Q JEE MAIN 2022

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

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Q JEE MAIN_2026

Let $x$ and $y$ be real numbers such that $50\left(\frac{2 x}{1+3 i}-\frac{y}{1-2 i}\right)=31+17 i, i=\sqrt{-1}$. Then the value of $...

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Q JEE MAIN 2026

The number of values of $z \in £$, satisfying the equations $|z-(4+8 i)|=\sqrt{10}$ and $|z-(3+5 i)|+|z-(5+11 i)|=4 \sqrt{5}$, is :

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Q JEE MAIN 2026

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

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Q JEE MAIN 2026

Let the set of all values of $k \in \mathrm{i}$ such that the equation $z(\bar{z}+2+i)+k(2+3 i)=0, z \in \mathrm{C}$, has...

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Q JEE MAIN 2026

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

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Q JEE MAIN 2026

If $\alpha=1$ and $\beta=1+i \sqrt{2}$, where $i=\sqrt{-1}$ are two roots of the equation $...

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Q JEE MAIN 2026

Above equation has one solution using graph. Let $\mathrm{S}=\left\{z \in £: z^2+4 z+16=0\right\}$. Then $\sum_{z \in \mathrm{~S}}|z+\sqrt{3} \mathrm{i}|^2$ is equal...

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Q JEE MAIN 2026

Let $a, b \in £$. Let $\alpha, \beta$ be the roots of the equation $x^2+a x+b=0$. If $\beta-\alpha=\sqrt{11}$ and $...

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Q Jee Main 2026

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

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Q Jee Main 2026

Let the circles $\mathrm{C}_1:|z|=\mathrm{r}$ and $\mathrm{C}_2:|z-3-4 \mathbf{i}|=5, z \in \mathbb{C}$, be such that $\mathrm{C}_2$ lies within $\mathrm{C}_1$. If $z_1$ moves...

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Q JEE Main 2024

Let $S_1=\{z \in C:|z| \leq 5\}, S_2=\left\{z \in C: \operatorname{Im}\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right\}$ and $...

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Q JEE Main 2024

Let z be a complex number such that the real part of $\frac{z-2 i}{z+2 i}$ is zero. Then, the maximum...

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Q JEE Main 2024

The area (in sq. units) of the region $S=\{z \in \mathbb{C} ;|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{lm}(z) \geq 0\}$...

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Q Jee Main 2024

Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{6} x+3=0$ such that $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Let $a, b$ be integers not...

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Q JEE-Main 2024

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 $\mathrm{z}_0=1+\mathrm{i}$. Then, the value...

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Q JEE MAIN 2024

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

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Q Jee Main 2024

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z=2-i\left(2 \tan \frac{5 \pi}{8}\right)$, then $...

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Q Jee Main 2024

Let $z_1$ and $z_2$ be two complex number 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-

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Q JEE Main 2024

Let z=(1+i)(1+2i)(1+3i)...(1+n i), where $i=\sqrt{-1}$. If $|z|^2=44200$, then $n$ is equal to

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Q JEE Main 2019

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3\left|z_1\right|=4\left|z_2\right|$. If $z=\frac{3 z_1}{2 z_2}+\frac{2 z_2}{3 z_1}$ then

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Q JEE MAIN 2026

Let $$ \begin{aligned} & A=\{z \in \mathbb{C}:|z-2| \leqslant 4\} \text { and } \\ & B=\{z \in \mathbb{C}:|z-2|+|z+2|=5\} \end{aligned} $$...

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Q JEE MAIN 2026

Let $z$ be a complex number such that $|z-6|=5$ and $|z+2-6 i|=5$. Then the value of $z^3+3 z^2-15 z+141$ is...

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Q JEE MAIN 2026

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}, i=\sqrt{-1}$, then $\left(z^{201}-i\right)^8$ is equal to

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Q JEE MAIN 2026

Let $\mathrm{S}=\left\{z \in \mathbb{C}:\left|\frac{z-6 i}{z-2 i}\right|=1\right.$ and $\left.\left|\frac{z-8+2 i}{z+2 i}\right|=\frac{3}{5}\right\}$. Then $\sum_{z \in \mathrm{~S}}|z|^2$ is equal to

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Q JEE MAIN_2026

Let $\mathrm{S}=\{z: 3 \leq 2 z-3(1+\mathrm{i}) \mid \leq 7\}$ be a set of complex numbers. Then $...

