Complex Numbers — JEE Main PYQs

Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha }}$ lie on the circles ${\left|z-z_0\right|}^2=4$ and ${\left|z-z_0\right|}^2=16$ respectively, where $z_0=1+i$. Then, the value of $100|\alpha {|}^2$ is

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

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

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

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 …

Let $S=\{z\in C:|z-1|=1$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2\sqrt{2}\}$. Let $z_1,z_2\in S$ be such that $\left|z_1\right|={\max }_{z\in S}|z|$ and $\left|z_2\right|={\min }_{z\in S}|z|$. Then ${\left|\sqrt{2}{z}_1-z_2\right|}^2$ equals :

Let $\mathrm{P}=\{\mathrm{z}\in \mathbb{C}:|\mathrm{z}+2-3\mathrm{i}|\le 1\}$ and $\mathrm{Q}=\{\mathrm{z}\in \mathbb{C}:\mathrm{z}(1+\mathrm{i})+\overline{\mathrm{z}}(1-\mathrm{i})\le -8\}$. Let in $\mathrm{P}\cap \mathrm{Q}$, $|\mathrm{z}-3+2\mathrm{i}|$ be max …

If $z=\frac{1}{2}-2i$, 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 $\mathrm{Im}(\alpha )>\mathrm{Im}(\beta )$. Then ${\alpha }^6+{\alpha }^4+{\beta }^4-5{\alpha }^2$ is equal to

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

Let $a\ne b$ be two non-zero real numbers. Then the number of elements in the set $X=\{z\in \mathbb{C}:\mathrm{Re}(a{z}^2+bz)=a\text{ and }\mathrm{Re}(b{z}^2+az)=b\}$ is equal to

For $\alpha ,\beta ,z\in \mathbb{C}$ and $\lambda >1$, if $\sqrt{\lambda -1}$ is the radius of the circle $|z-\alpha {|}^2+|z-\beta {|}^2=2\lambda$, then $|\alpha -\beta |$ is equal to ____.

If for $\mathrm{z}=\alpha +\mathrm{i}\beta ,|\mathrm{z}+2|=\mathrm{z}+4(1+\mathrm{i})$, then $\alpha +\beta$ and $\alpha \beta$ are the roots of the equation

Let $\mathrm{A}=\{\theta \in (0,2\pi ):\frac{1+2\mathrm{i}\sin \theta }{1-\mathrm{i}\sin \theta }$ is purely imaginary $\}$.

Then the sum of the elements in A is

Let $S=\{z=x+iy:\frac{2z-3i}{4z+2i}$ is a real number $\}$.

Then which of the following is NOT correct?

Let the complex number $\mathrm{z}=\mathrm{x}+$ iy be such that $\frac{2z-3i}{2z+i}$ is purely imaginary. If $x+y^2=0$, then $y^4+y^2-y$ is equal to :

Let C be the circle in the complex plane with centre ${\mathrm{z}}_0=\frac{1}{2}(1+3\mathrm{i})$ and radius $\mathrm{r}=1$. Let ${\mathrm{z}}_1=1+\mathrm{i}$ and the complex number ${\mathrm{z}}_2$ be outside the circle C such that …

Let $S=\left\{Z\in C:\bar{Z}=i\left(Z^2+\mathrm{Re}(\bar{Z})\right)\right\}$. Then ${\sum }_{z\in S}|z{|}^2$ is equal to

Let $\omega =z\overline{z}+k_{1}z+k_{2}iz+\lambda (1+i)$, $k_1,k_2\in \mathbb{R}$. Let $\mathrm{Re}(\omega )=0$ be the circle C of radius 1 in the first quadrant touching the line $y=1$ and the y-axis. If the curve $\mathrm{Im}(\omega )=0$ intersects C at …

If the set $\left\{\mathrm{Re}\left(\frac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}\right):z\in \mathbb{C},\mathrm{Re}(z)=3\right\}$ is equal to the interval $(\alpha ,\beta ]$, then $24(\beta -\alpha )$ is equal to

