Complex Numbers Questions - JEE

If $ \omega $ is a primitive $n$th root of unity, then $ 1 + \omega + \omega^2 + \cdots + \omega^{n-1} $ equals:

The solutions of the equation z^4 + 4iz^3 - 6z^2 - 4iz - i = 0 represent vertices of a convex polygon in the complex plane. The area of the polygon is:

A rectangle of maximum area is inscribed in the circle $|z - 3 - 4i| = 1$. If one vertex of the rectangle is $4 + 4i$, then another adjacent vertex of this rectangle can be

For the regular hexagon ABCDEF with $z_1 = -2$ and $z_3 = 1 - \sqrt{3}i$, the square of the inradius of the hexagon is equal to

Locus of the point z satisfying the equation |iz - 1| + |z - 1| = 2, is (where i = \sqrt{-1})

Let $z = r(\cos\theta + i\sin\theta)$. The least value of $\dfrac{\sin 5\theta}{(\sin\theta)^5}$ expressed in terms of $\cot\theta$ is:

Let the product of$\$omega = (8 + i)$sin$$\$theta + (7 + 4i)$cos$$\theta and$$\$omega = (1 + 8i)$sin$$\$theta + (4 + 7i)$cos$$\theta be$$\alpha + i$$\beta, 1 2 i =$$\$sqrt-1$. Let p and q be the maximum and the minimum values of$$\alpha +$$\beta respectively.$

If $z_1$ and $z_2$ are the complex roots of the equation $(x-3)^3 + 1 = 0$, then $z_1 + z_2$ equals

If $z$ is a complex number having least modulus and $z - 2 - 2i| = 1$, then $z =$

Given $\alpha, \beta$, respectively, the fifth and the fourth non-real roots of unity, respectively, then find the value of $(1 + \alpha)(1 + \beta)(1 + \alpha^2)(1 + \beta^2)(1 + \alpha^4)(1 + \beta^4)$.

If $\dfrac{z-i}{z+i}$ is a purely imaginary number (where $z \neq -i$), then which of the following is true?

Find $a^{10} + \bar{a}^{10}$ where $a = \frac{2-\sqrt{-3}}{3}$

(D) $2, 3 + 2$

If $|z_2 + iz_1| = |z_1| + |z_2|$ and $|z_1| = 3$ and $|z_2| = 4$, then the area of $\triangle ABC$, if affixes of $A$, $B$, and $C$ are $z_1, z_2$, and $[(z_2 - iz_1)/(1-i)]$ respectively, is

For Problems 26–28: Complex numbers $z$ satisfy the equation $|z - (4/z)| = 2$.Locus of $z$ if $|z - z_1| = |z - z_2|$, where $z_1$ and $z_2$ are complex numbers with the greatest and the least moduli, is

Sum of common roots of the equations $z^3 + 2z^2 + 2z + 1 = 0$ and $z^{1985} + z^{100} + 1 = 0$ is

For the circle with center $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, find the condition on $(Z-1)^n = Z^n \Rightarrow |Z-1| = |Z| \Rightarrow x = \frac{1}{2}$. Given $\frac{1}{2} + y^2 = 1 \Rightarrow y = \pm\frac{\sqrt{3}}{2}$. Which would exist when $n$ is a multiple of $6$: $\arg\left(\frac{1}{2} + \frac{\sqrt{3}}{2}i\right) = \tan^{-1}\sqrt{3} = \frac{\pi}{3}$; least value of $n$ is equal to $\frac{2\pi}{6}$

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

One root lies inside the unit circle and one outside if

If $a$ and $b$ are complex and one of the roots of the equation $x^2 + ax + b = 0$ is purely real whereas the other is purely imaginary, then

A complex number $z$ is said to be unimodular if $|z| = 1$. Let $z_1$ and $z_2$ are complex numbers such that $\frac{z_2 - z_1}{z_2 - z_3}$ is unimodular and $z_2$ is not real. Then $z_1$ lies on a

If $|z - i\,\text{Re}(z)| = |z - \text{Im}(z)|$, then prove that $z$ lies on the bisectors of the quadrants.

