How many questions from Complex Numbers appear in JEE?

Complex Numbers has appeared in 165 of the last 186 JEE Main and JEE Advanced papers — about 89% of them — contributing 191 questions in total across those papers.

Is Complex Numbers an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 89% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Complex Numbers questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Complex Numbers JEE Previous Year Questions with Answers

Q1·MathematicsSingle correctJEE Main 2026

The number of values of

z \in \mathbb{C}

, satisfying the equations

|z - (4 + 8i)| = \sqrt{10}

and

|z - (3 + 5i)| + |z - (5 + 11i)| = 4\sqrt{5}

, is:

(A)0
(B)2
(C)1
(D)4

Q2·MathematicsSingle correctJEE Main 2026

Let the set of all values of k ∈ ℝ such that the equation

z(\bar{z} + 2 + i) + k(2 + 3i) = 0

, z ∈ ℂ, has at least one solution, be the interval [α, β]. Then 9(α + β) is equal to:

(A)−10
(B)−8
(C)10√13
(D)8√13

Q3·MathematicsSingle correctJEE Main 2026

Let

S = \{z \in \mathbb{C} : z^{2} + \sqrt{6}\,iz - 3 = 0\}

. Then

\sum_{z \in S} z^{8}

is equal to :

(A)162
(B)184
(C)262
(D)324

Q4·MathematicsSingle correctJEE Main 2026

Let

z_1, z_2

∈ ℂ be the distinct solutions of the equation

z^2 + 4z - (1 + 12i) = 0

. Then

|z_1|^2 + |z_2|^2

is equal to :

(A)18
(B)22
(C)29
(D)34

Q5·MathematicsSingle correctJEE 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

|z|^2

is equal to:

(A) $9$
(B) $4$
(C) $5$
(D) $1$

Q6·MathematicsSingle correctJEE Main 2026

Let

S = \{z \in \mathbb{C} : z^{2} + 4z + 16 = 0\}

. Then

\sum_{z \in S} |z + \sqrt{3}i|^{2}

is equal to:

(A)42
(B)23
(C)27
(D)38

Q7·MathematicsSingle correctJEE Main 2026

If

\alpha = 1

and

\beta = 1 + i\sqrt{2}

, where

i = \sqrt{-1}

are two roots of the equation

x^{3} + ax^{2} + bx + c = 0

,

a, b, c \in \mathbb{R}

, then

\int_{-1}^{1}(x^{3} + ax^{2} + bx + c)dx

is equal to:

(A)−2
(B)−4
(C)−8
(D)−10

Q8·MathematicsSingle correctJEE Main 2026

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) $24$

Q9·MathematicsSingle correctJEE Main 2026

Let x and y be real numbers such that

50\left(\frac{2x}{1+3i}-\frac{y}{1-2i}\right)=31+17i

,

i=\sqrt{-1}

. Then the value of

10(x-3y)

is :

(A)20
(B)31
(C)35
(D)75

Q10·MathematicsSingle correctJEE Main 2026

Let

A = \left\{z \in \mathbb{C} : |z - 2| \leq 4\right\}

and

B = \left\{z \in \mathbb{C} : |z - 2| + |z + 2| = 5\right\}

. Then the max

\left\{|z_1 - z_2| : z_1 \in A \text{ and } z_2 \in B\right\}

is

(A) $\frac{15}{2}$
(B)8
(C) $\frac{17}{2}$
(D)9

Q11·MathematicsSingle correctJEE Main 2026

Let

z

be a complex number such that

|z - 6| = 5

and

|z + 2 - 6i| = 5

. Then the value of

z^3 + 3z^2 - 15z + 141

is equal to

(A)42
(B)37
(C)50
(D)61

Q12·MathematicsNumericalJEE Main 2026

Let

z = (1 + i)(1 + 2i)(1 + 3i) \ldots (1 + ni)

, where

i = \sqrt{-1}

. If

|z|^{2} = 44200

, then n is equal to

-

Q13·MathematicsSingle correctJEE Main 2026

Let

S=\left\{z\in\mathbb{C}:\left|\frac{z-6i}{z-2i}\right|=1\ \text{and}\ \left|\frac{z-8+2i}{z+2i}\right|=\frac{3}{5}\right\}

