Complex Numbers — JEE Main PYQs

Let the circles $C_1 : |z| = r$ and $C_2 : |z - 3 - 4i| = 5$, $z \in \mathbb{C}$, be such that $C_2$ lies within $C_1$. If $z_1$ moves on $C_1$, $z_2$ moves on $C_2$ and $\min |z_1 - z_2| = 2$, then $\max |z_1 - z_2|$ is equal to :

Let $z$ be a complex number such that $|z+2| = |z-2|$ and $\arg\left(\frac{z+3}{z-i}\right) = \frac{\pi}{4}$. Then $|z|^2$ is equal to:

Let $S = \{z \in \mathbb{C} : z^2 + 4z + 16 = 0\}$. Then $\displaystyle\sum_{z \in S} |z + \sqrt{3}\,i|^2$ is equal to:

Let $a, b \in \mathbb{C}$. Let $\alpha, \beta$ be the roots of the equation $x^2 + ax + b = 0$. If $\beta - \alpha = \sqrt{11}$ and $\beta^2 - \alpha^2 = 3i\sqrt{11}$, then $(\beta^3 - \alpha^3)^2$ is equal to:

Let $z_1, z_2 \in \mathbf{C}$ be the distinct solutions of the equation $z^2 + 4z - (1 + 12i) = 0$. Then $|z_1|^2 + |z_2|^2$ is equal to :

Let the set of all values of $k \in \mathbb{R}$ such that the equation $z\left(\bar{z}+2+i\right)+k\left(2+3i\right)=0$, $z \in \mathbb{C}$, has at least one solution, be the interval $[\alpha, \beta]$. Then $9(\alpha+\beta)$ is equal to:

Let $S = \left\{z \in \mathbb{C} : z^2 + \sqrt{6}\,iz - 3 = 0\right\}$. Then $\sum_{z \in S} z^8$ is equal to :

The number of values of $z \in \mathbb{C}$, satisfying the equations

$$|z - (4 + 8i)| = \sqrt{10} \text{ and } |z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5},$$

is :

Let $z$ be a complex number such that $|z| = 1$. If $\frac{2 + k^2 z}{k + \bar{z}} = kz$, $k \in \mathbf{R}$, then the maximum distance of $k + ik^2$ from the circle $|z - (1 + 2i)| = 1$ is :

Let $z \in \mathbb{C}$ be such that $\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i$. Then the sum of all possible values of $z^2$ is

If $z_1, z_2, z_3 \in \mathbb{C}$ are the vertices of an equilateral triangle, whose centroid is $z_0$, then $\displaystyle\sum_{k=1}^{3} (z_k - z_0)^2$ is equal to

Let $\mathrm{A} = \{z \in \mathbb{C} : |z - 2 - i| = 3\}$, $\mathrm{B} = \{z \in \mathbb{C} : \mathrm{Re}(z - iz) = 2\}$ and $\mathrm{S} = \mathrm{A} \cap \mathrm{B}$. Then $\sum\limits_{z \in S} |z|^2$ is equal to ________

Let the product of $\omega_1=(8+i)\sin\theta+(7+4i)\cos\theta$ and $\omega_2=(1+8i)\sin\theta+(4+7i)\cos\theta$ be $\alpha+i\beta$, $i=\sqrt{-1}$. Let $p$ and $q$ be the maximum and the minimum values of $\alpha+\beta$ respectively. Then $p+q$ is equal to :

If $\alpha$ is a root of the equation $x^2+x+1=0$ and $\displaystyle\sum_{\mathrm{k}=1}^{\mathrm{n}}\left(\alpha^{\mathrm{k}}+\dfrac{1}{\alpha^{\mathrm{k}}}\right)^2=20$, then $\mathrm{n}$ is equal to __________.

Among the statements

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

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

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

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

Let $\alpha$ be a solution of $x^2 + x + 1 = 0$, and for some $a$ and $b$ in $\mathbb{R}$, $\begin{bmatrix} 4 & a & b \end{bmatrix} \begin{bmatrix} 1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$. If $\frac{4}{\alpha^4} + \frac{m}{\alpha^a} + \frac{n}{\alpha^b} = 3$, then $m + n$ is equal to ________

Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg(z_1)=\frac{-\pi}{4}$, $\arg(z_2)=0$ and $\arg(z_3)=\frac{\pi}{4}$. If $\left|z_1\bar{z}_2+z_2\bar{z}_3+z_3\bar{z}_1\right|^2=\alpha+\beta\sqrt{2}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2+\beta^2$ is :

Let the curve $z(1 + i) + \bar{z}(1 - i) = 4$, $z \in \mathbb{C}$, divide the region $|z - 3| \leq 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha - \beta|$ equals :

Let $\left|\frac{\bar{z} - i}{2\bar{z} + i}\right| = \frac{1}{3}$, $z \in \mathbb{C}$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $(0, 0)$, $C$ and $(\alpha, 0)$ is 11 square units, then $\alpha^2$ equals:

The number of complex numbers $z$, satisfying $|z| = 1$ and $\left|\dfrac{z}{\bar{z}} + \dfrac{\bar{z}}{z}\right| = 1$, is :

If $\alpha$ and $\beta$ are the roots of the equation $2z^2-3z-2i=0$, where $i=\sqrt{-1}$, then $16 \cdot \mathrm{Re}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \cdot \mathrm{Im}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)$ is equal to