The minimum value of the expression $|3z - 3| + |2z - 4|$ is equal to (where, $z$ is a complex number)

Let $z_1$ and $z_2$ be two roots of the equation $z^2 + az + b = 0$, $z$ being complex. Further, assume that the origin, $z_1$ and $z_2$ form an equilateral triangle, then

The complex numbers $\sin x + i\sin 2x$ and $\cos x - i\cos 2x$ are conjugate to each other, for

If $|(z - z_1)/(z - z_2)| = 3$, where $z_1$ and $z_2$ are fixed complex numbers and $z$ is a variable complex number, then $z$ lies on a

If $\left(\dfrac{1+i}{1-i}\right)^x = 1$, then

A complex number $z$ is said to be unimodular if $|z| = 1$. Suppose $z_1$ and $z_2$ are complex numbers such that $\dfrac{z_1 - 2z_2}{2 - z_1\bar{z}_2}$ is unimodular and $z_2$ is non-unimodular. Then the point $z_1$ lies on a

The largest value of $r$ for which the region represented by the set $\{\omega \in \mathbb{C} : |\omega - 4 - i| \leq r\}$ is contained in the region represented by the set $\{z \in \mathbb{C} : |z - 1| \leq |z + i|\}$, is equal to

Find the locus of the points representing the complex number $z$ for which $|z+5|^2 - |z-5|^2 = 10$.

Find the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\]

If z = x + iy, then the equation |(2z − i)/(z + 1)| = m represents a circle, then m can be

Find the greatest and the least value of $|z_1 + z_2|$ if $z_1 = 24 + 7i$ and $|z_2| = 6$.

If $|z - 2 - 3i|^2 + |z - 5 - 7i|^2 = \lambda$ represents the equation of a circle with least radius, then find the value of $\lambda$.

For any complex number $z$, find the minimum value of $|z| + |z - 2i|$.

The point represented by $2+i$ in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $2\sqrt{2}$ units in the south-westwards direction. Then its new position in the Argand plane is at the point represented by

If

If 1, a_1, a_2, \ldots, a_{n-1} are n roots of unity, then the value of (1 - a_1)(1 - a_2)(1 - a_3) \cdots (1 - a_{n-1}) is equal to

What is the digit in the unit's place of $(1008)^{786}$?

Let \omega be the imaginary cube root of unity and (a + b\omega + c\omega^2)^{2015} = (a + b\omega^2 + c\omega) where a, b, c are unequal real numbers. Then the value of a^2 + b^2 + c^2 - ab - bc - ca equals:

If \text{Im}\left(\frac{z-1}{2z+1}\right) = -4, then locus of z is

If x^2 - x + 1 = 0, then the value of \sum_{n=1}^{5} x^n + x^{-n} is

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

Let four points $z_1, z_2, z_3, z_4$ be in complex plane such that \(|z_1|

All three roots of az3 + bz2 + cz + d = 0, have negative real part, where (a, b, c ∈ ℝ). Then:

If a = \cos \alpha + i \sin \alpha, b = \cos \beta + i \sin \beta, c = \cos \gamma + i \sin \gamma and \frac{b}{c} + \frac{c}{a} + \frac{a}{b} = 1, then \cos(\beta - \gamma) + \cos(\gamma - \alpha) + \cos(\alpha - \beta) is equal to

Let z be a complex number satisfying |z - 3| ≤ |z - 1|, |z - 3| ≤ |z - 5|, |z - i| ≤ |z + i| and |z - i| ≤ |z - 5i|. Then the area of region in which z lies is A square units, where A = ?

730. Let $z$ ($z \in$ complex number) be one of the roots of the equation $x^2 - (\log_2 \alpha - \log_2 \beta)x + \cos\alpha - \sin\beta = 0$. If the harmonic mean of the roots is 2 and $|z| = 1$, find the sum of all values of $\beta$ in degrees when \(0

The sequence S = i + 2i2 + 3i3 + 4i4 + ... up to 100 terms simplifies to, where i = $\sqrt{-1}$

Find the locus of a complex number z = x + iy, which satisfy the equation $\frac{z - 5i}{z + 5i} = 1$.

Complex Numbers Questions (654)