. Then

\sum_{z\in s}|z|^{2}

is equal to

(A) $398$
(B) $413$
(C) $423$
(D) $385$

Q14·MathematicsSingle correctJEE Main 2026

If

z = \frac{\sqrt{3}}{2} + \frac{i}{2}, i = \sqrt{-1}

, then

(z^{201} - i)^{8}

is equal to

(A)–1
(B)0
(C)1
(D)256

Q15·MathematicsSingle correctJEE Main 2026

Let

S = \{z : 3 \le | 2z - 3(1 + i)| \le 7\}

be a set of complex numbers. Then

\underset{z \in S}{\mathrm{Min}}\ \left|\left(z + \frac{1}{2}(5 + 3i)\right)\right|

is equal to :

(A) $\frac{1}{2}$
(B) $\frac{3}{2}$
(C)2
(D) $\frac{5}{2}$

Q16·MathematicsNumericalJEE 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}

, then m is _________.

Q17·MathematicsSingle correctJEE Main 2026

Let

S = \left\{ z \in \mathbb{C} : 4z^{2} + \bar{z} = 0 \right\}

. Then

\sum\limits_{z \in S} |z|^{2}

is equal to :

(A) $\frac{3}{16}$
(B) $\frac{7}{64}$
(C) $\frac{1}{16}$
(D) $\frac{5}{64}$

Q18·MathematicsSingle correctJEE 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}+...+\left(x^{25}+\frac{1}{x^{25}}\right)^{4}

is :

(A)128
(B)162
(C)175
(D)145

Q19·MathematicsSingle correctJEE Main 2026

Let z be the complex number satisfying

|z - 5| \le 3

and having maximum positive principal argument. Then

34\left|\frac{5z - 12}{5iz + 16}\right|^{2}

is equal to:

(A)16
(B)12
(C)26
(D)20

Q20·MathematicsNumericalJEE Advanced 2025

For a non-zero complex number z, let arg(z) denote the principal argument of z, with −π < arg(z) ≤ π. Let ω be the cube root of unity for which 0 < arg(ω) < π. Let

\alpha = \arg\left( \sum_{n=1}^{2025} (-\omega)^n \right)

. Then the value of

\frac{3\alpha}{\pi}

is ______

Q21·MathematicsMultiple correctJEE Advanced 2025

Let R denote the set of all real numbers. Let

z_1 = 1 + 2i

and

z_2 = 3i

be two complex numbers, where

i = \sqrt{-1}

. Let

S = \{(x, y) \in R \times R : |x + iy - z_1| = 2|x + iy - z_2|\}

. Then which of the following statements is(are) TRUE ?

(A)S is a circle with centre $\left(-\frac{1}{3}, \frac{10}{3}\right)$
(B)S is a circle with centre $\left(\frac{1}{3}, \frac{8}{3}\right)$
(C)S is a circle with radius $\frac{\sqrt{2}}{3}$
(D)S is a circle with radius $\frac{2\sqrt{2}}{3}$

Q22·MathematicsSingle correctJEE Main 2025

Let

A=\left\{\theta\in[0,2\pi]:1+10\,\mathrm{Re}\left(\dfrac{2\cos\theta+i\sin\theta}{\cos\theta-3i\sin\theta}\right)=0\right\}

. Then

\sum_{\theta\in A}\theta^2

is equal to:

(A) $\dfrac{21}{4}\pi^2$
(B) $8\pi^2$
(C) $\dfrac{27}{4}\pi^2$
(D) $6\pi^2$

Q23·MathematicsSingle correctJEE Main 2025

Let the locus of

z\in\mathbb{C}

, such that

\mathrm{Re}\left(\dfrac{z-1}{2z+i}\right)+\mathrm{Re}\left(\dfrac{\bar{z}-1}{2\bar{z}-i}\right)=2

, is a circle of radius

r

and center

(a,b)

, then

\dfrac{15ab}{r^2}

is equal to:

(A)24
(B)12
(C)18
(D)16

Q24·MathematicsSingle correctJEE Main 2025

Among the statements (S1): The set

\{z\in\mathbb{C}-\{-i\}:|z|=1

and

\dfrac{z-i}{z+i}

is purely real

\}

contains exactly two elements, and (S2): The set

\{z\in\mathbb{C}-\{-1\}:|z|=1

and

\dfrac{z-1}{z+1}

is purely imaginary

\}

contains infinitely many elements.

