Question 1: Palette 1/167

JEE Mains 2026 MCQ

Let

$A = \{ z \in \mathbb{C} : |z - 2| \leq 4 \}$ and

$B = \{ z \in \mathbb{C} : |z - 2| + |z + 2| = 5 \}$.

Then the max $\{|z_1 - z_2| : z_1 \in A \text{ and } z_2 \in B \}$ is :

A

$ \dfrac{17}{2} $

B

8

C

9

D

$ \dfrac{15}{2} $

Question 2: Palette 2/167

JEE Mains 2026 MCQ

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 equal to :

A

61

B

37

C

42

D

50

Question 3: Palette 3/167

JEE Mains 2026 MCQ

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 $\sum\limits_{z \in \mathrm{~s}}|z|^2$ is equal to :

A

413

B

398

C

385

D

423

Question 4: Palette 4/167

JEE Mains 2026 MCQ

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

A

1

B

0

C

-1

D

256

Question 5: Palette 5/167

JEE Mains 2026 MCQ

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

Then $\operatorname{Min}_{z \in S}\left|\left(z+\frac{1}{2}(5+3 i)\right)\right|$ is equal to :

A

$\frac{1}{2}$

B

$\frac{5}{2}$

C

2

D

$\frac{3}{2}$

Question 6: Palette 6/167

JEE Mains 2026 MCQ

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

A

$\frac{5}{64}$

B

$\frac{1}{16}$

C

$\frac{7}{64}$

D

$\frac{3}{16}$

Question 7: Palette 7/167

JEE Mains 2026 MCQ

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

Then $34 \left| \frac{5z - 12}{5iz + 16} \right|^2$ is equal to:

A

20

B

26

C

12

D

16

Question 8: Palette 8/167

JEE Mains 2026 MCQ

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:

A

162

B

145

C

128

D

175

Question 9: Palette 9/167

JEE Mains 2026 MCQ

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

A

0

B

2

C

1

D

4

Question 10: Palette 10/167

JEE Mains 2026 MCQ

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

A

162

B

184

C

262

D

324

Question 11: Palette 11/167

JEE Mains 2026 MCQ

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

A

-10

B

-8

C

$ 10 \sqrt{13} $

D

$ 8\sqrt{13} $

Question 12: Palette 12/167

JEE Mains 2026 MCQ

Let $z_1, z_2 \in \mathbb{C}$ 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 :

A

18

B

22

C

29

D

34

Question 13: Palette 13/167

JEE Mains 2026 MCQ

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

A

160

B

176

C

194

D

187

Question 14: Palette 14/167

JEE Mains 2026 MCQ

Let $\mathrm{S}=\left\{z \in \mathrm{C}: z^2+4 z+16=0\right\}$. Then $\sum\limits_{z \in \mathrm{~S}}|z+\sqrt{3} \mathrm{i}|^2$ is equal to :

A

42

B

23

C

27

D

38

Question 15: Palette 15/167

JEE Mains 2026 MCQ

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:

A

9

B

4

C

5

D

1

Question 16: Palette 16/167

JEE Mains 2026 MCQ

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 :

A

12

B

17

C

22

D

24

Question 17: Palette 17/167

JEE Mains 2026 MCQ

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 $10(x-3 y)$ is :

A

20

B

31

C

35

D

75

Question 18: Palette 18/167

JEE Mains 2025 MCQ

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

A

$ \frac{21}{4} \pi^2 $

B

$ 6\pi^2 $

C

$ \frac{27}{4} \pi^2 $

D

$ 8\pi^2 $

Question 19: Palette 19/167

JEE Mains 2025 MCQ

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

A

16

B

24

C

12

D

18

Question 20: Palette 20/167

JEE Mains 2025 MCQ

Among the statements

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

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

A

both are incorrect

B

both are correct

C

only (S2) is correct

D

only (S1) is correct

Question 21: Palette 21/167

JEE Mains 2025 MCQ

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 $\alpha+i \beta$, $i=\sqrt{-1}$. Let p and q be the maximum and the minimum values of $\alpha+\beta$ respectively. Then $\mathrm{p}+\mathrm{q}$ is equal to :

A

130

B

150

C

160

D

140

Question 22: Palette 22/167

JEE Mains 2025 MCQ

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

A

0

B

1

C

i

D

-i

Question 23: Palette 23/167

JEE Mains 2025 MCQ

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

A

$ -19+2 i $

B

$-19-2 i$

C

$19-2 i$

D

$19+2 i$

Question 24: Palette 24/167

JEE Mains 2025 MCQ

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

A

$\sqrt{5}+1$

B

3

C

$\sqrt{3}+1$

D

2

Question 25: Palette 25/167

JEE Mains 2025 MCQ

Let $ |z_1 − 8−2i| \leq 1 $ and $ |z_2−2+6i| \leq 2 $, $ z_1, z_2 \in \mathbb{C} $. Then the minimum value of $ |z_1 − z_2| $ is :

A

3

B

10

C

7

D

13

Question 26: Palette 26/167

JEE Mains 2025 MCQ

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:

A

2

B

-6

C

6

D

-2

Question 27: Palette 27/167

JEE Mains 2025 MCQ

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

A

area of triangle ABO is $\frac{11}{4}$

B

area of triangle ABO is $\frac{11}{\sqrt{3}}$

C

ABO is a scalene triangle

D

ABO is an obtuse angled isosceles triangle

Question 28: Palette 28/167

JEE Mains 2025 MCQ

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

A

441

B

312

C

409

D

398

Question 29: Palette 29/167

JEE Mains 2025 MCQ

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

A

8

B

10

C

4

D

6

Question 30: Palette 30/167

JEE Mains 2025 MCQ

Let $\left|\frac{\bar{z}-i}{2 \bar{z}+i}\right|=\frac{1}{3}, z \in C$, 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:

A

$\frac{121}{25}$

B

100

C

$\frac{81}{25}$

D

50

Question 31: Palette 31/167

JEE Mains 2025 MCQ

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

A

$1+\frac{\pi}{3}$

B

$1+\frac{\pi}{6}$

C

$1+\frac{\pi}{2}$

D

$1+\frac{\pi}{4}$

Question 32: Palette 32/167

JEE Mains 2025 MCQ

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

A

41

B

29

C

24

D

31

Question 33: Palette 33/167

JEE Mains 2024 MCQ

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

A

8

B

12

C

10

D

$\infty$

Question 34: Palette 34/167

JEE Mains 2024 MCQ

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

A

$4 \pi$

B

$3 \pi$

C

$2 \pi$

D

$5 \pi$

Question 35: Palette 35/167

JEE Mains 2024 MCQ

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

A

$\frac{2 \sqrt{6}}{5}$

B

$\frac{24}{5}$

C

$\frac{\sqrt{6}}{5}$

D

$\frac{1+\sqrt{6}}{5}$

Question 36: Palette 36/167

JEE Mains 2024 MCQ

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

A

12

B

4

C

8

D

5

Question 37: Palette 37/167

JEE Mains 2024 MCQ

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

A

either $z_1$ lies on a circle of radius $\frac{1}{2}$ or $z_2$ lies on a circle of radius 1.

