Question

Let $u=\frac{2z+i}{z-ki},z=x+iy$ and $k>0$. If the curve represented by$Re\left(u\right)+Im\left(u\right)=1$ intersects the $y$-axis at points $\mathrm{P}$ and $\mathrm{Q}$ where $\mathrm{PQ}=5$ then the value of $\mathrm{k}$ is

JEE Main 2020 (04 Sep Shift 1)

Options

• A: $\frac{3}{2}$
• B: $\frac{1}{2}$
• C: $4$
• D: $2$

Question

Let a complex number be $w=1-\sqrt{3}i$. Let another complex number $z$ be such that $\left|zw\right|=1$ and $\arg \left(z\right)-\arg \left(w\right)=\frac{\pi }{2}$. Then the area of the triangle (in sq. units) with vertices origin, $z$ and $w$ is equal to

JEE Main 2021 (18 Mar Shift 2)

Options

• A: $4$
• B: $\frac{1}{2}$
• C: $\frac{1}{4}$
• D: $2$

Question

Let the lines $\left(2-i\right)z=\left(2+i\right)\overline{z}$ and $\left(2+i\right)z+\left(i-2\right)\overline{z}-4i=0,$ (here $i^2=-1$) be normal to a circle $C$. If the line $iz+\overline{z}+1+i=0$ is tangent to this circle $C$, then its radius is :

JEE Main 2021 (25 Feb Shift 1)

Options

• A: $\frac{3}{\sqrt{2}}$
• B: $3\sqrt{2}$
• C: $\frac{3}{2\sqrt{2}}$
• D: $\frac{1}{2\sqrt{2}}$

Question

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

$S_1=\left\{z\in C| {\left|z–3–2i\right|}^2=8\right\},$

$S_2=z\in C| \mathrm{Re}\left(\mathrm{z}\right)\ge 5$ and

$S_3=\left\{z\in C| \left|z–\overline{z}\right|\ge 8\right\}.$

Then the number of elements in $S_1\cap S_2\cap S_3$ is equal to

JEE Main 2021 (27 Jul Shift 1)

Options

• A: $1$
• B: $0$
• C: $2$
• D: Infinite

Question

Let $z$ and $w$ be two complex numbers such that $w=z\overline{z}-2z+2,\left|\frac{z+i}{z-3i}\right|=1$ and $Re\left(w\right)$ has minimum value. Then, the minimum value of $n\in N$ for which $w^n$ is real, is equal to _______.

JEE Main 2021 (16 Mar Shift 1)

Enter your answer

Question

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

JEE Main 2022 (25 Jul Shift 1)

Options

• A: $4$
• B: $1$
• C: $-4$
• D: $-1$

Question

For $n\in N$, let $S_{\mathit{n}}=\left\{z\in C:\left|z-3+2i\right|=\frac{n}{4}\right\}$ and $T_n=\left\{z\in C:\left|z-2+3i\right|=\frac{1}{n}\right\}$. Then the number of elements in the set $\left\{n\in N:S_{\mathit{n}}\cap T_{\mathit{n}}=\varphi \right\}$ is

JEE Main 2022 (25 Jul Shift 1)

Options

• A: $0$
• B: $2$
• C: $3$
• D: $4$

Question

Let a circle $C$ in complex plane pass through the points $z_1=3+4i,z_2=4+3i$ and $z_3=5i$. If $z\left(\ne z_1\right)$ is a point on $C$ such that the line through $z$ and $z_1$ is perpendicular to the line through $z_2$ and $z_3$, then $\arg \left(z\right)$ is equal to

JEE Main 2022 (25 Jun Shift 1)

Options

• A: ${\tan }^{-1}\left(\frac{24}{7}\right)-\pi$
• B: ${\tan }^{-1}\left(\frac{2}{\sqrt{5}}\right)-\pi$
• C: ${\tan }^{-1}\left(3\right)-\pi$
• D: ${\tan }^{-1}\left(\frac{3}{4}\right)-\pi$

Question

Let $O$ be the origin and $A$ be the point $z_1=1+2i$. If $B$ is the point $z_2,Re\left(z_2\right)<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true?

JEE Main 2022 (26 Jul Shift 1)

Options

• A: $\arg z_2=\pi -{\tan }^{-1}3$
• B: $\arg \left(z_1-2z_2\right)=-{\tan }^{-1}\frac{4}{3}$
• C: $\left|z_2\right|=\sqrt{10}$
• D: $\left|2z_1-z_2\right|=5$

Question

Let $A=\left\{z\in C:\left|\frac{z+1}{z-1}\right|<1\right\}$ and $B=\left\{z\in C:\arg \left(\frac{z-1}{z+1}\right)=\frac{2\pi }{3}\right\}$. Then $A\cap B$ is

JEE Main 2022 (26 Jun Shift 1)

