JEE MAIN Complex Number Previous Year Questions (PYQs) – Page 2 of 14

If $z$ is a complex number such that $|z|\ge 2$, then the minimum value of $\left|z+\dfrac{1}{2}\right|$ :

Let z = x + iy be a non-zero complex numbersuch that ${z^2} = i{\left| z \right|^2}$, where i = $\sqrt { - 1} $ , then z lieson the :

The area (in sq. units) of the region $S = \{\, z \in \mathbb{C} : |z - 1| \le 2,\ (z + \bar{z}) + i(z - \bar{z}) \le 2,\ \operatorname{Im}(z) \ge 0 \,\}$ is :

$ \text{Let } f,g : \mathbb{N} - \{1\} \to \mathbb{N} \text{ be functions defined by } f(a) = \alpha, \text{ where } \alpha \text{ is the maximum of the powers of those primes } p \text{ such that } p^\alpha \text{ divides } a, \text{ and } g(a) = a+1, \text{ for all } a \in \mathbb{N} - \{1\}. \text{ Then, the function } f+g \text{ is} $

Consider the following two statements: Statement I: For any two non-zero complex numbers $z_1,z_2$, $(|z_1|+|z_2|)\left|\dfrac{z_1}{|z_1|}+\dfrac{z_2}{|z_2|}\right|\le 2(|z_1|+|z_2|)$. Statement II: If $x,y,z$ are three distinct complex numbers and $a,b,c$ are positive real numbers such that $\dfrac{a}{|,y-z,|}=\dfrac{b}{|,z-x,|}=\dfrac{c}{|,x-y,|}$, then $\dfrac{a^{2}}{,y-z,}+\dfrac{b^{2}}{,z-x,}+\dfrac{c^{2}}{,x-y,}=1$. Between the above two statements:

The set of all $\alpha \in \mathbb{R}$ for which $w = \dfrac{1 + (1-8\alpha)z}{1-z}$ is purely imaginary number, for all $z \in \mathbb{C}$ satisfying $|z| = 1$ and $\operatorname{Re} z \ne 1$, is :

If $S = \{ z \in \mathbb{C} : |z - i| = |z + i| = |z - 1| \}$, then $n(S)$ is:

Among the statements (S1): The set ${z\in\mathbb{C}\setminus{-i}:\ |z|=1\ \text{ and }\ \dfrac{z-i}{z+i}\ \text{is purely real}}$ contains exactly two elements and (S2): The set ${z\in\mathbb{C}\setminus{-1}:\ |z|=1\ \text{ and }\ \dfrac{z-1}{z+1}\ \text{is purely imaginary}}$ contains infinitely many elements.

Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m$, then $m$ is ______.

Let $A = \{ z \in C:1 \le |z - (1 + i)| \le 2\} $

and $B = \{ z \in A:|z - (1 - i)| = 1\} $. Then, B :