Complex Numbers Questions (654)

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 \(\frac{z - 5i}{z + 5i} = 1\).

Let i = √(-1), define a sequence of complex numbers by z1 = 0, zn+1 = zn2 + i for n ≥ 1. In the complex plane, how far from the origin is z111?

If a = cos θ + i sin θ, then \(\frac{1+a}{1-a}\) is equal to

The sum of all complex numbers \(z\) satisfying \(z^3-(\bar{z})^2=0\) is ___.

If |z₁| and |z₂| are the distances of points on the curve 5zz̄ - 2i(z² - z̄²) - 9 = 0 which are at maximum and minimum distance from the origin, then the value of |z₁| + |z₂| is equal to:

Let 1/(a₁ + ω) + 1/(a₂ + ω) + 1/(a₃ + ω) + ... + 1/(aₙ + ω) = i where a₁, a₂, a₃, ..., aₙ ∈ ℝ and ω is an imaginary cube root of unity. Then evaluate ∑ᵣ₌₁ⁿ (2aᵣ - 1)/(aᵣ² - aᵣ + 1).

The points A, B and C represent the complex numbers z1, z2, (1 - i)z1 + iz2 respectively, on the complex plane. The △ABC is

The centre of the circle represented by |z + 1| = 2|z - 1| on the complex plane, is

Find the range of real number \(\alpha\) for which the equation \(z + \alpha|z - 1| + 2i = 0\) has a solution.

If centre of a regular hexagon is at origin and one of the vertex on argand diagram is 1 + 2i, where i = \sqrt{-1}, its perimeter is

Let P and Q be two points on the circle \(|w| = r\) represented by \(w_1\) and \(w_2\) respectively. Then the complex number representing the point of intersection of the tangents at P and Q is:

If \alpha = \cos \frac{2\pi}{7} + i\sin \frac{2\pi}{7}, then the quadratic equation whose roots are \beta = \alpha + \alpha^2 + \alpha^4 and \gamma = \alpha^3 + \alpha^5 + \alpha^6 is

If \( z = x+iy \) and \( \arg\left(\dfrac{z-1}{z+1}\right) = \dfrac{\pi}{4} \), then the locus of \( z \) is:

Which of the following is true?

Find the principal argument of (c) \(\sin\alpha + i(1-\cos\alpha)\), \(0

If f(x) = g(x^3) + xh(x^3) is divisible by x^2 + x + 1, then

If \(\alpha^{14}+\alpha^{10}+\alpha^6+\alpha^2+1=0\), then \(\alpha\) can be:

If x = 4 + 3i\ (where i = \sqrt{-1}\), then the value of x^3 - 4x^2 - 7x + 12\ equals:

If \cos \alpha + \cos \beta + \cos \gamma = \sin \alpha + \sin \beta + \sin \gamma = 0, then \cos 3\alpha + \cos 3\beta + \cos 3\gamma is equal to

Let $z$ and $w$ be non-zero complex numbers such that $zw = |z^2|$ and $|w| = \left|\frac{z}{w}\right| = 4$. If $w$ varies, then the perimeter of the locus of $z$ is

If $z$ and $w$ are two non-zero complex numbers such that $|zw| = 1$ and $\arg(z) - \arg(w) = \frac{\pi}{2}$, then the value of $5iz + w$ is equal to

For a complex number $Z$, if all the roots of the equation $Z^3 + aZ^2 + bZ + c = 0$ are unimodular, then

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 __________.

If |z - 2 - 3i| - |z + 2 - 6i| = 4, where i = \sqrt{-1}, then locus of P(z) is

If w be an imaginary nth root of unity, then ∑r=1n(ar + b)wr-1 is equal to:

If |z₁| = 1, |z₂| = 2, |z₃| = 3 and |9z₁z₂ + 4z₁z₃ + z₂z₃| = 36, then |z₁ + z₂ + z₃| is equal to:

If α = \cos \frac{8π}{11} + i \sin \frac{8π}{11}, where i = \sqrt{-1}, then \text{Re}(α + α^2 + α^3 + α^4 + α^5) is

Let \(z\) lie on \(|z-1|+|z+1|=4\). Find \(7\left(\dfrac{x^2}{4}+\dfrac{y^2}{3}\right)\).

A complex number \(z\) satisfies a certain algebraic condition. It lies on:

If \( \arg\left(\dfrac{z-1}{z+1}\right) = \dfrac{\pi}{4} \), then the locus of \(z\) is:

If \( z = \dfrac{3+4i}{4-3i} \), the principal argument of \(z\) is:

The number of complex numbers \(z\) satisfying \(|z-3|=|z-7|\) and \(|z-3i|=4\) is ___.

If the imaginary part of \(\dfrac{2z+1}{iz+1}\) is \(-2\), then the locus of \(z=x+iy\) is a straight line. Find \(310a\) where \(a\) is its slope.

If \( \omega \) is a non-real cube root of unity, then \( (1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 \) is:

Let the two roots of $z^2 + 2z + 2 = 0$ be $\alpha$ and $\beta$.

995. If complex number \(z\) satisfies \((z - \bar{z})^2 = 12|z|^2 - 4\) then find the maximum value of \(3\sqrt{3}\,\text{Re}(z) + 8\,\text{Im}(z)\).

The set of points in an Argand diagram which satisfy both |z| ≤ 4 and 0 ≤ \arg(z) ≤ \frac{π}{3}, is

If α, β, γ are the roots of equation \(x^3 - 3x^2 + 3x + 7 = 0\) and ω is a cube root of unity, then find the value of \(\dfrac{\alpha - 1}{\beta - 1} + \dfrac{\beta - 1}{\gamma - 1} + \dfrac{\gamma - 1}{\alpha - 1}\).

If $\alpha + 1 + i = a$, $\alpha(1 + i) = a$, $\alpha + \alpha i = \alpha + 1 + i$, $\alpha = \frac{1+i}{i} = 1 - i$, $\alpha = 1 - i + 1 + i = 2$, and $\alpha^4 = (\alpha^2)^2 = (1 + (-1))^2 = 0^2 = (2i)^2 = -4$, then find the value of $\alpha^4$.

If the complex numbers $Z_1, Z_2$ and $Z_3$ are the vertices $A, B$ and $C$ respectively of an isosceles right-angled triangle $ABC$ with right angle at $C$, then the value of $-\frac{(Z_1 - Z_3)^2}{Z_2 - Z_1)(Z_2 - Z_3)}$ is equal to

Find the multiplicative inverse of the complex number \(4 - 3i\).

If \(Z = \frac{7 + i}{3 + 4i}\), then find \(Z^{14}\):

Let z be a complex number having the argument \(\theta\), \(0

If $\lambda \in \mathbb{R}$ such that the origin and the non-real roots of the equation $2z^2 + 2z + \lambda = 0$ form the vertices of an equilateral triangle in the argand plane, then $\lambda$ is equal to

Express the following in a + ib form:(a) \(\dfrac{(\cos\alpha + i\sin\alpha)^4}{(\sin\beta + i\cos\beta)^5}\)

If $zw = |z|^2$ and $zw = z\bar{z}$, find $|z|$ given that $|z| = 4$

If \(\sqrt{x+iy} = \pm(a+ib)\), then find \(\sqrt{-x-iy}\).