Let O be the origin, the point A be $z_1 = \sqrt{3} + 2\sqrt{2}\,i$, the point B$(z_2)$ be such that $\sqrt{3}\,|z_2| = |z_1|$ and $\arg(z_2) = \arg(z_1) + \frac{\pi}{6}$. Then

If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3-2i)x - (2i-2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to :

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^{2}+|z|=0$, $z \in \mathbf{C}$. Then $4(\alpha^{2}+\beta^{2})$ is equal to :

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

Consider the following two statements :

Statement I : For any two non-zero complex numbers $z_1, z_2$, $$(|z_1| + |z_2|)\left|\frac{z_1}{|z_1|} + \frac{z_2}{|z_2|}\right| \le 2\,(|z_1| + |z_2|),$$ and

Statement II : If $x, y, z$ are three distinct complex numbers and $a, b, c$ are three positive real numbers such that $\frac{a}{|y-z|} = \frac{b}{|z-x|} = \frac{c}{|x-y|}$, then $$\frac{a^2}{y-z} + \frac{b^2}{z-x} + \frac{c^2}{x-y} = 1.$$

Between the above two statements,

Let $S_1 = \{z \in \mathbf{C} : |z| \leq 5\}$, $S_2 = \left\{z \in \mathbf{C} : \operatorname{Im}\left(\dfrac{z + 1 - \sqrt{3}\,i}{1 - \sqrt{3}\,i}\right) \geq 0\right\}$ and $S_3 = \{z \in \mathbf{C} : \operatorname{Re}(z) \geq 0\}$. Then the area of the region $S_1 \cap S_2 \cap S_3$ is :

If $z_1, z_2$ are two distinct complex number such that $\left|\dfrac{z_1-2z_2}{\frac{1}{2}-z_1\bar{z}_2}\right|=2$, then

Let $z$ be a complex number such that $|z+2|=1$ and $\mathrm{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $\left|\mathrm{Re}\left(\overline{z+2}\right)\right|$ is

If the set $R=\{(a,b): a+5b=42, a,b\in\mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^{m}\left(1-i^{n!}\right)=x+iy$, where $i=\sqrt{-1}$, then the value of $m+x+y$ is

The sum of all possible values of $\theta \in [-\pi, 2\pi]$, for which $\dfrac{1 + i\cos\theta}{1 - 2i\cos\theta}$ is purely imaginary, is equal to :

The sum of the square of the modulus of the elements in the set $\{z=\mathrm{a}+\mathrm{ib} : \mathrm{a}, \mathrm{b}\in\mathbf{Z},\ z\in\mathbf{C},\ |z-1|\le 1,\ |z-5|\le|z-5\mathrm{i}|\}$ is ________.

Let $z$ be a complex number such that the real part of $\frac{z-2i}{z+2i}$ is zero. Then, the maximum value of $|z-(6+8i)|$ is equal to

Let $S=\{z \in \mathbb{C} : |z-1|=1 \text{ and } (\sqrt{2}-1)(z+\bar{z}) - i(z-\bar{z}) = 2\sqrt{2}\}$. Let $z_1, z_2 \in S$ be such that $|z_1| = \max\limits_{z \in S} |z|$ and $|z_2| = \min\limits_{z \in S} |z|$. Then $\left|\sqrt{2}\, z_1 - z_2\right|^2$ equals :

Let $P = \{z \in \mathbb{C} : |z + 2 - 3i| \le 1\}$ and $Q = \{z \in \mathbb{C} : z(1 + i) + \bar{z}(1 - i) \le -8\}$. Let in $P \cap Q$, $|z - 3 + 2i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $|z_1|^2 + 2|z_2|^2 = \alpha + \beta\sqrt{2}$, where $\alpha, \beta$ are integers, then $\alpha + \beta$ equals ________.

If $z$ is a complex number such that $|z| \geq 1$, then the minimum value of $\left|z+\frac{1}{2}(3+4i)\right|$ is :

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

If $\alpha$ satisfies the equation $x^2 + x + 1 = 0$ and $(1 + \alpha)^7 = A + B\alpha + C\alpha^2$, $A, B, C \geq 0$, then $5(3A - 2B - C)$ is equal to ________.

If $z=\dfrac{1}{2}-2i$ is such that $|z+1| = \alpha z + \beta(1+i)$, $i=\sqrt{-1}$ and $\alpha, \beta \in \mathbb{R}$, then $\alpha + \beta$ is equal to

Let $\alpha, \beta$ be the roots of the equation $x^2 - x + 2 = 0$ with $Im(\alpha) > Im(\beta)$.

Then $\alpha^6 + \alpha^4 + \beta^4 - 5\alpha^2$ is equal to __________.

If $z=x+\mathrm{i}y$, $xy\neq 0$, satisfies the equation $z^2+\mathrm{i}\bar{z}=0$, then $|z^2|$ is equal to :

If $z$ is a complex number, then the number of common roots of the equations $z^{1985}+z^{100}+1=0$ and $z^3+2z^2+2z+1=0$, is equal to

If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and $\beta=\left(\frac{|z|}{\arg(z)}\right)$, where $z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi}\,i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi}\,i}\right]$, $i=\sqrt{-1}$, then the distance of the point $(\alpha, \beta)$ from the line $4x-3y=7$ is _____

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $z_{1}+z_{2}=5$ and $z_{1}^{3}+z_{2}^{3}=20+15 i$. Then, $\left|z_{1}^{4}+z_{2}^{4}\right|$ equals -