B

$z_1$ lies on a circle of radius $\frac{1}{2}$ and $z_2$ lies on a circle of radius 1.

C

either $z_1$ lies on a circle of radius 1 or $z_2$ lies on a circle of radius $\frac{1}{2}$.

D

both $z_1$ and $z_2$ lie on the same circle.

Question 38: Palette 38/167

JEE Mains 2024 MCQ

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

A

$\frac{125 \pi}{24}$

B

$\frac{125 \pi}{6}$

C

$\frac{125 \pi}{12}$

D

$\frac{125 \pi}{4}$

Question 39: Palette 39/167

JEE Mains 2024 MCQ

Consider the following two statements :

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

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

Between the above two statements,

A

both Statement I and Statement II are incorrect.

B

Statement I is correct but Statement II is incorrect.

C

Statement I is incorrect but Statement II is correct.

D

both Statement I and Statement II are correct.

Question 40: Palette 40/167

JEE Mains 2024 MCQ

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

A

$\frac{7 \pi}{4}$

B

$\frac{3 \pi}{2}$

C

$\frac{7 \pi}{3}$

D

$\frac{17 \pi}{8}$

Question 41: Palette 41/167

JEE Mains 2024 MCQ

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 $4(\alpha^2+\beta^2)$ is equal to :

A

4

B

2

C

6

D

8

Question 42: Palette 42/167

JEE Mains 2024 MCQ

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

A

2

B

$\frac{5}{2}$

C

$\frac{3}{2}$

D

3

Question 43: Palette 43/167

JEE Mains 2024 MCQ

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

A

1

B

4

C

3

D

2

Question 44: Palette 44/167

JEE Mains 2024 MCQ

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 -

A

$15 \sqrt{15}$

B

$30 \sqrt{3}$

C

$25 \sqrt{3}$

D

75

Question 45: Palette 45/167

JEE Mains 2024 MCQ

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

A

0

B

2

C

1

D

3

Question 46: Palette 46/167

JEE Mains 2024 MCQ

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

A

9

B

$\frac{1}{4}$

C

4

D

1

Question 47: Palette 47/167

JEE Mains 2024 MCQ

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

A

$\left(2 \sec \frac{11 \pi}{8}, \frac{11 \pi}{8}\right)$

B

$\left(2 \sec \frac{3 \pi}{8}, \frac{3 \pi}{8}\right)$

C

$\left(2 \sec \frac{5 \pi}{8}, \frac{3 \pi}{8}\right)$

D

$\left(2 \sec \frac{3 \pi}{8}, \frac{5 \pi}{8}\right)$

Question 48: Palette 48/167

JEE Mains 2024 MCQ

If $z=\frac{1}{2}-2 i$ 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

A

2

B

$-$4

C

3

D

$-$1

Question 49: Palette 49/167

JEE Mains 2024 MCQ

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

A

1

B

2

C

3

D

0

Question 50: Palette 50/167

JEE Mains 2023 MCQ

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

the interval $(\alpha, \beta]$, then $24(\beta-\alpha)$ is equal to :

A

36

B

27

C

42

D

30

Question 51: Palette 51/167

JEE Mains 2023 MCQ

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

A

$\frac{7}{2}$

B

4

C

3

D

$\frac{5}{2}$

Question 52: Palette 52/167

JEE Mains 2023 MCQ

Let $\mathrm{C}$ be the circle in the complex plane with centre $\mathrm{z}_{0}=\frac{1}{2}(1+3 i)$ and radius $r=1$. Let $\mathrm{z}_{1}=1+\mathrm{i}$ and the complex number $z_{2}$ be outside the 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 :

A

$\frac{3}{2}$

B

$\frac{5}{2}$

C

$\frac{13}{2}$

D

$\frac{7}{2}$

Question 53: Palette 53/167

JEE Mains 2023 MCQ

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

(S1): If $\operatorname{Re}(a), \operatorname{Im}(a) > 0$, then the set A contains all the real numbers

(S2) : If $\operatorname{Re}(a), \operatorname{Im}(a) < 0$, then the set B contains all the real numbers,

A

both are false

B

only (S1) is true

C

only (S2) is true

D

both are true

Question 54: Palette 54/167

JEE Mains 2023 MCQ

Let $w_{1}$ be the point obtained by the rotation of $z_{1}=5+4 i$ about the origin through a right angle in the anticlockwise direction, and $w_{2}$ be the point obtained by the rotation of $z_{2}=3+5 i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_{1}-w_{2}$ is equal to :

A

$-\pi+\tan ^{-1} \frac{8}{9}$

B

$-\pi+\tan ^{-1} \frac{33}{5}$

C

$\pi-\tan ^{-1} \frac{8}{9}$

D

$\pi-\tan ^{-1} \frac{33}{5}$

Question 55: Palette 55/167

JEE Mains 2023 MCQ

Let $S = \left\{ {z = x + iy:{{2z - 3i} \over {4z + 2i}}\,\mathrm{is\,a\,real\,number}} \right\}$. Then which of the following is NOT correct?