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Q JEE MAIN_2026

Let $\alpha=\frac{-1+i \sqrt{3}}{2}$ and $\beta=\frac{-1-i \sqrt{3}}{2}, i=\sqrt{-1}$. If$(7-7 \alpha+9 \beta)^{20}+(9+7 \alpha-7 \beta)^{20}+(-7+9 \alpha+7 \beta)^{20}+(14+7 \alpha+7 \beta)^{20}=m^{10}$

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Q JEE MAIN 2026

Let $\mathrm{S}=\left\{z \in \mathbb{C}: 4 z^2+\bar{z}=0\right\}$. Then $\sum_{z \in \mathrm{~S}}|z|^2$ is equal to:

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Q JEE MAIN 2026

Let z be the complex number satisfying $|z-5| \leq 3$ and having maximum positive principal argument. Then $...

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Q JEE MAIN 2026

If $x^2+x+1=0$, then the value of $\left(x+\frac{1}{x}\right)^4+\left(x^2+\frac{1}{x^2}\right)^4+\left(x^3+\frac{1}{x^3}\right)^4+\ldots .+\left(x^{25}+\frac{1}{x^{25}}\right)^4$ is:

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Q JEE MAIN_2019

If $a>0$ and $z=\frac{(1+i)^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$, then $\bar{z}$ is equal to:

JEE Main Mathematics Medium

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Q JEE MAIN 2019

All the points in the set $\mathrm{S}=\left\{\frac{\alpha+\mathrm{i}}{\alpha-\mathrm{i}}: \alpha \in \mathrm{R}\right\}(\mathrm{i}=\sqrt{-1})$ lie on a :

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Q JEE MAIN 2019

Let $z \in C$ with $\operatorname{lm}(z)=10$ and it satisfies $\frac{2 z-n}{2 z+n}=2 i-1$ for some natural number $n$. Then :

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Q JEE MAIN 2020

The imaginary part of $(3+2 \sqrt{-54})^{1 / 2}-(3-2 \sqrt{-54})^{1 / 2}$ can be :

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Q JEE MAIN 2019

If $\frac{z-\alpha}{z+\alpha}(a \in R)$ is a purely imaginary number and $|z|=2$, then a value of $\alpha$ is

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Q JEE Main 2020

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

JEE Main Mathematics Medium

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Q JEE MAIN 2019

Let $z_0$ be a root of the quadratic equation, $x^2+x+1=0$. If $z=3+6 i z_0^{81}-3 i z_0^{93}$, then arg $z$ is...

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Q JEE MAIN 2020

If the four complex numbers $z, \bar{z}, \bar{z}-2 \operatorname{Re}(\bar{z})$ and $z-2 \operatorname{Re}(z)$ represent the vertices of a square of side...

JEE Main Mathematics Hard

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Q JEE MAIN 2019

Let $z_1$ and $z_2$ be two complex numbers satisfying $\left|z_1\right|=9$ and $\left|z_2-3-4 i\right|=4$. Then the minimum value of $...

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Q JEE MAIN 2020

Let $z=x+$ iy be a non-zero complex number such that $z^2=i|z|^2$, where $i=\sqrt{-1}$, then $z$ lies on the

JEE Main Mathematics Easy

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Q JEE MAIN 2020

The region represented by $\{z=x+i y \in C:|z|-\operatorname{Re}(z) \leq 1\}$ is also given by the inequality :

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Q JEE MAIN 2019

Let $z_0$ be a root of the quadratic equation, $x^2+x+1=0$. If $z=3+6 i z_0^{81}-3 i z_0^{93}$, then arg $z$ is...

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Q JEE MAIN 2020

If the equation, $x^2+b x+45=0(b \in R)$ has conjuate complex roots and they satisfy $|z+1|=2 \sqrt{10}$, then

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Q JEE MAIN 2020

Let $z$ be a complex number such that $\left|\frac{z-i}{z+2 i}\right|=1$ and $|z|=\frac{5}{2}$. Then the value of $|z+3 i|$ is

JEE Main Mathematics Medium

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Q JEE MAIN 2019

If $z$ and $w$ are two complex numbers such that $|z w|=1$ and $\arg (z)-\arg (w)=\frac{\pi}{2}$, then :

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Q JEE-MAIN 2020

The value of $\left(\frac{-1+i \sqrt{3}}{1-i}\right)^{30}$ is :

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Q JEE MAIN 2020

If $a$ and $b$ are real numbers such that $(2+\alpha)^4=a+b \alpha$, where $\alpha=\frac{-1+i \sqrt{3}}{2}$, then $a+b$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN 2020

Let $u=\frac{2 z+i}{z-k i}, z=x+i y$ and $k>0$. If the curve represented by $\operatorname{Re}(\mathrm{u})+\operatorname{Im}(\mathrm{u})=1$ intersects the y axis at the...

