text{Re}(z)+2|\}$, $S_2=\{z\in\mathbb{C}: z(1+i)+\bar{z}(1-i)-12\leq 0\}$, $S_3=\{z\in\mathbb{C}:\text{Re}(z)\geq 0,\text{Im}(z)\geq 0\}$ and $S=S_1\cap S_2\cap S_3$. The maximum value of $|z-2i|^2$; $z\in S$ is

Find all nonzero complex numbers $z$ satisfying $\bar{z} = iz^2$.

If n is a natural number ≥ 2, such that $z^n = (z+1)^n$, then

If $z = re^{i\theta}$, then prove that $|e^{iz}| = e^{-r\sin\theta}$.

If $|z_1| = |z_2| = |z_3| = 1$ and $z_1 + z_2 + z_3 = 0$, then the area of the triangle whose vertices are $z_1, z_2, z_3$ is

For Problems 20–22: Consider the equation of line $a\bar{z} + \bar{a}z + b = 0$, where $b$ is a real parameter and $a$ is fixed non-zero complex number.The locus of mid-point of the line intercepted between real and imaginary axis is given by

If $\sqrt{3} + i = (a + ib)(c + id)$, then find the value of $\tan^{-1}(b/a) + \tan^{-1}(d/c)$.

If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg\left(\dfrac{1+z}{1+\bar{z}}\right)$ equals

Find the common roots of $x^{12} - 1 = 0$ and $x^4 + x^2 + 1 = 0$.

Find nonzero integral solutions of $|1 - i|^x = 2^x$.

If $z_1, z_2, z_3$ are three distinct complex numbers such that $\dfrac{1}{|z_2 - z_3|} = \dfrac{2}{|z_3 - z_1|} = \dfrac{3}{|z_1 - z_2|}$, then $\dfrac{1}{(z_2 - z_3)} + \dfrac{4}{(z_3 - z_1)} + \dfrac{1}{(z_1 - z_2)} =$

Find the minimum value of $|z - 1|$ if $\||z - 3| - |z + 1|\| = 2$.

If a < 0, b > 0, then $\sqrt{a} \cdot \sqrt{b}$ is equal to

If α, β ∈ C are the distinct roots of the equation $x^2 - x + 1 = 0$, then $\alpha^{101} + \beta^{107}$ is equal to

If z is any complex number such that $|3z - 2| + |3z + 2| = 4$, then identify the locus of z.

For Problems 17–19: Suppose $z$ and $\omega$ are two complex numbers such that $|z| \leq 1$, $|\omega| \leq 1$, and $|z + i\omega| = |z - i\bar{\omega}| = 2$.Which of the following is true for $z$ and $\omega$?

Let $z = x - iy$ and $z^{1/3} = p + iq$. If $\dfrac{x}{p} + \dfrac{y}{q} = k(p^2 + q^2)$, then $k$ is:

Let $ x_1, x_2, \ldots, x_{10} $ be the roots of the polynomial equation $ x^{10} + x^9 + \cdots + x + 1 = 0 $. Then the value of \[ \sum_{n=1}^{10} \left( \frac{1}{1 - x_n} \right): \]

Let $z$ be a complex number satisfying $|z| = 3|z - 1|$. Then prove that $\left|z - \dfrac{9}{8}\right| = \dfrac{3}{8}$.

If $\text{Im}\left(\dfrac{z-1}{e^{\theta i}} + \dfrac{e^{\theta i}}{z-1}\right) = 0$, then find the locus of $z$.

For Problems 12 and 13: The roots of the equation $z^4 + az^3 + (12+9i)z^2 + bz = 0$ (where $a$ and $b$ are complex numbers) are the vertices of a square.The area of the square is

The number of solutions of the equation $z^2 + \bar{z} = 0$ is

Number of distinct quadratic equations with real roots such that the equation remains unchanged if their roots are cubed is equal to

If $|z - 1| + |z + 3| \leq 8$, then find the range of values of $|z - 4|$.

Given $\left(\dfrac{1+i\sqrt{3}}{1-i\sqrt{3}}\right)^n = 1$, find the least positive integer value of $n$.

Find the modulus, argument and the principal argument of the complex number $(\tan 1 - i)^2$.

Let $z$ be a complex number satisfying the equation $(z^3 + 3)^2 = -16$, then find the value of $|z|$.

For Problems 20–22: Consider the equation of line $a\bar{z} + \bar{a}z + b = 0$, where $b$ is a real parameter and $a$ is fixed non-zero complex number.The intercept of line on imaginary axis is given by

Let the set of complex numbers $(a_1, b_1), (a_2, b_2), (a_3, b_3)$........ denoting the points on the complex plane satisfying $(a_{n+1}, b_{n+1}) = (\sqrt{3}\, a_n - b_n,\, \sqrt{3}\, b_n + a_n)$ for $n = 1, 2, 3, \ldots$ Suppose $(a_{100}, b_{100}) = (2, 4)$, then the value of $(a_1 + b_1)$ is equal to:

If $\omega$ is a non-real cube root of unity, then the value of $\dfrac{a + b\omega + c\omega^2}{b + c\omega + a\omega^2} + \dfrac{a + b\omega + c\omega^2}{c + a\omega + b\omega^2}$ is equal to:

If the imaginary part of \ $\dfrac{2z+1}{iz+1}$ is \ $-2$, then find the locus of the point \ $z$ representing in the complex plane.

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

If z_1, z_2 ∈ ℂ, z_1^2 + z_2^2 ∈ ℝ, z_1(z_1^2 + 3z_2^2) = 2 and z_2(3z_1^2 + z_2^2) = 11, find the value of z_1^2 + z_2^2.

Find the number of solutions of the equations $|z - (4 + 8i)| = 10$ and $|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}$, where $i = \sqrt{-1}$.

Let w = (√3 + i)/2 and P = {wⁿ : n = 1, 2, 3, ...}. Further, H₁ = {z ∈ ℂ : Re(z) > -1/2} and H₂ = {z ∈ ℂ : Re(z) . Let S = P ∩ H₁ ∩ H₂. Then the number of elements in S

The value of $\sum_{k=1}^{10} \left(\sin\frac{2k\pi}{11} + i\cos\frac{2k\pi}{11}\right)$ (where $i = \sqrt{-1}$) is

Let z and w be complex numbers. If $\text{Re}(z) = |z - 2|$, $\text{Re}(w) = |w - 2|$, and $\arg(z - w) = \frac{\pi}{3}$, find the value of $\text{Im}(z + w)$.

What is the digit in the unit's place of $(13057)^{941120579}$?

What is the digit in the unit's place of $(1354)^{22222}$?

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

If z_1, z_2, z_3 and z_4 are the affixes of four points in the Argand plane and z is the affix of a point, such that |z - z_1| = |z - z_2| = |z - z_3| = |z - z_4|, then z_1, z_2, z_3 and z_4 are

If $\omega$ and $\omega^2$ are the nonreal cube roots of unity and $[1/(a+\omega)] + [1/(b+\omega)] + [1/(c+\omega)] = 2\omega^2$ and $[1/(a+\omega^2)] + [1/(b+\omega^2)] + [1/(c+\omega^2)] = 2\omega$, then find the value of $[1/(a+1)] + [1/(b+1)] + [1/(c+1)]$.

Complex Numbers Questions (654)