A

$y + {x^2} + {y^2} \ne - {1 \over 4}$

B

$(x,y) = \left( {0, - {1 \over 2}} \right)$

C

$x = 0$

D

$y \in \left( { - \infty , - {1 \over 2}} \right) \cup \left( { - {1 \over 2},\infty } \right)$

Question 56: Palette 56/167

JEE Mains 2023 MCQ

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

A

${4 \over 3}$

B

${3 \over 2}$

C

${3 \over 4}$

D

${2 \over 3}$

Question 57: Palette 57/167

JEE Mains 2023 MCQ

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 elements in $\mathrm{A}$ is :

A

$3 \pi$

B

$\pi$

C

$2 \pi$

D

$4 \pi$

Question 58: Palette 58/167

JEE Mains 2023 MCQ

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

A

$x^{2}+2 x-3=0$

B

$x^{2}+3 x-4=0$

C

$x^{2}+x-12=0$

D

$x^{2}+7 x+12=0$

Question 59: Palette 59/167

JEE Mains 2023 MCQ

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

A

0

B

2

C

1

D

Infinite

Question 60: Palette 60/167

JEE Mains 2023 MCQ

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 $|z-1|=|2z|$, then a possible value of $\frac{1+[a]}{4b}$, where $[t]$ is greatest integer function, is :

A

$\left(\frac{1+\sqrt{7}}{4}\right)$

B

$\frac{1}{2}$

C

0

D

$-$1

Question 61: Palette 61/167

JEE Mains 2023 MCQ

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

A

12

B

10

C

11

D

9

Question 62: Palette 62/167

JEE Mains 2023 MCQ

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

A

$\cos \frac{\pi}{12}-i \sin \frac{\pi}{12}$

B

$\sqrt{2}\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)$

C

$\sqrt{2} i\left(\cos \frac{5 \pi}{12}-i \sin \frac{5 \pi}{12}\right)$

D

$\sqrt{2}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)$

Question 63: Palette 63/167

JEE Mains 2023 MCQ

For all $z \in C$ on the curve $C_{1}:|z|=4$, let the locus of the point $z+\frac{1}{z}$ be the curve $\mathrm{C}_{2}$. Then :

A

the curves $C_{1}$ and $C_{2}$ intersect at 4 points

B

the curve $C_{2}$ lies inside $C_{1}$

C

the curve $C_{1}$ lies inside $C_{2}$

D

the curves $C_{1}$ and $C_{2}$ intersect at 2 points

Question 64: Palette 64/167

JEE Mains 2023 MCQ

For two non-zero complex numbers $z_{1}$ and $z_{2}$, if $\operatorname{Re}\left(z_{1} z_{2}\right)=0$ and $\operatorname{Re}\left(z_{1}+z_{2}\right)=0$, then which of the following are possible?

A. $\operatorname{Im}\left(z_{1}\right)>0$ and $\operatorname{Im}\left(z_{2}\right) > 0$

B. $\operatorname{Im}\left(z_{1}\right) < 0$ and $\operatorname{Im}\left(z_{2}\right) > 0$

C. $\operatorname{Im}\left(z_{1}\right) > 0$ and $\operatorname{Im}\left(z_{2}\right) < 0$

D. $\operatorname{Im}\left(z_{1}\right) < 0$ and $\operatorname{Im}\left(z_{2}\right) < 0$

Choose the correct answer from the options given below :

A

A and C

B

A and B

C

B and D

D

B and C

Question 65: Palette 65/167

JEE Mains 2023 MCQ

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

A

(0, $-$2)

B

(0, 0)

C

(0, 2)

D

(2, 0)

Question 66: Palette 66/167

JEE Mains 2023 MCQ

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

A

hyperbola with the length of the transverse axis 7

B

hyperbola with eccentricity 2

C

straight line with the sum of its intercepts on the coordinate axes equals $-18$

D

straight line with the sum of its intercepts on the coordinate axes equals $14$

Question 67: Palette 67/167

JEE Mains 2023 MCQ

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

A

$ - {1 \over 2}\left( {1 - i\sqrt 3 } \right)$

B

$ - {1 \over 2}\left( {\sqrt 3 - i} \right)$

C

${1 \over 2}\left( {1 - i\sqrt 3 } \right)$

D

${1 \over 2}\left( {\sqrt 3 + i} \right)$

Question 68: Palette 68/167

JEE Mains 2023 MCQ

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

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 69: Palette 69/167

JEE Mains 2022 MCQ

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

A

$\sqrt{2}$

B

1

C

$\sqrt{2}-1$

D

$\sqrt{2}+1$

Question 70: Palette 70/167

JEE Mains 2022 MCQ

Let $\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|<2,|z+i|=|z-1|\}$. Then the set of all values of $x$, for which $w=2 x+i y \in \mathrm{S}$ for some $y \in \mathbb{R}$, is :

A

$\left(-\sqrt{2}, \frac{1}{2 \sqrt{2}}\right]$

B

$\left(-\frac{1}{\sqrt{2}}, \frac{1}{4}\right]$

C

$\left(-\sqrt{2}, \frac{1}{2}\right]$

D

$\left(-\frac{1}{\sqrt{2}}, \frac{1}{2 \sqrt{2}}\right]$

Question 71: Palette 71/167

JEE Mains 2022 MCQ

If $z=2+3 i$, then $z^{5}+(\bar{z})^{5}$ is equal to :

A

244

B

224

C

245

D

265

Question 72: Palette 72/167

JEE Mains 2022 MCQ

Let $S_{1}=\left\{z_{1} \in \mathbf{C}:\left|z_{1}-3\right|=\frac{1}{2}\right\}$ and $S_{2}=\left\{z_{2} \in \mathbf{C}:\left|z_{2}-\right| z_{2}+1||=\left|z_{2}+\right| z_{2}-1||\right\}$. Then, for $z_{1} \in S_{1}$ and $z_{2} \in S_{2}$, the least value of $\left|z_{2}-z_{1}\right|$ is :

A

0

B

$\frac{1}{2}$

C

$\frac{3}{2}$

D

$\frac{5}{2}$

Question 73: Palette 73/167

JEE Mains 2022 MCQ

Let S be the set of all $(\alpha, \beta), \pi<\alpha, \beta<2 \pi$, for which the complex number $\frac{1-i \sin \alpha}{1+2 i \sin \alpha}$ is purely imaginary and $\frac{1+i \cos \beta}{1-2 i \cos \beta}$ is purely real. Let $Z_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in S$. Then $\sum\limits_{(\alpha, \beta) \in S}\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right)$ is equal to :

A

3

B

3 i

C

1

D

2 $-$ i

Question 74: Palette 74/167

JEE Mains 2022 MCQ

Let the minimum value $v_{0}$ of $v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}$ is attained at ${ }{z}=z_{0}$. Then $\left|2 z_{0}^{2}-\bar{z}_{0}^{3}+3\right|^{2}+v_{0}^{2}$ is equal to :

A

1000

B

1024

C

1105

D

1196

Question 75: Palette 75/167

JEE Mains 2022 MCQ

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

A

$x+2 y-4=0$

B

$x^{2}+y-4=0$

C

$x+2 y+4=0$

D

$x^{2}-y+3=0$

Question 76: Palette 76/167

JEE Mains 2022 MCQ

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

A

$\arg {z_2} = \pi - {\tan ^{ - 1}}3$

B

$\arg ({z_1} - 2{z_2}) = - {\tan ^{ - 1}}{4 \over 3}$

C

$|{z_2}| = \sqrt {10} $

D

$|2{z_1} - {z_2}| = 5$

Question 77: Palette 77/167

JEE Mains 2022 MCQ

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 Question: of $p$ is _____________.

A

3

B

$\frac{7}{2}$

C

4

D

$\frac{9}{2}$

Question 78: Palette 78/167

JEE Mains 2022 MCQ

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

A

0

B

2

C

3

D

4

Question 79: Palette 79/167

JEE Mains 2022 MCQ

The real part of the complex number ${{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }}$ is equal to :

A

${{500} \over {13}}$

B

${{110} \over {13}}$

C

${{55} \over {6}}$

D

${{550} \over {13}}$

Question 80: Palette 80/167

JEE Mains 2022 MCQ

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

A

exactly at one point.

B

exactly at two points.

C

nowhere.

D

at infinitely many points.

Question 81: Palette 81/167

JEE Mains 2022 MCQ

Let $\alpha$ and $\beta$ be the roots of the equation x² + (2i $-$ 1) = 0. Then, the value of |$\alpha$⁸ + $\beta$⁸| is equal to :

A

50

B

250

C

1250

D

1500

Question 82: Palette 82/167

JEE Mains 2022 MCQ

The number of points of intersection of

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

A

0

B

1

C

2

D

3

Question 83: Palette 83/167

JEE Mains 2022 MCQ

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

A

${{3\sqrt 3 } \over 4}$

B

${{3\sqrt 3 } \over 2}$

C

${3 \over 2}$

D

${3 \over 4}$

Question 84: Palette 84/167

JEE Mains 2022 MCQ

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

A

a portion of a circle centred at $\left( {0, - {1 \over {\sqrt 3 }}} \right)$ that lies in the second and third quadrants only

B

a portion of a circle centred at $\left( {0, - {1 \over {\sqrt 3 }}} \right)$ that lies in the second quadrant only

C

an empty

D

a portion of a circle of radius ${2 \over {\sqrt 3 }}$ that lies in the third quadrant only

Question 85: Palette 85/167

JEE Mains 2022 MCQ

Let z₁ and z₂ be two complex numbers such that ${\overline z _1} = i{\overline z _2}$ and $\arg \left( {{{{z_1}} \over {{{\overline z }_2}}}} \right) = \pi $. Then :

A

$\arg {z_2} = {\pi \over 4}$

B

$\arg {z_2} = - {{3\pi } \over 4}$

C

$\arg {z_1} = {\pi \over 4}$

D

$\arg {z_1} = - {{3\pi } \over 4}$

Question 86: Palette 86/167

JEE Mains 2022 MCQ

Let a circle C in complex plane pass through the points ${z_1} = 3 + 4i$, ${z_2} = 4 + 3i$ and ${z_3} = 5i$. If $z( \ne {z_1})$ is a point on C such that the line through z and z₁ is perpendicular to the line through z₂ and z₃, then $arg(z)$ is equal to :

A

${\tan ^{ - 1}}\left( {{2 \over {\sqrt 5 }}} \right) - \pi $

B

${\tan ^{ - 1}}\left( {{{24} \over 7}} \right) - \pi $

C

${\tan ^{ - 1}}\left( 3 \right) - \pi $

D

${\tan ^{ - 1}}\left( {{3 \over 4}} \right) - \pi $

Question 87: Palette 87/167

JEE Mains 2022 MCQ

Let $A = \{ z \in C:1 \le |z - (1 + i)| \le 2\} $

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

A

is an empty set

B

contains exactly two elements

C

contains exactly three elements

D

is an infinite set

Question 88: Palette 88/167

JEE Mains 2021 MCQ

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

A

$2\sqrt 2 - 1$

B

$3\sqrt 2 $

C

$6\sqrt 2 $

D

$2\sqrt 2 $

Question 89: Palette 89/167

JEE Mains 2021 MCQ

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

A

S contains exactly two elements

B

S contains only one element

C

S is a circle in the complex plane

D

S is a straight line in the complex plane

Question 90: Palette 90/167

JEE Mains 2021 MCQ

If ${\left( {\sqrt 3 + i} \right)^{100}} = {2^{99}}(p + iq)$, then p and q are roots of the equation :

A

${x^2} - \left( {\sqrt 3 - 1} \right)x - \sqrt 3 = 0$

B

${x^2} + \left( {\sqrt 3 + 1} \right)x + \sqrt 3 = 0$

C

${x^2} + \left( {\sqrt 3 - 1} \right)x - \sqrt 3 = 0$

D

${x^2} - \left( {\sqrt 3 + 1} \right)x + \sqrt 3 = 0$

Question 91: Palette 91/167

JEE Mains 2021 MCQ

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

A

centre at (0, $-$1) and radius $\sqrt 2 $

B

centre at (0, 1) and radius $\sqrt 2 $

C

centre (0, 0) and radius $\sqrt 2 $

D

centre at (0, 1) and radius 2

Question 92: Palette 92/167

JEE Mains 2021 MCQ

Let C be the set of all complex numbers. Let

S₁ = {z$\in$C : |z $-$ 2| $\le$ 1} and

S₂ = {z$\in$C : z(1 + i) + $\overline z $(1 $-$ i) $\ge$ 4}.

Then, the maximum value of ${\left| {z - {5 \over 2}} \right|^2}$ for z$\in$S₁ $\cap$ S₂ is equal to :

A

${{3 + 2\sqrt 2 } \over 4}$

B

${{5 + 2\sqrt 2 } \over 2}$

C

${{3 + 2\sqrt 2 } \over 2}$

D

${{5 + 2\sqrt 2 } \over 4}$

Question 93: Palette 93/167

JEE Mains 2021 MCQ

Let C be the set of all complex numbers. Let

${S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\} $

${S_2} = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\} $ and

${S_3} = \{ z \in C||z - \overline z | \ge 8\} $.

