List of top Mathematics Questions on Complex numbers asked in KEAM

The imaginary part of $\frac{\cos 50^{\circ}+i\sin 50^{\circ}}{\cos 50^{\circ}-i\sin 50^{\circ}}$ is equal to

The value of $\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)^{3}+\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)^{3}$ is equal to

The value of $\dfrac{(1+i)^n}{(1-i)^{n-4}}$, where $i=\sqrt{-1}$ and $n$ is an integer, is:

Let $x$ and $y$ be real numbers. If $(3+i)x + y + (1-i)y + 3i - 4 = (2x+1)i + (x-y+2)i$, where $i=\sqrt{-1}$, then the pair $(x,y)$ is equal to:

Let $z_1 = \dfrac{5+7i}{7-5i}, \, z_2 = \dfrac{3+2i}{3-2i}$ and $z_3 = \dfrac{1+11i}{11-i}$. Then $z_1\overline{z_1} + z_2\overline{z_2} + z_3\overline{z_3}$ is equal to:

The point \(z=\frac{1}{\sqrt{2}}(1+i)\) in the complex plane is rotated about the origin through an angle \(\frac{\pi}{4}\) in the clockwise direction, then the new position of \(z\) is

If \(z\) is a complex number, then the minimum value of \(|z-2|+|z-4|\) is

Real part of $\frac{1+\sin\frac{2\pi}{27}-i\cos\frac{2\pi}{27}}{1+\sin\frac{2\pi}{27}+i\cos\frac{2\pi}{27}}$ is equal to:

Let $z$ be a complex number such that $z^{3}+iz^{2}-iz+1=0$ where $i^{2}=-1$. Then $|z|=$

Let $z=x+iy$ be a complex number, where $i=\sqrt{-1}$ is the complex unit. Then $|z-1+i|=5$ is a circle with:

Imaginary parts of \(\left(\frac{3 - 2i}{2i}\right)^2\) is equal to

If $\arg(z_1) = \arg(z_2)$, then

For \( |z| \ge 2 \), if \( \left|z + \frac{1}{2}\right| \ge k \), the minimum possible value of \(k\) is

The area of the triangle in the complex plane formed by \( z, iz \) and \( z + iz \) is

The area of the triangle in the complex plane formed by \( z, iz \) and \( z + iz \) is

The area of the triangle in the complex plane formed by \( z, iz \) and \( z + iz \) is

Let $z, w$ be two nonzero complex numbers. If $\bar{z} + i\overline{w} = 0$ and $\arg(zw) = \pi$, then $\arg z =$

If $z = \frac{2 - i}{i}$, then $\text{Re}(z^2) + \text{Im}(z^2)$ is equal to: