List of top Mathematics Questions on Complex Numbers and Quadratic Equations asked in KEAM

Let \( \omega \ne 1 \) be a cube root of unity and \( (1+\omega)^7 = a + \omega \). Then the value of \( a \) is
Let \( w = \frac{1-iz}{z-i} \). If \( |w| = 1 \), which of the following must be true?
Let \( z_1 = 1 + i\sqrt{3} \) and \( z_2 = 1 + i \), then \( \arg\left( \frac{z_1}{z_2} \right) \) is
The complex number \( \sqrt{2}\left[ \sin \frac{\pi}{8} + i \cos \frac{\pi}{8} \right]^6 \) represents
If \( z^2 + z + 1 = 0 \), where \( z \) is a complex number, then the value of \[ \left( z + \frac{1}{z} \right)^2 + \left( z^2 + \frac{1}{z^2} \right)^2 + \left( z^3 + \frac{1}{z^3} \right)^2 + \cdots + \left( z^6 + \frac{1}{z^6} \right)^2 \] is
If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + \alpha x + \beta = 0 \), then
The real part of \( (i - \sqrt{3})^{13} \) is
The value of \( \frac{2(\cos 75^\circ + i \sin 75^\circ)}{0.2(\cos 30^\circ + i \sin 30^\circ)} \) is
The real part of \( (i - \sqrt{3})^{13} \) is
If \( z_1 = 2 - i \) and \( z_2 = 1 + i \), then \( \left|\frac{z_1 + z_2 + 1}{z_1 - z_2 + i}\right| \) is
The value of \( \left( i^{18} + \left(\frac{1}{i}\right)^{25} \right)^3 \) is equal to
The modulus of \( \frac{1+i}{1-i} - \frac{1-i}{1+i} \) is
If \( z = e^{i4\pi/3} \), then \( (z^{192} + z^{194})^3 \) is equal to
If \( a \) and \( b \) are real numbers and \( (a + ib)^{11} = 1 + 3i \), then \( (b + ia)^{11} \) is equal to
If \( \alpha \neq \beta \), \( \alpha^2 = 5\alpha - 3 \), \( \beta^2 = 5\beta - 3 \), then the equation having \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \) as its roots is
The value of \( \frac{2(\cos 75^\circ + i \sin 75^\circ)}{0.2(\cos 30^\circ + i \sin 30^\circ)} \) is
The real part of \( (i - \sqrt{3})^{13} \) is
If \( z_1 = 2 - i \) and \( z_2 = 1 + i \), then \( \left|\frac{z_1 + z_2 + 1}{z_1 - z_2 + i}\right| \) is
The value of \( \left( i^{18} + \left(\frac{1}{i}\right)^{25} \right)^3 \) is equal to
The modulus of \( \frac{1+i}{1-i} - \frac{1-i}{1+i} \) is
If \( z = e^{i4\pi/3} \), then \( (z^{192} + z^{194})^3 \) is equal to
If \( a \) and \( b \) are real numbers and \( (a + ib)^{11} = 1 + 3i \), then \( (b + ia)^{11} \) is equal to
If \( \alpha \neq \beta \), \( \alpha^2 = 5\alpha - 3 \), \( \beta^2 = 5\beta - 3 \), then the equation having \( \frac{\alpha}{\beta} \) and \( \frac{\beta}{\alpha} \) as its roots is
The value of \( \frac{2(\cos 75^\circ + i \sin 75^\circ)}{0.2(\cos 30^\circ + i \sin 30^\circ)} \) is
Let $f (x) = px^2 + qx + r$ , where $p, q, r$ are constants and $p \neq 0$ . If $f (5) = -3 f (2)$ and $f (-4) = 0$ , then the other root of $f$ is