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

If $z(3 - i) = 2 + i$, then $z^2 =$

If $|z + 4| = 2|z + 1|$, where $z$ is a complex number then $|z|$ is equal to

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 modulus of \((\frac{1+i}{1-i})^{75}-(\frac{1-i}{1+i})^{75}\)is

If $2+i$ is a root of $x^2-4x+c=0$, where $c$ is a real number, then the value of $c$ is

Let $z_1$ and $z_2$ be complex numbers satisfying $|z_1|=|z_2|=2$ and $|z_1+z_2|=3$. Then $\left|\frac{1}{z_1}+\frac{1}{z_2}\right|=$

The principal argument of the complex number $z=\frac{1+\sin\pi-i\cos\pi}{1+\sin\pi+i\cos\pi}$ is

The real part of $e^{(3+4i)x}$ is

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