For $a\in \mathbb{C}$, let $A=\{z\in \mathbb{C}:\mathrm{Re}(a+\bar{z})>\mathrm{Im}(\bar{a}+z)\}$ and $B=\{z\in \mathbb{C}:\mathrm{Re}(a+\bar{z})<\mathrm{Im}(\bar{a}+z)\}$. Then among the two statements:
(S1): If …

Let $S=\left\{z\in \mathbb{C}-\{i,2i\}:\frac{z^2+8iz-15}{z^2-3iz-2}\in \mathbb{R}\right\}$. If $\alpha -\frac{13}{11}i\in S$, $\alpha \in \mathbb{R}-\{0\}$, then $242{\alpha }^2$ is equal to

Let $w_1$ be the point obtained by the rotation of $z_1=5+4i$ about the origin through a right angle in the anticlockwise direction, and $w_2$ be the point obtained by the rotation of $z_2=3+5i$ about the origin through a right angle in the clockwise direction. Th …

Let $\mathrm{p},\mathrm{q}\in \mathbb{R}$ and $(1-\sqrt{3}\mathrm{i}{)}^{200}=2^{199}(\mathrm{p}+\mathrm{iq})$, $\mathrm{i}=\sqrt{-1}$. Then $\mathrm{p}+\mathrm{q}+{\mathrm{q}}^2$ and $\mathrm{p}-\mathrm{q}+{\mathrm{q}}^2$ are roots of the equation.

Let $\mathrm{z}=1+\mathrm{i}$ and ${\mathrm{z}}_1=\frac{1+\mathrm{i}\overline{\mathrm{z}}}{\overline{\mathrm{z}}(1-\mathrm{z})+\frac{1}{\mathrm{z}}}$. Then $\frac{12}{\pi }\arg \left({\mathrm{z}}_1\right)$ is equal to ______.

Let $\alpha =8-14\mathrm{i}$, $\mathrm{A}=\left\{\mathrm{z}\in \mathbb{C}:\frac{\alpha \mathrm{z}-\bar{\alpha }\overline{\mathrm{z}}}{{\mathrm{z}}^2-(\overline{\mathrm{z}}{)}^2-112\mathrm{i}}=1\right\}$ and …

For two non-zero complex number ${\mathrm{z}}_1$ and ${\mathrm{z}}_2$, if $\mathrm{Re}({\mathrm{z}}_1{\mathrm{z}}_2)=0$ and $\mathrm{Re}({\mathrm{z}}_1+{\mathrm{z}}_2)=0$, then which of the following are possible ?
(A) $\mathrm{Im}({\mathrm{z}}_1)>0$ …

Let $z$ be a complex number such that $\left|\frac{\mathrm{z}-2\mathrm{i}}{\mathrm{z}+\mathrm{i}}\right|=2,\mathrm{z}\ne -\mathrm{i}$. Then z lies on the circle of radius 2 and centre

The value of ${\left(\frac{1+\sin \frac{2\pi }{9}+i\cos \frac{2\pi }{9}}{1+\sin \frac{2\pi }{9}-i\cos \frac{2\pi }{9}}\right)}^3$ is

The complex number $z=\frac{i-1}{\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}}$ is equal to:

For all $z\in \mathbb{C}$ on the curve $C_1:|z|=4$, let the locus of the point $z+\frac{1}{z}$ be the curve $C_2$. Then

If the center and radius of the circle $\left|\frac{z-2}{z-3}\right|=2$ are respectively ( $\alpha ,\beta$ ) and $\gamma$, then $3(\alpha +\beta +\gamma )$ is equal to

Let ${\mathrm{z}}_1=2+3\mathrm{i}$ and ${\mathrm{z}}_2=3+4\mathrm{i}$. The set
$S=\left\{\mathrm{z}\in \mathrm{C}:{\left|\mathrm{z}-{\mathrm{z}}_1\right|}^2-{\left|\mathrm{z}-{\mathrm{z}}_2\right|}^2={\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}^2\right\}$ represents a

