Complex Numbers Questions - JEE

Find the sum of squares of all roots of the equation $x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = 0$.

Let A, B, C, D be four concyclic points in order in which AD:AB = CD:CB. If A, B, C are represented by complex numbers a, b, c, respectively, find the complex number associated with point D.

Find the real numbers $x$ and $y$ if $(x - iy)(3 + 5i)$ is the conjugate of $-6 - 24i$.

If $z \leq 4$, then find the maximum value of $|iz + 3 - 4i|$.

If $\omega = \alpha + i\beta$, where $\beta \neq 0$, $i = \sqrt{-1}$ and z \neq 1\), satisfies the condition that $\frac{\omega - \omega z}{1 - z}$ is purely real, the set of values of z\) is

If z1, z2, z3 and z4 are the roots of the equation z4 + 1 = 0, the value of \frac{1}{z_1 + i} + \frac{1}{z_2 + i} + \frac{1}{z_3 + i} + \frac{1}{z_4 + i} is

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

The triangle formed by the complex numbers $z$, $iz$, $i^2z$ is:

If the arguments of $(1-i)(\sqrt{3}+i)(1+\sqrt{3}i)$ and $(z-2)\left(\frac{-1}{z-1}\right)$ are equal, then the locus of $Z$ is part of a circle with centre $(a,b)$. The value of $\frac{a}{b+1}$ is

The conjugate of a complex number is $\dfrac{1}{i-1}$. Then the complex number is

We have $ S = \dfrac{\alpha + i}{\alpha - i} $ If $ S = x + iy $, find $ x^2 + y^2 $.

If $z$ is a complex number such that $|z|\ge 1$, then the minimum value of $\left|z+\frac{1}{2}(3+4i)\right|$ is: [Note: In official NTA paper no option was correct.]

Given $|z - 3 + 2i| \leq 4$, find $|z|_{\max} - |z|_{\min}$.

If $P(z)$ is a variable point in the complex plane such that $\tan\left(-\frac{1}{z}\right) = \frac{1}{2}$, then the value of the perimeter of the locus of $P(z)$ is (use $\pi = 3.14$)

The values of (16)^{1/4} are

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

If ω is a complex cube root of unity, then find the value of $(1 + \omega)(1 + \omega^2)(1 + \omega^4)(1 + \omega^8) \cdots$ to 2n factors.

If w = a + i b, where b ≠ 0 and z ≠ 1, satisfies the condition that $\frac{w - wz}{1 - z}$ is purely real, then the set of values of z is

Find the imaginary part of (3 + 2√(-54))1/2 - (3 - 2√(-54))1/2.

If $z_1, z_2$ and $z_3, z_4$ are 2 pairs of complex conjugate numbers, then $\arg\left(\dfrac{z_1}{z_4}\right)+\arg\left(\dfrac{z_2}{z_3}\right)$ equals

For all complex numbers z of the form $1 + i\alpha,\ \alpha \in R$, if $z^2 = x + iy$, then

If $z$ is a nonreal root of $\sqrt[7]{-1}$, then find the value of $z^{86} + z^{175} + z^{289}$.

If $|z_1| = \sqrt{2}$, $|z_2| = \sqrt{3}$ and $|z_1 + z_2| = \sqrt{5 - 2\sqrt{3}}$, then $\arg\left(\dfrac{z_1}{z_2}\right)$ (not necessarily principal) is

In a triangle ABC, the side lengths BC, CA and AB are consecutive positive integers in increasing order. Let $z_1$, $z_2$ and $z_3$ be the affixes of vertices A, B and C respectively in argand plane, such that $\arg\left(\frac{z_1 - z_3}{z_2 - z_3}\right) = 2\arg\left(\frac{z_3 - z_1}{z_2 - z_1}\right)$. Find the biggest side of the triangle.