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

A

1

B

0

C

2

D

Infinite

Question 94: Palette 94/167

JEE Mains 2021 MCQ

Let n denote the number of solutions of the equation z² + 3$\overline z $ = 0, where z is a complex number. Then the value of $\sum\limits_{k = 0}^\infty {{1 \over {{n^k}}}} $ is equal to :

A

1

B

${4 \over 3}$

C

${3 \over 2}$

D

2

Question 95: Palette 95/167

JEE Mains 2021 MCQ

If z and $\omega$ are two complex numbers such that $\left| {z\omega } \right| = 1$ and $\arg (z) - \arg (\omega ) = {{3\pi } \over 2}$, then $\arg \left( {{{1 - 2\overline z \omega } \over {1 + 3\overline z \omega }}} \right)$ is :

(Here arg(z) denotes the principal argument of complex number z)

A

${\pi \over 4}$

B

$ - {{3\pi } \over 4}$

C

$ - {\pi \over 4}$

D

${{3\pi } \over 4}$

Question 96: Palette 96/167

JEE Mains 2021 MCQ

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

A

4

B

${1 \over 4}$

C

2

D

${1 \over 2}$

Question 97: Palette 97/167

JEE Mains 2021 MCQ

If the equation $a|z{|^2} + \overline {\overline \alpha z + \alpha \overline z } + d = 0$ represents a circle where a, d are real constants then which of the following condition is correct?

A

|$\alpha$|² $-$ ad $\ne$ 0

B

|$\alpha$|² $-$ ad > 0 and a$\in$R $-$ {0}

C

|$\alpha$|² $-$ ad $ \ge $ 0 and a$\in$R

D

$\alpha$ = 0, a, d$\in$R⁺

Question 98: Palette 98/167

JEE Mains 2021 MCQ

Let S₁, S₂ and S₃ be three sets defined as

S₁ = {z$\in$C : |z $-$ 1| $ \le $ $\sqrt 2 $}

S₂ = {z$\in$C : Re((1 $-$ i)z) $ \ge $ 1}

S₃ = {z$\in$C : Im(z) $ \le $ 1}

Then the set S₁ $\cap$ S₂ $\cap$ S₃ :

A

has exactly three elements

B

is a singleton

C

has infinitely many elements

D

has exactly two elements

Question 99: Palette 99/167

JEE Mains 2021 MCQ

The area of the triangle with vertices A(z), B(iz) and C(z + iz) is :

A

1

B

${1 \over 2}$| z |²

C

${1 \over 2}$| z + iz |²

D

${1 \over 2}$

Question 100: Palette 100/167

JEE Mains 2021 MCQ

The least value of |z| where z is complex number which satisfies the inequality $\exp \left( {{{(|z| + 3)(|z| - 1)} \over {||z| + 1|}}{{\log }_e}2} \right) \ge {\log _{\sqrt 2 }}|5\sqrt 7 + 9i|,i = \sqrt { - 1} $, is equal to :

A

8

B

3

C

2

D

$\sqrt 5 $

Question 101: Palette 101/167

JEE Mains 2021 MCQ

Let a complex number z, |z| $\ne$ 1,

satisfy ${\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2$. Then, the largest value of |z| is equal to ____________.

A

5

B

8

C

6

D

7

Question 102: Palette 102/167

JEE Mains 2021 MCQ

If $\alpha$, $\beta$ $\in$ R are such that 1 $-$ 2i (here i² = $-$1) is a root of z² + $\alpha$z + $\beta$ = 0, then ($\alpha$ $-$ $\beta$) is equal to :

A

$-$7

B

7

C

3

D

$-$3

Question 103: Palette 103/167

JEE Mains 2021 MCQ

Let the lines (2 $-$ i)z = (2 + i)$\overline z $ and (2 $+$ i)z + (i $-$ 2)$\overline z $ $-$ 4i = 0, (here i² = $-$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 :

A

${3 \over {2\sqrt 2 }}$

B

$3\sqrt 2 $

C

${1 \over {2\sqrt 2 }}$

D

${3 \over {\sqrt 2 }}$

Question 104: Palette 104/167

JEE Mains 2020 MCQ

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

A

line, y = –x

B

real axis

C

line, y = x

D

imaginary axis

Question 105: Palette 105/167

JEE Mains 2020 MCQ

The region represented by
{z = x + iy $ \in $ C : |z| – Re(z) $ \le $ 1} is also given by the
inequality : {z = x + iy $ \in $ C : |z| – Re(z) $ \le $ 1}

A

y² $ \le $ $2\left( {x + {1 \over 2}} \right)$

B

y² $ \le $ ${x + {1 \over 2}}$

C

y² $ \ge $ 2(x + 1)

D

y² $ \ge $ x + 1

Question 106: Palette 106/167

JEE Mains 2020 MCQ

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

A

–2¹⁵i

B

–2¹⁵

C

2¹⁵i

D

6⁵

Question 107: Palette 107/167

JEE Mains 2020 MCQ

If the four complex numbers $z,\overline z ,\overline z - 2{\mathop{\rm Re}\nolimits} \left( {\overline z } \right)$ and $z-2Re(z)$ represent the vertices of a square of side 4 units in the Argand plane, then $|z|$ is equal to :

A

4$\sqrt 2 $

B

4

C

2

D

2$\sqrt 2 $

Question 108: Palette 108/167

JEE Mains 2020 MCQ

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

A

33

B

9

C

24

D

57

Question 109: Palette 109/167

JEE Mains 2020 MCQ

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

A

2

B

4

C

1/2

D

3/2

Question 110: Palette 110/167

JEE Mains 2020 MCQ

If z₁ , z₂ are complex numbers such that
Re(z₁) = |z₁ – 1|, Re(z₂) = |z₂ – 1| , and
arg(z₁ - z₂) = ${\pi \over 6}$, then Im(z₁ + z₂ ) is equal to :

A

${{\sqrt 3 } \over 2}$

B

${1 \over {\sqrt 3 }}$

C

${2 \over {\sqrt 3 }}$

D

${2\sqrt 3 }$

Question 111: Palette 111/167

JEE Mains 2020 MCQ

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

A

-2$\sqrt 6 $

B

6

C

$\sqrt 6 $

D

-$\sqrt 6 $

Question 112: Palette 112/167

JEE Mains 2020 MCQ

The value of

${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}$ is :