(A)both are incorrect
(B)only (S1) is correct
(C)only (S2) is correct
(D)both are correct

Q25·MathematicsIntegerJEE Main 2025

Let

A=\{z\in\mathbb{C}:|z-2-i|=3\}

,

B=\{z\in\mathbb{C}:\mathrm{Re}(z-iz)=2\}

and

S=A\cap B

. Then

\displaystyle\sum_{z\in S}|z|^2

is equal to ______.

Q26·MathematicsIntegerJEE Main 2025

If

\alpha

is a root of the equation

x^2+x+1=0

and

\displaystyle\sum_{k=1}^{n}\left(\alpha^k+\dfrac{1}{\alpha^k}\right)^2=20

, then

n

is equal to ______.

Q27·MathematicsSingle correctJEE 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

\displaystyle\sum_{k=1}^{3}(z_k-z_0)^2

is equal to:

(A)0
(B)1
(C) $i$
(D) $-i$

Q28·MathematicsSingle correctJEE Main 2025

Let

z\in\mathbb{C}

be such that

\dfrac{z^2+3i}{z-2+i}=2+3i

. Then the sum of all possible values of

z^2

is:

(A) $19-2i$
(B) $-19-2i$
(C) $19+2i$
(D) $-19+2i$

Q29·MathematicsSingle correctJEE Main 2025

Let

z

be a complex number such that

|z|=1

. If

\dfrac{2+k^2z}{k+\bar{z}}=kz

,

k\in R

, then the maximum distance of

k+ik^2

from the circle

|z-(1+2i)|=1

is:

(A) $\sqrt{5}+1$
(B)2
(C)3
(D) $\sqrt{3}+1$

Q30·MathematicsIntegerJEE Main 2025

Let integers

a, b\in[-3, 3]

be such that

a+b\ne 0

. Then the number of all possible ordered pairs

(a, b)

, for which

\left|\dfrac{z-a}{z+b}\right|=1

and

\begin{vmatrix}z+1 & \omega & \omega^2\\ \omega & z+\omega^2 & 1\\ \omega^2 & 1 & z+\omega\end{vmatrix}=1

,

z\in C

, where

\omega

and

\omega^2

are the roots of

x^2+x+1=0

, is equal to ______.

Q31·MathematicsSingle correctJEE Main 2025

Let

|z_1-8-2i|\le 1

and

|z_2-2+6i|\le 2

,

z_1,z_2\in C

. Then the minimum value of

|z_1-z_2|

is:

(A)3
(B)7
(C)13
(D)10

Q32·MathematicsSingle correctJEE Main 2025

Let O be the origin, the point A be

z_1=\sqrt{3}+2\sqrt{2}i

, the point

B(z_2)

be such that

\sqrt{3}|z_1|=|z_2|

and

\arg(z_2)=\arg(z_1)+\frac{\pi}{6}

. Then

(A)area of triangle ABO is $\frac{11}{\sqrt{3}}$
(B)ABO is a scalene triangle
(C)area of triangle ABO is $\frac{11}{4}$
(D)ABO is an obtuse angled isosceles triangle

Q33·MathematicsSingle correctJEE Main 2025

If

\alpha

and

\beta

are the roots of the equation

2z^2-3z-2i=0

, where

i=\sqrt{-1}

, then

16\cdot\mathrm{Re}\left(\dfrac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)\cdot\mathrm{Im}\left(\dfrac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)

is equal to

(A) $398$
(B) $312$
(C) $409$
(D) $441$

Q34·MathematicsSingle correctJEE Main 2025

The number of complex numbers

z

, satisfying

|z|=1

and

\left|\dfrac{z}{\bar z}+\dfrac{\bar z}{z}\right|=1

, is :

(A)6
(B)4
(C)10
(D)8

Q35·MathematicsSingle correctJEE Main 2025

Let

\left|\dfrac{\bar z-i}{2\bar z+i}\right|=\dfrac{1}{3}

,

z\in\mathbb{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) $100$
(B) $50$
(C) $\dfrac{121}{25}$
(D) $\dfrac{81}{25}$