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Q JEE MAIN 2020

If z be a complex number satisfying |Re(z)| + |km(z)| = 4, then |z| cannot be

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Q JEE MAIN 2020

Let $\alpha=\frac{-1+\mathrm{i} \sqrt{3}}{2}$. If $\mathrm{a}=(1+\alpha) \sum_{\mathrm{k}=0}^{100} \alpha^{2 \mathrm{k}}$ and $\mathrm{b}=\sum_{\mathrm{k}=0}^{100} \alpha^{3 \mathrm{k}}$, then a and b are the roots of...

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Q JEE MAIN 2019

Let $z$ be a complex number such that $|z|+z=3+i($ where $i=\sqrt{-1})$. Then $|z|$ is equal to :

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Q JEE-MAIN 2020

If $z_1, z_2$ are complex numbers such that $\operatorname{Re}\left(z_1\right)=\left|z_1-1\right|, \operatorname{Re}\left(z_2\right)=\left|z_2-1\right|$, and $\arg \left(z_1-z_2\right)=\frac{\pi}{6}$, then $\operatorname{Im}\left(z_1+z_2\right)$ is equal to

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Q JEE MAIN 2019

Let $z \in C$ be such that $|z|

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Q JEE MAIN 2021

A point $Z$ moves in the complex plane such that arg $\left(\frac{z-2}{z+2}\right)-\frac{\pi}{4}$, then the minimum value of $|Z-9 \sqrt{2}-2 \mathrm{i}|^2$...

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Q JEE MAIN 2020

If $\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)=1$, where $z=x+i y$, then the point $(x, y)$ lies on a

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If $a_r=\cos \frac{2 r \pi}{9}+i \sin \frac{2 r \pi}{9}, r=1,2,3, \ldots, \mathrm{i}=\sqrt{-1}$ then the determinant $...

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Q JEE Main 2021

If $z$ is a complex number such that $\frac{z-1}{z-1}$ is purely imaginary, then the minimum value of $|z-(3+3 i)|$ is...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

Let $\mathrm{z}=\left(\frac{\sqrt{3}}{2}+\frac{\mathrm{i}}{2}\right)^5+\left(\frac{\sqrt{3}}{2}-\frac{\mathrm{i}}{2}\right)^5$. If $\mathrm{R}(\mathrm{z})$ and $\mathrm{I}(\mathrm{z})$ respectively denote the real and imaginary parts of z , then

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Q JEE MAIN 2021

If $S=\left\{z \in \mathbb{C}: \frac{z-i}{z+2 i} \in \mathbb{R}\right\}$, then:

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Q JEE Main 2021

Let $z=\frac{1-i \sqrt{3}}{2}, i=\sqrt{-1}$. Then the value of $21+\left(z+\frac{1}{z}\right)^3+\left(z^2+\frac{1}{z^2}\right)^3+\left(z^3+\frac{1}{z^3}\right)^3+\ldots+\left(z^{21}+\frac{1}{z^{21}}\right)^3$ is $\_\_\_\_$ .

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Q JEE Main 2021

The equation arg $\left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}$ represents a circle with:

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Q JEE MAIN 2021

The least positive integer n such that $\frac{(2 \mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}, \mathrm{i}=\sqrt{-1}$ is a positive integer, is

JEE Main Mathematics Medium

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Q JEE MAIN 2019

If $\alpha$ and $\beta$ be the roots of the equation $x^2-2 x+2=$ 0 , then the least value of $n$...

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Q JEE MAIN 2021

If $(\sqrt{3}+i)^{100}=9^{99}(p+i q)$, then $p$ and $q$ are roots of the equation :

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Q JEE Main 2019

Let $\alpha$ and $\beta$ be two roots of the equation $x^2+2 x+2=0$, then $\alpha^{15}+\beta^{15}$ is equal to

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Q JEE Main 2019

Let $A=\left\{\theta \in\left(-\frac{\pi}{2}, \pi\right): \frac{3+2 i \sin \theta}{1-2 i \sin \theta}\right.$ is purely imaginary $\}$ Then the sum of the...