A

${1 \over 2}\left( {\sqrt 3 - i} \right)$

B

-${1 \over 2}\left( {\sqrt 3 - i} \right)$

C

$ - {1 \over 2}\left( {1 - i\sqrt 3 } \right)$

D

${1 \over 2}\left( {1 - i\sqrt 3 } \right)$

Question 113: Palette 113/167

JEE Mains 2020 MCQ

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

A

$\sqrt {10} $

B

$\sqrt {7} $

C

$\sqrt {{{17} \over 2}} $

D

$\sqrt {8} $

Question 114: Palette 114/167

JEE Mains 2020 MCQ

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

A

$2\sqrt 3 $

B

$\sqrt {10} $

C

${{15} \over 4}$

D

${7 \over 2}$

Question 115: Palette 115/167

JEE Mains 2020 MCQ

If the equation, x² + bx + 45 = 0 (b $ \in $ R) has conjugate complex roots and they satisfy |z +1| = 2$\sqrt {10} $ , then :

A

b² – b = 42

B

b² + b = 12

C

b² + b = 72

D

b² – b = 30

Question 116: Palette 116/167

JEE Mains 2020 MCQ

If ${{3 + i\sin \theta } \over {4 - i\cos \theta }}$, $\theta $ $ \in $ [0, 2$\theta $], is a real number, then an argument of
sin$\theta $ + icos$\theta $ is :

A

$\pi - {\tan ^{ - 1}}\left( {{3 \over 4}} \right)$

B

$ - {\tan ^{ - 1}}\left( {{3 \over 4}} \right)$

C

${\tan ^{ - 1}}\left( {{4 \over 3}} \right)$

D

$\pi - {\tan ^{ - 1}}\left( {{4 \over 3}} \right)$

Question 117: Palette 117/167

JEE Mains 2020 MCQ

If ${\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {2z + i}}} \right) = 1$, where z = x + iy, then the point (x, y) lies on a :

A

straight line whose slope is ${3 \over 2}$

B

straight line whose slope is $-{2 \over 3}$

C

circle whose diameter is ${{\sqrt 5 } \over 2}$

D

circle whose centre is at $\left( { - {1 \over 2}, - {3 \over 2}} \right)$

Question 118: Palette 118/167

JEE Mains 2019 MCQ

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

A

n = 20 and Re(z) = –10

B

n = 40 and Re(z) = 10

C

n = 40 and Re(z) = –10

D

n = 20 and Re(z) = 10

Question 119: Palette 119/167

JEE Mains 2019 MCQ

The equation |z – i| = |z – 1|, i = $\sqrt { - 1} $, represents :

A

a circle of radius 1

B

the line through the origin with slope – 1

C

a circle of radius ${1 \over 2}$

D

the line through the origin with slope 1

Question 120: Palette 120/167

JEE Mains 2019 MCQ

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

A

$z\overline w = {{1 - i} \over {\sqrt 2 }}$

B

$\overline z w = i$

C

$z\overline w = {{ - 1 + i} \over {\sqrt 2 }}$

D

$\overline z w = -i$

Question 121: Palette 121/167

JEE Mains 2019 MCQ

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

A

$ - {1 \over 5} + {3 \over 5}i$

B

$ - {1 \over 5} - {3 \over 5}i$

C

${1 \over 5} - {3 \over 5}i$

D

$ - {3 \over 5} - {1 \over 5}i$

Question 122: Palette 122/167

JEE Mains 2019 MCQ

Let z $ \in $ C be such that |z| < 1.

If $\omega = {{5 + 3z} \over {5(1 - z)}}$z, then :

A

4Im( $\omega$) > 5

B

5Im( $\omega$) < 1

C

5Re( $\omega$) > 4

D

5Re( $\omega$) > 1

Question 123: Palette 123/167

JEE Mains 2019 MCQ

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

A

straight line whose slope is –1

B

straight line whose slope is 1.

C

circle whose radius is 1.

D

circle whose radius is $\sqrt 2$ .

Question 124: Palette 124/167

JEE Mains 2019 MCQ

If $z = {{\sqrt 3 } \over 2} + {i \over 2}\left( {i = \sqrt { - 1} } \right)$,

then (1 + iz + z⁵ + iz⁸)⁹ is equal to :

A

1

B

–1

C

0

D

(-1 + 2i)⁹

Question 125: Palette 125/167

JEE Mains 2019 MCQ

If $\alpha $ and $\beta $ be the roots of the equation x² – 2x + 2 = 0, then the least value of n for which ${\left( {{\alpha \over \beta }} \right)^n} = 1$ is :

A

2

B

5

C

4

D

3

Question 126: Palette 126/167

JEE Mains 2019 MCQ

Let z₁ and z₂ be two complex numbers satisfying | z₁ | = 9 and | z₂ – 3 – 4i | = 4. Then the minimum value of | z₁ – z₂ | is :

A

0

B

1

C

2

D

$\sqrt 2 $

Question 127: Palette 127/167

JEE Mains 2019 MCQ

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

A

${1 \over 2}$

B

$\sqrt 2 $

C

2

D

1

Question 128: Palette 128/167

JEE Mains 2019 MCQ

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

A

${{\sqrt {34} } \over 3}$

B

${5 \over 3}$

C

${5 \over 4}$

D

${{\sqrt {41} } \over 4}$

Question 129: Palette 129/167

JEE Mains 2019 MCQ

Let ${\left( { - 2 - {1 \over 3}i} \right)^3} = {{x + iy} \over {27}}\left( {i = \sqrt { - 1} } \right),\,\,$ where x and y are real numbers, then y $-$ x equals :

A

$-$ 85

B

85

C

$-$ 91

D

91

Question 130: Palette 130/167

JEE Mains 2019 MCQ

Let $z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5}.$ If R(z) and 1(z) respectively denote the real and imaginary parts of z, then :

A

R(z) = $-$ 3

B

R(z) < 0 and I(z) > 0

C

I(z) = 0

D

R(z) > 0 and I(z) > 0

Question 131: Palette 131/167

JEE Mains 2019 MCQ

Let z₁ and z₂ be any two non-zero complex numbers such that   $3\left| {{z_1}} \right| = 4\left| {{z_2}} \right|.$  If  $z = {{3{z_1}} \over {2{z_2}}} + {{2{z_2}} \over {3{z_1}}}$  then :