Q36·MathematicsSingle correctJEE Main 2025

Let

z_1,z_2,z_3

lie on

|z|=1

with

\arg z_1=-\tfrac{\pi}{4},\ \arg z_2=0,\ \arg z_3=\tfrac{\pi}{4}

. If

|z_1\bar z_2+z_2\bar z_3+z_3\bar z_1|^2=\alpha+\beta\sqrt2\ (\alpha,\beta\in\mathbb{Z})

, then

\alpha^2+\beta^2

is:

(A)24
(B)41
(C)33
(D)29

Q37·MathematicsSingle correctJEE Main 2025

Let the curve

z(1+i)+\bar{z}(1-i)=4

,

z\in\mathbb{C}

, divide the region

|z-3|\le1

into two parts of areas

\alpha

and

\beta

. Then

|\alpha-\beta|

equals:

(A) $1+\dfrac{\pi}{2}$
(B) $1+\dfrac{\pi}{3}$
(C) $1+\dfrac{\pi}{4}$
(D) $1+\dfrac{\pi}{6}$

Q38·MathematicsNumericalJEE Main 2024

The sum of the square of the modulus of the elements in the set

\{z = a + ib : a, b \in \mathbb{Z}, z \in \mathbb{C}, |z-1| \le 1, |z-5| \le |z-5i|\}

is ________.

Q39·MathematicsSingle correctJEE Main 2024

Let z be a complex number such that the real part of

\dfrac{z-2i}{z+2i}

is zero. Then, the maximum value of

|z-(6+8i)|

is equal to:

(A) $12$
(B) $\infty$
(C) $10$
(D) $8$

Q40·MathematicsSingle correctJEE Main 2024

Let

z

be a complex number such that

|z+2|=1

and

\mathrm{Im}\left(\dfrac{z+1}{z+2}\right)=\dfrac15

. Then the value of

\left|\mathrm{Re}\left(\overline{z+2}\right)\right|

is:

(A) $\dfrac{\sqrt{6}}{5}$
(B) $\dfrac{1+\sqrt{6}}{5}$
(C) $\dfrac{24}{5}$
(D) $\dfrac{2\sqrt{6}}{5}$

Q41·MathematicsSingle correctJEE Main 2024

The sum of all possible values of

\theta\in[-\pi,2\pi]

, for which

\dfrac{1+i\cos\theta}{1-2i\cos\theta}

is purely imaginary, is equal to

(A) $2\pi$
(B) $3\pi$
(C) $5\pi$
(D) $4\pi$

Q42·MathematicsSingle correctJEE Main 2024

If the set

R=\{(a,b):a+5b=42,\ a,b\in\mathbb{N}\}

has

m

elements and

\displaystyle\sum_{n=1}^{m}i^{n!}=x+iy

, where

i=\sqrt{-1}

, then the value of

m+x+y

is:

(A) $8$
(B) $12$
(C) $4$
(D) $5$

Q43·MathematicsSingle correctJEE Main 2024

If

z_1, z_2

are two distinct complex number such that

\left|\frac{z_1 - 2z_2}{\frac{1}{2} - z_1\bar{z_2}}\right| = 2

, then

(A)either $z_1$ lies on a circle of radius 1 or $z_2$ lies on a circle of radius $\tfrac{1}{2}$
(B)either $z_1$ lies on a circle of radius $\tfrac{1}{2}$ or $z_2$ lies on a circle of radius 1.
(C) $z_1$ lies on a circle of radius $\tfrac{1}{2}$ and $z_2$ lies on a circle of radius 1.
(D)both $z_1$ and $z_2$ lie on the same circle.

Q44·MathematicsSingle correctJEE Main 2024

Consider the following two statements: Statement I: For any two non-zero complex numbers

z_1, z_2

,

(|z_1| + |z_2|)\left|\dfrac{z_1}{|z_1|} + \dfrac{z_2}{|z_2|}\right| \le 2(|z_1| + |z_2|)

and Statement II: If

x, y, z

are three distinct complex numbers and

a, b, c

are three positive real numbers such that

\dfrac{a}{|y-z|} = \dfrac{b}{|z-x|} = \dfrac{c}{|x-y|}

, then

\dfrac{a^2}{y-z} + \dfrac{b^2}{z-x} + \dfrac{c^2}{x-y} = 1

. Between the above two statements,

(A)both Statement I and Statement II are incorrect.
(B)Statement I is incorrect but Statement II is correct.
(C)Statement I is correct but Statement II is incorrect.
(D)both Statement I and Statement II are correct.

