List of top Mathematics Questions on Algebra of Complex Numbers asked in KEAM

If $\alpha$ is a real number satisfying $\alpha^2 - \frac{1}{\alpha^2} = 2$, then the value of $(\alpha + \frac{i}{\alpha})^{16}$ is equal to

The complex number $z$ satisfying the equation $\frac{Re(z)}{2+i} + \frac{Im(z)}{1+2i} = \frac{3}{1-2i}$, is

Let $z = 1 + i$, where $i = \sqrt{-1}$. If $z - \frac{24\bar{z}}{z^2} = \lambda z$, then the value of $\lambda$ is equal to

If the complex number $z = x + iy$ satisfies the equation $5z - 2\bar{z} = \frac{7 - 7i}{1 + i}$, then the value of $x + y$ is equal to

The value of \[ \left[\frac{5i}{(3+i)(3-i)}\right]^{2026} \] is equal to:

If \(z |z| = 24 + 7i\), where \(z\) is a complex number, then the value of \(|z|\) is equal to

Let \(z = \frac{a - \frac{i}{2}}{i - 2}\), where \(a\) is a real number and \(i = \sqrt{-1}\). If \(\text{Im}(z) = 0\), then the value of \(a\) is equal to

In a complex plane, if two vertices of an equilateral triangle are at \(-3(1+i)\) and \(3(1-i)\), then the area of the triangle (in sq.units) is equal to

The equation $\text{Im}(1-i)z=1$ represents the line

Let $z=x+iy,$ where $x,y \in \mathbb{R}$ and $i^{2}=-1$. If $|z-i|=|z-1|$, then $y=$ ________.

If $x,y \in \mathbb{R}$ and $x+iy=-(6+i)^{3}, i^{2}=-1$, then $x-y$ is equal to ________.

$\sum_{n=1}^{2025}i^{n}(1+i), i^{2}=-1$ is equal to ________.

Let \( w \neq \pm 1 \) be a complex number. If \( |w| = 1 \) and \( z = \frac{w - 1}{w + 1} \), then \( \text{Re}(z) \) is equal to:

If \( z = e^{2\pi i / 3} \), then \( 1 + z + 3z^2 + 2z^3 + 2z^4 + 3z^5 \) is equal to:

The range of the function \( f(x) = x^2 + 2x + 2 \) is:

If \( f(x) = \sqrt{x \) and \( g(x) = 2x - 3 \), then \( (f \circ g)(x) \) is:}

If \( z = \frac{(\sqrt{3} + i)^3 (3i + 4)^2{(8 + 6i)^2} \), then \( |z| \) is equal to:}

If \( z = \frac{(\sqrt{3} + i)^3 (3i + 4)^2{(8 + 6i)^2} \), then \( |z| \) is equal to:}

Let \( w \neq \pm 1 \) be a complex number. If \( |w| = 1 \) and \( z = \frac{w - 1}{w + 1} \), then \( \text{Re}(z) \) is equal to:

If \( z = e^{2\pi i / 3} \), then \( 1 + z + 3z^2 + 2z^3 + 2z^4 + 3z^5 \) is equal to:

The range of the function \( f(x) = x^2 + 2x + 2 \) is:

If \( f(x) = \sqrt{x \) and \( g(x) = 2x - 3 \), then \( (f \circ g)(x) \) is:}

The range of the function \( f(x) = x^2 + 2x + 2 \) is:

If \( f(x) = \sqrt{x \) and \( g(x) = 2x - 3 \), then \( (f \circ g)(x) \) is:}

If \( z = \frac{(\sqrt{3} + i)^3 (3i + 4)^2{(8 + 6i)^2} \), then \( |z| \) is equal to:}