A

${\rm I}m\left( z \right) = 0$

B

$\left| z \right| = \sqrt {{17 \over 2}} $

C

$\left| z \right| =$ ${1 \over 2}\sqrt {9 + 16{{\cos }^2}\theta } $

D

Re(z) $=$ 0

Question 132: Palette 132/167

JEE Mains 2019 MCQ

Let z₀ be a root of the quadratic equation, x² + x + 1 = 0, If z = 3 + 6iz$_0^{81}$ $-$ 3iz$_0^{93}$, then arg z is equal to :

A

${\pi \over 4}$

B

${\pi \over 6}$

C

${\pi \over 3}$

D

0

Question 133: Palette 133/167

JEE Mains 2019 MCQ

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

A

${5\pi \over 6}$

B

$\pi $

C

${3\pi \over 4}$

D

${{2\pi } \over 3}$

Question 134: Palette 134/167

JEE Mains 2019 MCQ

Let $\alpha $ and $\beta $ be two roots of the equation x² + 2x + 2 = 0 , then $\alpha ^{15}$ + $\beta ^{15}$ is equal to :

A

-256

B

512

C

-512

D

256

Question 135: Palette 135/167

JEE Mains 2018 MCQ

The least positive integer n for which ${\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1,$ is :

A

2

B

3

C

5

D

6

Question 136: Palette 136/167

JEE Mains 2018 MCQ

If $\alpha ,\beta \in C$ are the distinct roots of the equation
x² - x + 1 = 0, then ${\alpha ^{101}} + {\beta ^{107}}$ is equal to :

A

2

B

-1

C

0

D

1

Question 137: Palette 137/167

JEE Mains 2018 MCQ

If |z $-$ 3 + 2i| $ \le $ 4 then the difference between the greatest value and the least value of |z| is :

A

$2\sqrt {13} $

B

8

C

4 + $\sqrt {13} $

D

$\sqrt {13} $

Question 138: Palette 138/167

JEE Mains 2018 MCQ

The set of all $\alpha $ $ \in $ R, for which w = ${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$ is purely imaginary number, for all z $ \in $ C satisfying |z| = 1 and Re z $ \ne $ 1, is :

A

an empty set

B

{0}

C

$\left\{ {0,{1 \over 4}, - {1 \over 4}} \right\}$

D

equal to R

Question 139: Palette 139/167

JEE Mains 2017 MCQ

The equation
Im $\left( {{{iz - 2} \over {z - i}}} \right)$ + 1 = 0, z $ \in $ C, z $ \ne $ i
represents a part of a circle having radius equal to :

A

2

B

1

C

${3 \over 4}$

D

${1 \over 2}$

Question 140: Palette 140/167

JEE Mains 2017 MCQ

Let z$ \in $C, the set of complex numbers. Then the equation, 2|z + 3i| $-$ |z $-$ i| = 0 represents :

A

a circle with radius ${8 \over 3}.$

B

a circle with diameter ${{10} \over 3}.$

C

an ellipse with length of major axis ${{16} \over 3}.$

D

an ellipse with length of minor axis ${{16} \over 9}.$

Question 141: Palette 141/167

JEE Mains 2017 MCQ

Let $\omega $ be a complex number such that 2$\omega $ + 1 = z where z = $\sqrt {-3} $. If

$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$,

then k is equal to :

A

z

B

-1

C

1

D

-z

Question 142: Palette 142/167

JEE Mains 2016 MCQ

The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $2\sqrt 2 $ units in the south-westwardsdirection. Then its new position in the Argand plane is at the point represented by :

A

2 + 2i

B

1 + i

C

$-$1 $-$ i

D

$-$2 $-$2i

Question 143: Palette 143/167

JEE Mains 2016 MCQ

A value of $\theta \,$ for which ${{2 + 3i\sin \theta \,} \over {1 - 2i\,\,\sin \,\theta \,}}$ is purely imaginary, is :

A

${\sin ^{ - 1}}\left( {{{\sqrt 3 } \over 4}} \right)$

B

${\sin ^{ - 1}}\left( {{1 \over {\sqrt 3 }}} \right)\,$

C

${\pi \over 3}$

D

${\pi \over 6}$

Question 144: Palette 144/167

JEE Mains 2015 MCQ

A complex number z is said to be unimodular if $\,\left| z \right| = 1$. Suppose ${z_1}$ and ${z_2}$ are complex numbers such that ${{{z_1} - 2{z_2}} \over {2 - {z_1}\overline {{z_2}} }}$ is unimodular and ${z_2}$ is not unimodular. Then the point ${z_1}$ lies on a :

A

circle of radius 2.

B

circle of radius ${\sqrt 2 }$.

C

straight line parallel to x-axis

D

straight line parallel to y-axis.

Question 145: Palette 145/167

JEE Mains 2014 MCQ

If z is a complex number such that $\,\left| z \right| \ge 2\,$, then the minimum value of $\,\,\left| {z + {1 \over 2}} \right|$ :

A

is strictly greater that ${{5 \over 2}}$

B

is strictly greater that ${{3 \over 2}}$ but less than ${{5 \over 2}}$

C

is equal to ${{5 \over 2}}$

D

lie in the interval (1, 2)

Question 146: Palette 146/167

JEE Mains 2013 MCQ

If z is a complex number of unit modulus and argument $\theta $, then arg $\left( {{{1 + z} \over {1 + \overline z }}} \right)$ equals :

A

$ - \theta \,\,$

B

${\pi \over 2} - \theta \,$

C

$\theta \,$

D

$\,\pi - \theta \,\,$

Question 147: Palette 147/167

JEE Mains 2012 MCQ

If $z \ne 1$ and $\,{{{z^2}} \over {z - 1}}\,$ is real, then the point represented by the complex number z lies :

A

either on the real axis or a circle passing through the origin.

B

on a circle with centre at the origin

C

either on real axis or on a circle not passing through the origin.

D

on the imaginary axis.