$z_1, z_2, z_3, z_4$ are distinct complex numbers representing the vertices of a quadrilateral $ABCD$ taken in order. If $z_1 - z_4 = z_2 - z_3$ and $\arg[(z_4 - z_1)/(z_2 - z_1)] = \pi/2$, then the quadrilateral is

If $\frac{z - a}{z + a}$ (a ∈ ℝ) is a purely imaginary number and |z| = 2, then a value of a is:

Let complex numbers α and 1 lie on circles (x - x₀)² + (y - y₀)² = r² and (x - x₀)² + (y - y₀)² = 4r², respectively. If z₀ = x₀ + iy₀ satisfies the equation 2|z₀|² = r² + 2, then |α| equals to

261. Let $z$ be the complex number satisfying $|z + 16| = 4|z + 1|$, then:

For any integer k, let $\alpha_k = \cos\dfrac{k\pi}{7} + i\sin\dfrac{k\pi}{7}$, where $i = \sqrt{-1}$. Value of the expression $\dfrac{\displaystyle\sum_{k=1}^{12}|\alpha_{k+1} - \alpha_k|}{\displaystyle\sum_{k=1}^{3}|\alpha_{4k-1} - \alpha_{4k-2}|}$ is _______.(JEE Advanced 2015)

Solve the equation $|z| = z + 1 + 2i$.

If $z^2 + z + 1 = 0$, where $z$ is a complex number, then the value of $\left(z + \dfrac{1}{z}\right)^2 + \left(z^2 + \dfrac{1}{z^2}\right)^2 + \left(z^3 + \dfrac{1}{z^3}\right)^2 + \cdots + \left(z^6 + \dfrac{1}{z^6}\right)^2$ is

Let z_1 and z_2 be two complex numbers satisfying |z_1| = 9 and |z_2 - 3 - 4i| = 4. Then, the minimum value of |z_1 - z_2| is

Given $z = 1 + i\alpha \Rightarrow z^2 = 1 - \alpha^2 + 2i\alpha$If $x + iy = z^2$, then which of the following is satisfied?(2) $y^2 + 4x - 4 = 0$

Find the value of $x^4 + 9x^3 + 35x^2 - x + 4$ for $x = -5 + 2\sqrt{-4}$.

All the points in the set $S = \left\{\dfrac{\alpha+i}{\alpha-i}; \alpha \in R\right\}$ $(i=\sqrt{-1})$ lie on a

If $ |z-1-2i|+|z+1+2i|=4 $, then $z$ lies on:

Let $z, w$ be complex numbers such that $\bar{z}+i\bar{w}=0$ and $\arg\, zw = \pi$. Then $\arg\, z$ equals

We have $S = \dfrac{\alpha + i}{\alpha - i}$ where $\alpha$ is a real number. Then $x^2 + y^2$ (where $S = x + iy$) equals:

If 1, \omega, \omega^2, \omega^3, \ldots, \omega^{n-1} are n nth roots of unity, then (1 - \omega)(1 - \omega^2)(1 - \omega^3)\cdots(1 - \omega^{n-1}) equals

If $|z^2 - 1| = |z|^2 + 1$, then $z$ lies on:

Let $z = x + iy$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation $z\bar{z}^3 + \bar{z}z^3 = 350$ is

If $a, b, c$ are nonzero real numbers and $az^2 + bz + c + i = 0$ has purely imaginary roots, then prove that $a = b^2c$.

If z1 and z2 be the nth root of unity which subtend a right angle at the origin, then n must be of the form

The polynomial $x^6 + 4x^5 + 3x^4 + 2x^3 + x + 1$ is divisible by (where $\omega$ is the cube root of unity)

The value of $(1 + \omega - \omega^2)^7$ is:

Simplify $\dfrac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}$.

Given equation $\frac{z^7 + 1}{z^3 + 1} = 0$ where $z \neq -1$. Find $z$ and the sum of all values.

Substituting z = x + iy, the equation 2|z + 3i| − |z − i| = 0 represents a circle. Find the radius of this circle.

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

If $J$ and $K$ are represented by points A and B in argand plane, then circumradius of $\triangle OAB$, where O is origin, is

Complex Numbers Questions (654)