Complex Numbers — JEE Advanced PYQs

Let

Then …

Let $\mathbb{R}$ denote the set of all real numbers. Let $z_1=1+2i$ and $z_2=3i$ be two complex numbers, where $i=\sqrt{-1}$. Let

Then which of the fo …

For a non-zero complex number $z$, let $\arg (z)$ denote the principal argument of $z$, with $-\pi <\arg (z)\le \pi$. Let $\omega$ be the cube root of unity for which $0<\arg (\omega )<\pi$. Let

Then the …

Let $\mathrm{S}=\{\mathrm{a}+\mathrm{b}\sqrt{2}:\mathrm{a},\mathrm{b}\in \mathbb{Z}\},{\mathrm{T}}_1=\left\{(-1+\sqrt{2}{)}^{\mathrm{n}}:\mathrm{n}\in \mathbb{N}\right\}$ and ${\mathrm{T}}_2=\left\{(1+\sqrt{2}{)}^{\mathrm{n}}:\mathrm{n}\in \mathbb{N}\right\}$. Then which …

Let ${\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3,\dots ,{\mathrm{A}}_8$ be the vertices of a regular octagon that lie on a circle of radius 2. Let P be a point on the circle and let ${\mathrm{PA}}_{\mathrm{i}}$ denote the distance between the points P and …

Let $\mathrm{A}=\left\{\frac{1967+1686\mathrm{i}\sin \theta }{7-3\mathrm{i}\cos \theta }:\theta \in \mathbb{R}\right\}$. If A contains exactly one positive integer n, then the value of n is

Let z be complex number satisfying $|z{|}^3+2z^2+4\bar{z}-8=0$, where $\bar{z}$ denotes the complex conjugate of z. Let the imaginary part of z be nonzero.
Match each entry in List-I to the correct entries in List-II.

List-I
(P) $|z{|}^2$ is equal to
(Q) $|z-\bar{z}{|}^2$ is …

Let $z$ be a complex number with non-zero imaginary part. If

is a real number, then the value of $|z{|}^2$ is ____.

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of complex numbers, the number of distinct roots of the equation

is ____.

Let $\bar{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of

are integers, then which of the following is/are possible value(s) of $|z|$?

Let ${\theta }_1,{\theta }_2,\dots ,{\theta }_{10}$ be positive valued angles (in radian) such that ${\theta }_1+{\theta }_2+\dots +{\theta }_{10}=2\pi$. Define the complex numbers $z_1=e^{i{\theta }_1},z_k=z_{k-1}{e}^{i{\theta }_k}$ for $k=2,3,\dots ,10$, where $i=\sqrt{-1}$. Consid …

For any complex number $w=c+id$, let $\arg (w)\in (-\pi ,\pi ]$, where $i=\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+iy$ satisfying $\arg \left(\frac{z+\alpha }{z+\beta }\right)=\frac{\pi }{4}$, the ordered pair $(x,y)$ lies on the circl …

Let S be the set of all complex numbers z satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are TRUE?

For a complex number $z$, let $\mathrm{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4-|z{|}^4=4i{z}^2$, where $i=\sqrt{-1}$. Then the minimum possible value of ${\left|z_1-z_2\right|}^2$, where …

Let S be the set of all complex numbers z satisfying $|\mathrm{z}-2+\mathrm{i}|\ge \sqrt{5}$. If the complex number ${\mathrm{z}}_0$ is such that $\frac{1}{|z_0-1|}$ is the maximum of the set $\left\{\frac{1}{|z-1|}:z\in S\right\}$, then the principal argument of …

Let $\omega \ne 1$ be a cube root of unity. Then the minimum of the set

equals ____

For a non-zero complex number $z$, let $\arg (z)$ denote the principal argument with $-\pi <\arg (z)\le \pi$. Then, which of the following statement(s) is (are) FALSE?

Let $s,t,r$ be non-zero complex numbers and $L$ be the set of solutions $z=x+iy$ $(x,y\in \mathbb{R},i=\sqrt{-1})$ of the equation $sz+t\bar{z}+r=0$, where $\bar{z}=x-iy$. Then, which of the following statement(s) is (are) TRUE?

Let $a,b,x$ and $y$ be real numbers such that $a-b=1$ and $y\ne 0$. If the complex number $z=x+iy$ satisfies $\mathrm{Im}\left(\frac{az+b}{z+1}\right)=\mathrm{y}$, then which of the following is(are) possible value(s) of $x$?