Question 148: Palette 148/167

JEE Mains 2011 MCQ

If $\omega ( \ne 1)$ is a cube root of unity, and ${(1 + \omega )^7} = A + B\omega \,$. Then $(A,B)$ equals :

A

(1 ,1)

B

(1, 0)

C

(- 1 ,1)

D

(0 ,1)

Question 149: Palette 149/167

JEE Mains 2011 MCQ

Let $\alpha \,,\beta $ be real and z be a complex number. If ${z^2} + \alpha z + \beta = 0$ has two distinct roots on the line Re z = 1, then it is necessary that :

A

$\beta \, \in ( - 1,0)$

B

$\left| {\beta \,} \right| = 1$

C

$\beta \, \in (1,\infty )$

D

$\beta \, \in (0,1)$

Question 150: Palette 150/167

JEE Mains 2010 MCQ

The number of complex numbers z such that $\left| {z - 1} \right| = \left| {z + 1} \right| = \left| {z - i} \right|$ equals :

A

1

B

2

C

$\infty $

D

0

Question 151: Palette 151/167

JEE Mains 2009 MCQ

If $\,\left| {z - {4 \over z}} \right| = 2,$ then the maximum value of $\,\left| z \right|$ is equal to :

A

$\sqrt 5 + 1$

B

2

C

$2 + \sqrt 2 $

D

$\sqrt 3 + 1$

Question 152: Palette 152/167

JEE Mains 2008 MCQ

The conjugate of a complex number is ${1 \over {i - 1}}$ then that complex number is :

A

${{ - 1} \over {i - 1}}$

B

${1 \over {i + 1}}\,$

C

${{ - 1} \over {i + 1}}$

D

${1 \over {i - 1}}$

Question 153: Palette 153/167

JEE Mains 2007 MCQ

If $\,\left| {z + 4} \right|\,\, \le \,\,3\,$, then the maximum value of $\left| {z + 1} \right|$ is :

A

6

B

0

C

4

D

10

Question 154: Palette 154/167

JEE Mains 2006 MCQ

If ${z^2} + z + 1 = 0$, where z is complex number, then value of ${\left( {z + {1 \over z}} \right)^2} + {\left( {{z^2} + {1 \over {{z^2}}}} \right)^2} + {\left( {{z^3} + {1 \over {{z^3}}}} \right)^2} + .......... + {\left( {{z^6} + {1 \over {{z^6}}}} \right)^2}$ is :

A

18

B

54

C

6

D

12

Question 155: Palette 155/167

JEE Mains 2006 MCQ

The value of $\sum\limits_{k = 1}^{10} {\left( {\sin {{2k\pi } \over {11}} + i\,\,\cos {{2k\pi } \over {11}}} \right)} $ is :

A

i

B

1

C

- 1

D

- i

Question 156: Palette 156/167

JEE Mains 2005 MCQ

If the cube roots of unity are 1, $\omega \,,\,{\omega ^2}$ then the roots of the equation ${(x - 1)^3}$ + 8 = 0, are :

A

$ - 1, - 1 + 2\,\,\omega , - 1 - 2\,\,{\omega ^2}$

B

$ - 1, - 1, - 1$

C

$ - 1,1 - 2\omega ,1 - 2{\omega ^2}$

D

$ - 1,1 + 2\omega ,1 + 2{\omega ^2}$

Question 157: Palette 157/167

JEE Mains 2005 MCQ

If $\,\omega = {z \over {z - {1 \over 3}i}}\,$ and $\left| \omega \right| = 1$, then $z$ lies on :

A

an ellipse

B

a circle

C

a straight line

D

a parabola

Question 158: Palette 158/167

JEE Mains 2005 MCQ

If ${z_1}$ and ${z_2}$ are two non-zero complex numbers such that $\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|$, then arg ${z_1}$ - arg ${z_2}$ is equal to :

A

${\pi \over 2}\,$

B

$ - \pi $

C

0

D

${{ - \pi } \over 2}$

Question 159: Palette 159/167

JEE Mains 2004 MCQ

Let z and w be complex numbers such that $\overline z + i\overline w = 0$ and arg zw = $\pi $. Then arg z equals :

A

${{5\pi } \over 4}$

B

${{\pi } \over 2}$

C

${{3\pi } \over 4}$

D

${{\pi } \over 4}$

Question 160: Palette 160/167

JEE Mains 2004 MCQ

If $\,\left| {{z^2} - 1} \right| = {\left| z \right|^2} + 1$, then z lies on :

A

an ellipse

B

the imaginary axis

C

a circle

D

the real axis

Question 161: Palette 161/167

JEE Mains 2004 MCQ

If $z = x - iy$ and ${z^{{1 \over 3}}} = p + iq$, then

${{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}$ is equal to :

A

- 2

B

- 1

C

2

D

1

Question 162: Palette 162/167

JEE Mains 2003 MCQ

If $z$ and $\omega $ are two non-zero complex numbers such that $\left| {z\omega } \right| = 1$ and $Arg(z) - Arg(\omega ) = {\pi \over 2},$ then $\,\overline {z\,} \omega $ is equal to

A

$- i$

B

1

C

- 1

D

$i$

Question 163: Palette 163/167

JEE Mains 2003 MCQ

Let ${Z_1}$ and ${Z_2}$ be two roots of the equation ${Z^2} + aZ + b = 0$, Z being complex. Further , assume that the origin, ${Z_1}$ and ${Z_2}$ form an equilateral triangle. Then :

A

${a^2} = 4b$

B

${a^2} = b$

C

${a^2} = 2b$

D

${a^2} = 3b$

Question 164: Palette 164/167

JEE Mains 2003 MCQ

If ${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$ then :

A

x = 2n + 1, where n is any positive integer

B

x = 4n , where n is any positive integer

C

x = 2n, where n is any positive integer

D

x = 4n + 1, where n is any positive integer.

Question 165: Palette 165/167

JEE Mains 2002 MCQ

z and w are two nonzero complex numbers such that $\,\left| z \right| = \left| w \right|$ and Arg z + Arg w =$\pi $ then z equals

A

$\overline \omega $

B

$ - \overline \omega $

C

$\omega $

D

$ - \omega $

Question 166: Palette 166/167

JEE Mains 2002 MCQ

If $\left| {z - 4} \right| < \left| {z - 2} \right|$, its solution is given by :

A

${\mathop{\rm Re}\nolimits} (z) > 0$

B

${\mathop{\rm Re}\nolimits} (z) < 0$

C

${\mathop{\rm Re}\nolimits} (z) > 3$

D

${\mathop{\rm Re}\nolimits} (z) > 2$

Question 167: Palette 167/167

JEE Mains 2002 MCQ

The locus of the centre of a circle which touches the circle $\left| {z - {z_1}} \right| = a$ and$\left| {z - {z_2}} \right| = b\,$ externally

($z,\,{z_1}\,\& \,{z_2}\,$ are complex numbers) will be :

A

an ellipse

B

a hyperbola

C

a circle

D

none of these