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Q JEE MAIN 2021

If for the complex numbers $z$ satisfying $|z-2-2 i| \leq 1$, the maximum value of $|3 i z+6|$ is attained...

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Q JEE MAIN_2021

Let $...

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Q JEE MAIN 2021 S2

If the real part of the complex number $z=\frac{3+2 i \cos \theta}{1-3 i \cos \theta}, \theta \in\left(0, \frac{\pi}{2}\right)$ is zero,...

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Q JEE MAIN 2021 S2

Let be the set of all complex numbers. Let $$ \begin{aligned} & S_1=\{z \in \mathbb{C}:|z-2| \leq 1\} \text { and...

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Q JEE MAIN 2019

Let $f:[-1,3] \rightarrow R$ be defined as $$ f(x)=\left\{\begin{array}{cc} |x|+[x], & -1 \leq x

JEE Main Mathematics Easy

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Q JEE MAIN 2021

The equation of a circle is $\operatorname{Re}\left(z^2\right)+2(\operatorname{lm}(z))^2+2 \operatorname{Re}(z)=0$, where $z=x+i y$. A line which passes through the center of the...

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Q JEE MAIN 2019

If $z=\frac{\sqrt{3}}{2}+\frac{i}{2}(i=\sqrt{-1})$, then $\left(1+i z+z^5+i z^8\right)^9$ is equal to :

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Q JEE MAIN 2021

Let zbe those complex numbers which satisfy $|z+5| \leq 4$ and $z(1+i)+\bar{z}(1-i) \geq-10, i-\sqrt{-1}$. If the maximum value of $|z+1|^2$...

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Q JEE MAIN 2021

Let C be the set of all complex numbers. Let $...

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Q JEE MAIN 2021

If $z$ and $\omega$ are two complex numbers such that $|z \omega|=1$ and arg $(z)-\arg (\omega) \frac{3 \pi}{2}$, then $...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x)=f\left(x^3\right)+x g\left(x^3\right)$ is divisible by $x^2+x$ +1 , then...

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Q JEE-MAIN 2021

Let $S_1, S_2$ and $S_3$ be three sets defined as $$ \begin{aligned} & S_1=\{z \in \mathbb{C}:|z-1| \leq \sqrt{2}\} \\ &...

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Q JEE MAIN 2021

Let a complex number be $\mathrm{w}=1-\sqrt{3} i$. Let another complex number $z$ be such that $|z w d|=1$ and $...

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Q JEE MAIN 2021

If $a, \beta \in R$ are such that $1-2 i$ (here $i^2=-1$ ) is a root of $z^2+\alpha z+\beta=0$, then...

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Q JEE MAIN 2021

If the real part of the complex number $(1-\cos \theta+2 i \sin \theta)^{-1}$ is $\frac{1}{5}$ for $\theta \in(0, \pi)$, then...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If the least and the largest real values of $a$, for which the equation $z+\alpha|z-1|+2 i=0(z \in C$ and $i=\sqrt{-1})$...

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Q JEE MAIN 2021

The least value of $|z|$ where $z$ is complex number which satisfies the inequality $$ \exp \left(\frac{(|z|+3)(|z|-1)}{||z|+1|} \log _e 2\right)...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $\mathrm{i}=\sqrt{-1}$. If $\frac{(-1+\mathrm{i} \sqrt{3})^{21}}{(1-\mathrm{i})^{24}}+\frac{(1+\mathrm{i} \sqrt{3})^{21}}{(1+\mathrm{i})^{24}}=\mathrm{k}$, and $\mathrm{n}=$ [| $\mathrm{k} \mid]$ be the greatest integral part of $|\mathrm{k}|$. Then $\sum_{j=0}^{n+5}(j+5)^2-\sum_{j=0}^{n+5}(j+5)$...

JEE Main Mathematics Hard

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Q JEE MAIN 2021

Let $z_1, z_2$ be the roots of the equation $z^2+a z+12 =0$ and $\mathrm{Z}_1, \mathrm{Z}_2$ form an equilateral triangle with...