Q45·MathematicsSingle correctJEE Main 2024

Let

S_1=\{z\in\mathbb{C}:|z|\le 5\}

,

S_2=\left\{z\in\mathbb{C}:\operatorname{Im}\left(\dfrac{z+1-\sqrt{3}\,i}{1-\sqrt{3}\,i}\right)\ge 0\right\}

and

S_3=\{z\in\mathbb{C}:\operatorname{Re}(z)\ge 0\}

. Then the area of

S_1\cap S_2\cap S_3

is:

(A) $\dfrac{125\pi}{6}$
(B) $\dfrac{125\pi}{24}$
(C) $\dfrac{125\pi}{4}$
(D) $\dfrac{125\pi}{12}$

Q46·MathematicsSingle correctJEE Main 2024

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\mathbb{C}

. Then

4(\alpha^2+\beta^2)

is equal to:

(A) $6$
(B) $4$
(C) $8$
(D) $2$

Q47·MathematicsSingle correctJEE Main 2024

Let

S=\{z\in\mathbb{C}:|z-1|=1

and

(\sqrt2-1)(z+\bar z)-i(z-\bar z)=2\sqrt2\}

. Let

z_1,z_2\in S

be such that

|z_1|=\max\limits_{z\in S}|z|

and

|z_2|=\min\limits_{z\in S}|z|

. Then

\left|\sqrt2\,z_1-z_2\right|^2

equals:

(A) $1$
(B) $4$
(C) $3$
(D) $2$

Q48·MathematicsSingle correctJEE Main 2024

If

z

is a complex number such that

|z|\ge 1

, then the minimum value of

\left|z+\dfrac{1}{2}(3+4i)\right|

is:

(A) $\dfrac{5}{2}$
(B)2
(C)3
(D) $\dfrac{3}{2}$

Q49·MathematicsSingle correctJEE Main 2024

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+15i

. Then

\left|z_1^4+z_2^4\right|

equals

(A) $30\sqrt{3}$
(B) $75$
(C) $15\sqrt{15}$
(D) $25\sqrt{3}$

Q50·MathematicsNumericalJEE Main 2024

If

\alpha

denotes the number of solutions of

|1-i|^x=2^x

and

\beta=\dfrac{|z|}{\arg(z)}

, where

z=\dfrac{\pi}{4}(1+i)^4\left[\dfrac{1-\sqrt{\pi}i}{\sqrt{\pi}+i}+\dfrac{\sqrt{\pi}-i}{1+\sqrt{\pi}i}\right]

,

i=\sqrt{-1}

, then the distance of the point

(\alpha,\beta)

from the line

4x-3y=7

is ______.

Q51·MathematicsSingle correctJEE Main 2024

If

z=x+iy

,

x\ne0

, satisfies the equation

z^2+i\bar z=0

, then

|z|^2

is equal to:

(A) $9$
(B) $1$
(C) $4$
(D) $\dfrac14$

Q52·MathematicsSingle correctJEE Main 2024

If

z

is a complex number, then the number of common roots of the equation

z^{1985}+z^{100}+1=0

and

z^3+2z^2+2z+1=0

, is equal to:

(A) $1$
(B) $2$
(C) $0$
(D) $3$

Q53·MathematicsSingle correctJEE Main 2024

If

z=\dfrac{1}{2}-2i

, is such that

|z+1|=\alpha z+\beta(1+i)

,

i=\sqrt{-1}

and

\alpha,\beta\in R

, then

\alpha+\beta

is equal to

(A) $-4$
(B) $3$
(C) $2$
(D) $-1$

Q54·MathematicsSingle correctJEE Main 2024

Let

r

and

\theta

respectively be the modulus and amplitude of the complex number

z=2-i\left(2\tan\dfrac{5\pi}{8}\right)

, then

(r,\theta)

is equal to:

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

Q55·MathematicsSingle correctJEE Main 2024

If

S=\{z\in\mathbb C:|z-i|=|z+i|=|z-1|\}

, then

n(S)

is:

(A)1
(B)0
(C)3
(D)2

Q56·MathematicsNumericalJEE Main 2024

Let the complex numbers

\alpha

and

\dfrac{1}{\bar\alpha}

lie on the circles

|z-z_0|^2=4

and

|z-z_0|^2=16

respectively, where

z_0=1+i

. Then, the value of

100|\alpha|^2

is __________.