If $\mathrm{z}\ne 0$ be a complex number such that $\left|\mathrm{z}-\frac{1}{\mathrm{z}}\right|=2$, then the maximum value of |z| is:

Let $\mathrm{S}=\{\mathrm{z}=\mathrm{x}+\mathrm{iy}:|\mathrm{z}-1+\mathrm{i}|\ge |\mathrm{z}|,|\mathrm{z}|<2,|\mathrm{z}+\mathrm{i}|=|\mathrm{z}-1|\}$. Then the set of all values of $x$, for which $\mathrm{w}=2\mathrm{x}+\mathrm{iy}\in \mathrm{S}$ for some …

Let $S_1=\left\{z_1\in C:\left|z_1-3\right|=\frac{1}{2}\right\}$ and $S_2=\left\{z_2\in C:\left|z_2-\left|z_2+1\right|\right|=\left|z_2+\left|z_2-1\right|\right|\right\}$. Then, for ${\mathrm{z}}_1\in {\mathrm{S}}_1$ and ${\mathrm{z}}_2\in {\mathrm{S}}_2$, …

Let $z=a+ib,b\ne 0$ be complex numbers satisfying $z^2=\bar{z}\cdot 2^{1-|z|}$. Then the least value of $n\in \mathrm{N}$, such that ${\mathrm{z}}^{\mathrm{n}}=(\mathrm{z}+1{)}^{\mathrm{n}}$, is equal to

Let the minimum value $v_0$ of $v=|z{|}^2+|z-3{|}^2+|z-6i{|}^2$, $z\in \mathbb{C}$ is attained at $z=z_0$. Then ${\left|2z_0^2-{\bar{z}}_0^3+3\right|}^2+v_0^2$ is equal to

Let $S=\left\{z\in \mathbb{C}:z^2+\bar{z}=0\right\}$. Then ${\sum }_{z\in S}(\mathrm{Re}(z)+\mathrm{Im}(z))$ is equal to ______.

If $\alpha ,\beta ,\gamma ,\delta$ are the roots of the equation $x^4+x^3+x^2+x+1=0$, then
${\alpha }^{2021}+{\beta }^{2021}+{\gamma }^{2021}+{\delta }^{2021}$ is equal to :

If $z=x+iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$, then

Let S be the set of all $(\alpha ,\beta ),\pi <\alpha ,\beta <2\pi$, for which the complex number $\frac{1-i\sin \alpha }{1+2i\sin \alpha }$ is purely imaginary and $\frac{1+i\cos \beta }{1-2i\cos \beta }$ is purely real. Let …

For $z\in \mathbb{C}$ if the minimum value of $(|z-3\sqrt{2}|+|z-p\sqrt{2}i|)$ is $5\sqrt{2}$, then a value of $p$ is

Let O be the origin and A be the point ${\mathrm{z}}_1=1+2i$. If $B$ is the point $z_2,\mathrm{Re}\left(z_2\right)<0$, such that $OAB$ is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?

Sum of squares of modulus of all the complex numbers z satisfying $\overline{\mathrm{z}}=i{\mathrm{z}}^2+{\mathrm{z}}^2-\mathrm{z}$ is equal to

The number of points of intersection of $|z-(4+3i)|=2$ and $|z|+|z-4|=6,z\in C$ is :

The number of elements in the set $\{z=a+ib\in \mathbb{C}:a,b\in \mathbb{Z}$ and $1<|z-3+2i|<4\}$ is ____.

Let $\alpha$ be a root of the equation $1+x^2+x^4=0$. Then the value of ${\alpha }^{1011}+{\alpha }^{2022}-{\alpha }^{3033}$ is equal to:

Let arg$(z)$ represent the principal argument of the complex number $z$. Then, $|z|=3$ and $\arg (z-1)-\arg (z+1)=\frac{\pi }{4}$ intersect:

If $z^2+z+1=0,z\in \mathbb{C}$, then $\left|{\sum }_{n=1}^{15}{\left(z^n+(-1{)}^n\frac{1}{z^n}\right)}^2\right|$ is equal to ______.