Complex Numbers JEE Main PYQs, 2020 to 2025

Let $A = \left\{ \theta \in [0, 2\pi] : 1 + 10\operatorname{Re}\left( \frac{2\cos\theta + i\sin\theta}{\cos\theta - 3i\sin\theta} \right) = 0 \right\}$. Then $\sum\limits_{\theta \in A} \theta^2$ is equal to

Let α be a solution of $x^2 + x + 1 = 0$, and for some a and b in

$R, \begin{bmatrix} 4 & a & b \end{bmatrix} \begin{bmatrix} 1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$. If $\frac{4}{\alpha^4} + \frac{m}{\alpha^a} + \frac{n}{\alpha^b} = 3$, then m + n is equal to _______

If the locus of z ∈ ℂ, such that Re$\left( \frac{z - 1}{2z + i} \right) + \text{Re} \left( \frac{\overline{z} - 1}{2\overline{z} - i} \right) = 2$, is a circle of radius r and center $(a, b)$, then $\frac{15ab}{r^2}$ is equal to :

Among the statements

(S1) : The set $\left\{z \in \mathbb{C}-\{-i\}:|z|=1\right.$ and $\frac{z-i}{z+i}$ is purely real $\}$ contains exactly two elements, and

(S2) : The set $\left\{z \in \mathbb{C}-\{-1\}:|z|=1\right.$ and $\frac{z-1}{z+1}$ is purely imaginary $\}$ contains infinitely many elements.

Let $\alpha, \beta$ be the roots of the equation $x^{2}-\sqrt{2} x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to