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

Given that $i^2 = -1$. Then $i^{13} + i^{14} + i^{15} + \ldots + i^{2026}$ is equal to:

If \( -1 + 7i \), \( -1 + xi \) and \( 3 + 3i \) are the three vertices of an isosceles triangle which is right angled at \( -1 + xi \), then the value of \( x \) is equal to

If \( z = \frac{3+i}{2-i} \), then \( z^{-1} \) is equal to

If \( z = 1 + i \tan \theta \), where \( \pi < \theta < \dfrac{3\pi}{2} \), then \( |z| \) is equal to

Let $s, t, r$ be non-zero distinct positive real numbers. If the complex number $z=x+iy$ satisfies $sz+t\overline{z}+r=0$, then $z$ lies on:

Let \( z = \frac{2 - i}{\alpha + i} \), where \( \alpha \) is a real number. If \( 4\text{Re}(z) = 3\text{Im}(\bar{z}) \), then the value of \( \alpha \) is

If \( z + \bar{z} = 6 \) and \( z - \bar{z} = 4i \), then \( |z|^2 = \)

The modulus of the complex number \( (2\sqrt{2} + i2\sqrt{2})^2 \) is equal to

Let \( z = 1 - i \). Then the value of \( z^4 \) is equal to

The value of \((1 + i)^{10}\) is equal to

The value of \(i^{3} + i^{4} + i^{5} + \ldots i^{93}\) , where \(i = \sqrt{-1}\) , is equal to

The value of $\left[\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}\right]^{-4}$ is

If $(-\sqrt{3} - i)^{30} = -4^k$, then the value of $k$ is

If $\omega$ is an imaginary cube root of unity, then $(1+\omega-\omega^2)^7$ is equal to

Given that the equation $x^2 - (2a + b)x + \left(2a^2 + b^2 - b + \frac{1}{2}\right) = 0$ has two real roots. The value of $b$ is

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