Let $z=\frac{-1+\sqrt{3}i}{2}$, where $i=\sqrt{-1}$, and $r,s\in \{1,2,3\}$. Let $P=\left[\begin{array}{cc}(-z{)}^r& z^{2s}\\ z^{2s}& z^r\end{array}\right]$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2=-I$ is

Let $a,b\in \mathbb{R}$ and $a^2+b^2\ne 0$. Suppose $S=\left\{z\in \mathbb{C}:z=\frac{1}{a+ibt},t\in \mathbb{R},t\ne 0\right\}$, where $i=\sqrt{-1}$. If $z=x+iy$ and $z\in S$, then $(x,y)$ lies on

For any integer $k$, let ${\alpha }_k=\cos \left(\frac{k\pi }{7}\right)+i\sin \left(\frac{k\pi }{7}\right)$, where $i=\sqrt{-1}$. The value of the expression …

Let $z_k=\cos \left(\frac{2k\pi }{10}\right)+i\sin \left(\frac{2k\pi }{10}\right);k=1,2,\dots ,9$.
| Q. | There exists a …

Let complex numbers $\alpha$ and $\frac{1}{\bar{\alpha }}$ lie on circles ${\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2=r^2$ and ${\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2=4r^2$, respectively. If $z_0=x_0+i{y}_0$ satisfies the equation …

Let $w=\frac{\sqrt{3}+\mathrm{i}}{2}$ and $P=\left\{w^n:n=1,2,3,\dots \right\}$. Further $H_1=\left\{\mathrm{z}\in \mathbb{C}:\mathrm{Re}z>\frac{1}{2}\right\}$ and $H_2=\left\{\mathrm{z}\in \mathbb{C}:\mathrm{Re}z<\frac{-1}{2}\right\}$, where …

Let $\omega$ be a complex cube root of unity with $\omega \ne 1$ and $P=\left[p_{ij}\right]$ be a $n\times n$ matrix with $p_{ij}={\omega }^{i+j}$. Then $P^2\ne 0$, when $n=$

Let $S=S_1\cap S_2\cap S_3$, where
$S_1=\{z\in \mathbb{C}:|z|<4\}$, $S_2=\left\{z\in \mathbb{C}:\mathrm{Im}\left[\frac{z-1+\sqrt{3}i}{1-\sqrt{3}i}\right]>0\right\}$ and
$S_3=\{z\in \mathbb{C}:\mathrm{Re}z>0\}$.

Area of $S=$

Let $S=S_1\cap S_2\cap S_3$, where
$S_1=\{z\in \mathbb{C}:|z|<4\}$, $S_2=\left\{z\in \mathbb{C}:\mathrm{Im}\left[\frac{z-1+\sqrt{3}i}{1-\sqrt{3}i}\right]>0\right\}$ and
$S_3=\{z\in \mathbb{C}:\mathrm{Re}z>0\}$.

${\min }_{z\in S}|1-3i-z|=$

Let $z$ be a complex number such that the imaginary part of $z$ is nonzero and $a=z^2+z+1$ is real. Then $a$ cannot take the value

Let $a,b$ and $c$ be three real numbers satisfying

Let $\omega$ be a solution of …

If $z$ is any complex number satisfying $|z-3-2i|\le 2$, then the minimum value of $|2z-6+5i|$ is

Match the statements in Column-I with those in Column-II.

[Note: Here $z$ takes values in the complex plane and $\mathrm{Im}z$ and $\mathrm{Re}z$ denote, respectively, the imaginary part and the real part of z.]
| A) | The set of …

Let $\omega$ be a complex cube root of unity with $\omega \ne 1$. A fair die is thrown three times. If $r_1,r_2$ and $r_3$ are the numbers obtained on the die, then the probability that ${\omega }^{r_1}+{\omega }^{r_2}+{\omega }^{r_3}=0$ is

Let $\omega$ be the complex number $\cos \frac{2\pi }{3}+\mathrm{i}\sin \frac{2\pi }{3}$. Then the number of distinct complex numbers $z$ satisfying …

Let $z=x+iy$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation

is

Let $z=\cos \theta +i\sin \theta$. Then the value of

at $\theta =2^{\circ }$ is

Let $A,B,C$ be three sets of complex numbers as defined below


The number of elements in the set $A\cap B\cap C$ is

Let $A,B,C$ be three sets of complex numbers as defined below


Let $z$ be any point in $A\cap B\cap C$. Then, …

A particle $P$ starts from the point $z_0=1+2i$, where $i=\sqrt{-1}$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_1$. From $z_1$ the particle moves $\sqrt{2}$ units in the direction …

A man walks a distance of 3 units from the origin towards the north-east $(N45^{\circ }E)$ direction. From there, he walks a distance of 4 units towards the north-west $(N45^{\circ }W)$ direction to reach a point $P$. Then the position of $P$ in the Argand plane is

If $|z|=1$ and $z\ne \pm 1$, then all the values of $\frac{z}{1-z^2}$ lie on

If $w=\alpha +i\beta$, where $\beta \ne 0$ and $z\ne 1$, satisfies the condition that $\left(\frac{w-\overline{w}z}{1-z}\right)$ is purely real, then the set of values of z is