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Q JEE MAIN 2021

If the equation $\mathrm{a}|\mathrm{z}|^2+\overline{\alpha \overline{\mathrm{z}}+\alpha \overline{\mathrm{z}}}+\mathrm{d}=0$ represents a circle where $a, d$ are real constant when which of the following...

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Q JEE-MAIN 2021

Let $z$ and $\omega$ be two complex numbers such that $\omega=z \bar{z}-2 z+2,\left|\frac{z+i}{z-3 i}\right|=1$ and $\operatorname{Re}(\omega)$ has minimum value. Then,...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let a complex number $\mathrm{z},|\mathrm{z}| \neq 1$, satisfy $\log _{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^2}\right) \leq 2$. Then, the largest value of $|z|$ is equal...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

The sum of 162th power of the roots of equation $x^3-2 x^2+2 x-1=0$ is

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let the lines $(2-i) z=(2+i) \bar{z}$ and $(2+i) z+(i-2) \bar{z}-4 i=0$, (here $i^2=-1$ ) be normal to a circle $C$....

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Q JEE MAIN_2022

Let $O$ be the origin and $A$ be the point $z 1=1+2 i$. If $B$ is the point $...

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Q JEE MAIN 2022

For $\mathrm{n} \in \mathrm{N}$, let $\mathrm{S}_{\mathrm{n}}=\left\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-3+2 \mathrm{i}|=\frac{\mathrm{n}}{4}\right\}$ and $\mathrm{T}_{\mathrm{n}}=\left\{\mathrm{z} \in \mathrm{C}:|\mathrm{z}-2+3 \mathrm{i}|=\frac{1}{4}\right\}$. Then the number of elements in...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

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

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $S=\{z \in \mathbb{C}:|z-3| \leq$ and $z(4+3 i)+\bar{z}(4-3 i) \leq 24\}$. if $\alpha+\mathrm{i} \beta$ is the point in S which...

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

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:

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Q JEE MAIN 2022

Let for some real numbers $\alpha$ and $\beta, \mathrm{a}=\alpha-\mathrm{i} \beta$. If the system of equations 4ix $+(1+\mathrm{i}) \mathrm{y}=0$ and $...

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

Let $S=\{z \in C:|z-2| \leq 1, z(1+i)+\bar{z}(1-i) \leq 2\}$. Let $|z-4 i|$ attains minimum and maximum values, respectively, at $...

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Q JEE MAIN 2022

Let $z_1$ and $z_2$ be two complex numbers such that $\bar{z}_1=i \bar{z}_2$ and arg $\left(\frac{z_1}{\bar{z}_2}\right)=\pi$. Then

JEE Main Mathematics Medium

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Q JEE MAIN 2022

The number of elements in the set $\{z=a+i b \in \mathbb{C}: a, b \in \mathbb{Z}$ and $1

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+(2 i-1)=0$. Then, the value of $\left|\alpha^8+\beta^8\right|$ is equal to...

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Q JEE MAIN 2022

The area of the polygon, whose vertices are the non-real roots of the equation $\bar{z}=i z^2$ is:

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Q JEE MAIN 2022

Let a circle C in complex plane pass although the points $z_1=3+4 i, z_2=4+3 i$ and $z_3=5 i$. If $...

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Q JEE MAIN 2022

Let: $A=\left\{z \in C:\left|\frac{z+1}{z-1}

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Q JEE MAIN 2022

Let a circle $C$ touch the lines $L_1: 4 x-3 y+K_1=0$ and $L_2: 4 x-3 y+K_2=0, K_1, K_2 \in R$....

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Q JEE MAIN 2022

Let $A=\{z \in C: 1 \leq|z-(1+i)| \leq 2\}$ and $B=\{z \in A:|z-(1-i)|=1\}$. Then, $B:$

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Q JEE MAIN 2025

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

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Q JEE MAIN 2023

Let $S=\left\{z=x+i y: \frac{2 z-3 i}{4 z+2 i}\right.$ is a real number $\}$. Then which of the following is NOT...

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Q JEE MAIN 2023

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

JEE Main Mathematics Hard

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Q JEE MAIN 2023

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

JEE Main Mathematics Medium

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Q JEE-Main 2023

Let A $=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.$ is purely imaginary $\}$. Then the sum of the...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

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

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $a \neq b$ be two non-zero real numbers. Then the number of elements in the set $...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

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

JEE Main Mathematics Medium

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Q JEE MAIN_2025

If the locus of $z \in \mathrm{C}$, such that $\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)+\operatorname{Re}\left(\frac{\bar{z}-1}{2 \bar{z}-i}\right)=2$, is a circle of radius r and center...

