Complex Numbers Questions (654)

If \(\omega \neq 1\) is a cube root of unity, and \((1 + \omega)^7 = A + B\omega\). Then \((A, B)\) equals

If \(z^2 + az + b = 0\) has roots \(z_1\) and \(z_2\), and \(O\) (origin), \(z_1\), \(z_2\) form an equilateral triangle, then which of the following is true?

Find the principal argument of (d) \((1+i\sqrt{3})^2\)

Since \(\omega^3 = 1\) and \(\omega^2 + \omega + 1 = 0\), find the value of \((1+\omega)(1+\omega^2)(1+\omega^3)\cdots(1+\omega^{1987})\), given that \(1987 = 662 \times 3 + 1\).

Let the set of complex numbers \((a_1, b_1), (a_2, b_2), (a_3, b_3)\ldots\) 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:

The locus of \(z\) is \(x^2 + y^2 = 1\). If \(\lambda\) is the maximum value of \(2\left(|z_1 - 2| + \left|z_2 + \dfrac{1}{2}\right|\right)\) and \(4\lambda = 36\), find \(\lambda\).

If \(|Z_1 + Z_2| = |Z_1| + |Z_2|\), then which of the following is correct?

Let \(z = x + iy\). The maximum distance between the two points on the curve \(2|2x| = 32 + \left(\dfrac{2y}{i}\right)^2\) is ___.

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

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

Find roots of the equation \((z + 1)^5 = (z - 1)^5\).

If \(z\) is a complex number, then find the minimum value of \(|z| + |z - 1| + |2z - 3|\).

System of equations |z + 3| − |z − 3| = 6 and |z − 4| = r, where r ∈ ℝ⁺ has

Find the value of the following expression:\[\left[\frac{1-\cos\dfrac{\pi}{10}+i\sin\dfrac{\pi}{10}}{1-\cos\dfrac{\pi}{10}-i\sin\dfrac{\pi}{10}}\right]^{10}\]

If 1, \(z_1, z_2, z_3, \ldots, z_{n-1}\) be the nth roots of unity and \(\omega\) be a non-real complex cube root of unity, then \(\displaystyle\prod_{r=1}^{n-1}(\omega - z_r)\) can be equal to

All the points in the set \(S = \left\{\dfrac{\alpha + i}{\alpha - i}; \alpha \in \mathbb{R}\right\}\) \((i = \sqrt{-1})\) lie on a (JEE Main 2019, April)

If \(z = (\lambda + 3) - i\sqrt{5 - \lambda^2}\), then the locus of \(z\) is

Given \(\dfrac{z - \alpha}{z + \alpha}\) (\(\alpha \in \mathbb{R}\)) is a purely imaginary number. Its real part is equal to zero. Then which of the following holds?

If \(z_1\) and \(z_2\) are two non-zero complex numbers such that \(|z_1+z_2|=|z_1|+|z_2|\) then \(\arg z_1 - \arg z_2\) is equal to

Find the real part of \(e^{e^{i\theta}}\).

If P(z) be any point on the ellipse, then equation of the ellipse is \[|z - z_1| + |z - z_2| = \frac{|z_1 - z_2|}{e}\] It is given that origin is an interior point of the ellipse. Then the eccentricity \(e \in\):

Dividing \(f(z)\) by \(z - i\), we obtain the remainder \(i\) and dividing it by \(z + i\), we get the remainder \(1 + i\), then the remainder upon the division of \(f(z)\) by \(z^2 + 1\) is

If \(z = x - iy\) and \(z^{1/3} = p + iq\), then \(\dfrac{\left(\dfrac{x}{p} + \dfrac{y}{q}\right)}{(p^2 + q^2)}\) is equal to

Find the square root of \(9 + 40i\).

If \(z\) is a complex number such that \(|z| \geq 2\), then the minimum value of \(\left|z + \dfrac{1}{2}\right|\)

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

Solve: \(z^2 + |z| = 0\).

If \(|z^2 - 1| = |z|^2 + 1\), then \(z\) lies on

Given \(z\) is a complex number with modulus 1. Then the equation \([(1 + ia)/(1 - ia)]^4 = z\) has

If \(Z\) is a non-real complex number, then find the minimum value of \(\dfrac{\text{Im}\, Z^5}{(\text{Im}\, Z)^5}\).

Express the following in a + ib form:(b) \(\left(\dfrac{1+\cos\phi + i\sin\phi}{1+\cos\phi - i\sin\phi}\right)^n\)

If \(z_r, r = 1, 2, 3, \ldots, 50\) are the roots of the equation \(\displaystyle\sum_{r=0}^{50} z^r = 0\), then find the value of \(\displaystyle\sum_{r=1}^{50} \frac{1}{(z_r - 1)}\).

Let \(z\) be a complex number such that the imaginary part of \(z\) is nonzero and \(a = z^2 + z + 1\) is real. Then \(a\) cannot take the value

If the cube roots of unity are \(1, \omega, \omega^2\) then the roots of the equation \((x-1)^3 + 8 = 0\) are

If \(z_1, z_2, z_3\) lie on a circle with center origin and radius 1 unit and \(\dfrac{z_1^2}{z_2 z_3} + \dfrac{z_2^2}{z_3 z_1} + \dfrac{z_3^2}{z_1 z_2} = -1\), then sum of all the possible values of \(|z_1 + z_2 + z_3|\) is

Find the complex number \(z\) satisfying \(\text{Re}(z^2) = 0\), \(|z| = \sqrt{3}\).

Given Im \ w ≠ 0. If \ w - \(\bar{w}\)z = k(1-z), then which of the following is true about z?More specifically: Let \ w - \bar{w}z = k(1-z) where k is real. Then \(|z|\) equals:

For Problems 14–16: Consider a quadratic equation \(az^2 + bz + c = 0\), where \(a, b, c\) are complex numbers.The condition that the equation has one purely imaginary root is

The number of complex numbers \(z\) such that \(|z - 1| = |z + 1| = |z - i|\) is

If \(z_1, z_2, z_3\) are the vertices of an equilateral triangle \(ABC\) such that \(|z_1 - i| = |z_2 - i| = |z_3 - i|\), then \(|z_1 + z_2 + z_3|\) equals

Let \(z_1, z_2\) and \(z_3\) be three complex numbers satisfying \(|z| = 1\) and \(4z_3 = 3(z_1 + z_2)\), then \(|z_1 - z_2|\) is equal to

Find the values of \ \(\theta\) if \ \(\dfrac{3 + 2i\sin\theta}{1 - 2i\sin\theta}\) is purely real or purely imaginary.

If complex number \(z(z \neq 2)\) satisfies the equation \(z^2 = 4z + |z|^2 + \dfrac{16}{|z|^3}\), then the value of \(|z|^4\) is ___.

Which of the following is equal to \(\sqrt[3]{-1}\)?

If \(z\) lies on the circle \(|z - 2i| = 2\sqrt{2}\), then the value of \(\arg\left[\frac{z-2}{z+2}\right]\) is equal to

The maximum value of arg\(\left(\frac{1}{1-z}\right)\) for |z| = 1, z ≠ 1, is

Least positive argument of the 4th root of the complex number 2 - i\sqrt{12} is:

Find the number of integral solutions for n such that (n + i)4 has zero imaginary part.

The real value of θ for which \(\operatorname{Re}\left(\dfrac{2+3i\sin\theta}{1-2i\sin\theta}\right)=0\) is:

Let \omega_n = \cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n}, where i = \sqrt{-1}. Then