Options

• A: a portion of a circle centred at $\left(0,-\frac{1}{\sqrt{3}}\right)$ that lies in the second and third quadrants only
• B: a portion of a circle centred at $\left(0,-\frac{1}{\sqrt{3}}\right)$ that lies in the second quadrant only
• C: an empty set
• D: a portion of a circle of radius $\frac{2}{\sqrt{3}}$ that lies in the third quadrant only

Question

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 $z_1\in S_1$ and $z_2\in S_2$, the least value of $\left|z_2-z_1\right|$ is

JEE Main 2022 (28 Jul Shift 1)

Options

• A: $0$
• B: $\frac{1}{2}$
• C: $\frac{3}{2}$
• D: $\frac{5}{2}$

Question

Let $S=${$z\in \mathrm{ℂ}:\left|z-3\right|\le 1$ and$z\left(4+3i\right)+\overline{z}\left(4-3i\right)\le 24$}. If $\alpha +i\beta$ is the point in $S$ which is closest to $4i$, then $25\left(\alpha +\beta \right)$ is equal to ______.

JEE Main 2022 (24 Jun Shift 2)

Enter your answer

Question

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

JEE Main 2022 (28 Jul Shift 2)

Enter your answer

Question

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

JEE Main 2022 (28 Jun Shift 2)

Enter your answer

Question

Let $a,b$ be two real numbers such that $ab<0$. If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $\left|z-1\right|=\left|2z\right|$, then a possible value of $\frac{1+\left[a\right]}{4b}$, where $\left[t\right]$ is greatest integer function, is :

JEE Main 2023 (01 Feb Shift 2)

Options

• A: $0$
• B: $-1$
• C: $1$
• D: $\frac{1}{2}$

Question

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

JEE Main 2023 (24 Jan Shift 1)

Options

• A: $x^2+4x-1=0$
• B: $x^2-4x+1=0$
• C: $x^2+4x+1=0$
• D: $x^2-4x-1=0$

Question

Let $z_1=2+3i$ and $z_2=3+4i$. The set $\mathrm{S}=\left\{\mathrm{z}\in \mathrm{C}:{\left|\mathrm{z}-{\mathrm{z}}_1\right|}^2-{\left|\mathrm{z}-{\mathrm{z}}_2\right|}^2={\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}^2\right\}$ represents a

JEE Main 2023 (25 Jan Shift 1)

Options

• A: straight line with sum of its intercepts on the coordinate axes equals $14$
• B: hyperbola with the length of the transverse axis $7$
• C: straight line with the sum of its intercepts on the coordinate axes equals $-18$
• D: hyperbola with eccentricity $2$

Question

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

JEE Main 2023 (25 Jan Shift 2)

Options

• A: $(2,0)$
• B: $(0,2)$
• C: $(0,0)$
• D: $(0,-2)$

Question

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

JEE Main 2023 (06 Apr Shift 2)

Options

• A: $0$
• B: $1$
• C: $3$
• D: $2$

Question

Let $C$ be the circle in the complex plane with centre $z_0=\frac{1}{2}\left(1+3i\right)$ and radius $r=1$. Let $z_1=1+i$ and the complex number $z_2$ be outside circle $C$ such that $\left|z_1-z_0\right|\left|z_2-z_0\right|=1$. If $z_0, z_1$ and $z_2$ are collinear, then the smaller value of ${\left|z_2\right|}^2$ is equal to

JEE Main 2023 (12 Apr Shift 1)

Options

• A: $\frac{5}{2}$
• B: $\frac{7}{2}$
• C: $\frac{13}{2}$
• D: $\frac{3}{2}$

Question

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

JEE Main 2023 (13 Apr Shift 2)

Options

• A: $\frac{5}{2}$
• B: $4$
• C: $\frac{7}{2}$
• D: $3$

Question

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

JEE Main 2024 (04 Apr Shift 1)

Options

• A: 6
• B: 8
• C: 2
• D: 4

Question

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

JEE Main 2024 (04 Apr Shift 2)

Options

• A: $\frac{7 \pi}{3}$
• B: $\frac{7 \pi}{4}$
• C: $\frac{17 \pi}{8}$
• D: $\frac{3 \pi}{2}$

Question

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 $S_3=\{z \in C: \operatorname{Re}(z) \geq 0\}$. Then the area of the region $S_1 \cap S_2 \cap S_3$ is :

JEE Main 2024 (05 Apr Shift 2)

Options

• A: $\frac{125 \pi}{12}$
• B: $\frac{125 \pi}{4}$
• C: $\frac{125 \pi}{24}$
• D: $\frac{125 \pi}{6}$

Question

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 $(r,\theta )$ is equal to

JEE Main 2024 (29 Jan Shift 2)

Options

• A: $\left(2\sec \frac{3\pi }{8},\frac{3\pi }{8}\right)$
• B: $\left(2\sec \frac{3\pi }{8},\frac{5\pi }{8}\right)$
• C: $\left(2\sec \frac{5\pi }{8},\frac{3\pi }{8}\right)$
• D: $\left(2\sec \frac{11\pi }{8},\frac{11\pi }{8}\right)$