JEE Main Physics Medium

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Q JEE MAIN 2025

Let the product of $\omega_1=(8+i) \sin \theta+(7+4 i) \cos \theta$ and $\omega_2=(1+8 i) \sin \theta+(4+7 i) \cos \theta$ be $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

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

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $\omega=\mathrm{z} \overline{\mathrm{z}}+\mathrm{k} 1 \mathrm{z}+\mathrm{k} 2 \mathrm{z}+\lambda(1+\mathrm{j}), \mathrm{k}_1, \mathrm{k} 2 \in \mathbb{R}$. Let $\operatorname{Re}(\omega)=0$ be the circle C of radius...

JEE Main Mathematics Medium

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Q jee main 2023

$$ \text { If the set }\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in \mathbb{C}, \operatorname{Re}(z)=3\right\} \text { is equal to the...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let C be the circle in the complex plane with centre $z_0=\frac{1}{2}(1+3 i)$ and radius $r=1$. Let $z_1=1+i$ and the...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

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

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let w1 be the point obtained by the rotation of z1 = 5 + 4i about the origin through a...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

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

JEE Main Mathematics Easy

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Q JEE MAIN

The sum of the square of the modulus of the elements in the set {z=a+ib:a,b∈Z,z∈C,|z-1|≤1,|z-5|≤|z-5i|} is

JEE Main Mathematics Medium

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Q JEE MAIN 2024

If the set $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}) ; \mathrm{a}+5 \mathrm{~b}=42, \mathrm{a}, \mathrm{b} \in \mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^m\left(1+i^{n!}\right)=x+i y$, where $\mathrm{I}=\sqrt{-1}$,...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

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

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Consider the following two statements Statement I: For any two non-zero complex numbers $z_1, z_2\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)$ and Statement II:...

JEE Main Mathematics Hard

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Q JEE-Main 2024

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

JEE Main Mathematics Hard

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Q JEE MAIN 2024

If $\alpha$ denotes the number of solutions of $|1-\mathrm{i}|^{\mathrm{x}}=2^{\mathrm{x}}$ and $\beta=\left(\frac{|z|}{\arg (z)}\right)$, where $\mathrm{z}=\frac{\pi}{4}(1+\mathrm{i})^4\left(\frac{1-\sqrt{\pi} \mathrm{i}}{\sqrt{\pi}+\mathrm{i}}+\right. \left.\frac{\sqrt{\pi}-\mathrm{i}}{1+\sqrt{\pi} \mathrm{i}}\right), \mathrm{i}=\sqrt{-1}$, then the...

JEE Main Mathematics Medium

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Q JEE-Main 2024

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

JEE Main Mathematics Easy

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Q JEE MAIN

$\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^2} \int_{x^3}^{\left(\frac{\pi}{2}\right)^3} \cos \left(\frac{1}{t^3}\right) d t\right)$ is equal to

JEE Main Mathematics Easy

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Q JEE MAIN

If z=x+iy,xy≠0, satisfies the equation z^2+iz ‾=0, then |z^2 | is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN

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

JEE Main Mathematics Easy

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Q JEE MAIN 2024

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

JEE Main Mathematics Easy

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Q JEE MAIN 2024

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

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let integers $a, b \in[-3,3]$ be such that $a+b \neq 0$. Then the number of all possible ordered pairs (a,...

JEE Main Physics Easy

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Q JEE MAIN 2025

Among the statements (S1) : The set $\left\{ {z \in - \{ - i\} :|z| = 1} \right.$ and $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $z \in \mathbb{C}$ be such that $\frac{{{z^2} + 3i}}{{z - 2 + i}} = 2 + 3i$. Then the...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let z be a complex number such that $|z| = 1$. If $...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

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

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $\left| {{z_1} - 8 - 2i} \right| \le 1$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let O be the origin, the point A be ${z_1} = \sqrt 3 + 2\sqrt 2 i$, the point $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

If $\alpha $ and $\beta $ are the roots of the equation $2{z^2} - 3z - 2i = 0$, where...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Let $\left| {\frac{{\bar z - i}}{{2\bar z + i}}} \right| = \frac{1}{3},z \in C$ , be the equation of a...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

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

JEE Main Mathematics Medium

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