Q57·MathematicsNumericalJEE Main 2024

If

\alpha

satisfies the equation

x^2+x+1=0

and

(1+\alpha)^7=A+B\alpha+C\alpha^2

,

A,B,C\ge 0

, then

5(3A-2B-C)

is equal to __________.

Q58·MathematicsIntegerJEE Advanced 2023

Let

A_{1}

,

A_{2}

,

A_{3}

, .....,

A_{8}

be the vertices of a regular octagon that lie on a circle of radius 2. Let P be a point on the circle and let

PA_{i}

denote the distance between the points

P

and

A_{i}

for

i = 1, 2, ....., 8

. If

P

varies over the circle, then the maximum value of the product

PA_{1} \cdot PA_{2} ..... PA_{8}

, is

Q59·MathematicsSingle correctJEE Advanced 2023

Let z be a complex number satisfying

|z|^3 + 2z^2 + 4\bar{z} - 8 = 0

, where

\bar{z}

denotes the complex conjugate of z . Let the imaginary part of z be nonzero. Match each entry in List-I to the correct entries in List-II. The correct option is:

List – I	List – II
P. $\|z\|^2$ is equal to	1.12
Q. $\left\|z - \bar{z}\right\|^2$ is equal to	2.4
R. $\|z\|^2 + \left\|z + \bar{z}\right\|^2$ is equal to	3.8
S. $\left\|z + 1\right\|^2$ is equal to	4.10
	5.7

(A)(P) → (1) (Q) → (3) (R) → (5) (S) → (4)
(B)(P) → (2) (Q) → (1) (R) → (3) (S) → (5)
(C)(P) → (2) (Q) → (4) (R) → (5) (S) → (1)
(D)(P) → (2) (Q) → (3) (R) → (5) (S) → (4)

Q60·MathematicsNumericalJEE Advanced 2023

Let

A = \left\{\frac{1967 + 1686\,i\sin\theta}{7 - 3i\cos\theta} : \theta \in R\right\}

. If A contains exactly one positive integer n, then the value of n is

Q61·MathematicsSingle correctJEE Main 2023

If the set

\left\{\mathrm{Re}\left(\frac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}\right):z\in\mathbb{C},\mathrm{Re}(z)=3\right\}

is equal to the interval

(\alpha,\beta]

, then

24(\beta-\alpha)

is equal to

(A)36
(B)42
(C)27
(D)30

Q62·MathematicsSingle correctJEE Main 2023

Let

\alpha, \beta

be the roots of the equation

x^2 - \sqrt{2}x + 2 = 0

. Then

\alpha^{14} + \beta^{14}

is equal to

(A) $-64\sqrt{2}$
(B) $-128\sqrt{2}$
(C) $-64$
(D) $-128$

Q63·MathematicsNumericalJEE Main 2023

Let

\omega=z\bar z+k_{1}z+\bar k_{1}\bar z+\lambda(1+i),\,k_{1}\in\mathbb{C}

.

\operatorname{Re}(\omega)=0

be the circle

C

of radius

1

in the first quadrant touching the line

y=1

and the

y

-axis. If the curve

\operatorname{Im}(\omega)=0

intersects

C

at

A

and

B

, then

30(AB)^{2}

is equal to _____.

Q64·MathematicsSingle correctJEE Main 2023

Let

S = \left\{ z \in \mathbb{C} : \bar{z} = i(z^2 + \text{Re}(\bar{z})) \right\}

. Then

\sum\limits_{z \in S} |z|^2

is equal to

(A) $\dfrac{7}{2}$
(B) $4$
(C) $\dfrac{5}{2}$
(D) $3$

Q65·MathematicsSingle correctJEE Main 2023

Let C be the circle in the complex plane with centre

z_0=\dfrac12(1+3i)

and radius

r=1

. Let

z_1=1+i

and the complex number

z_2

be outside the circle C such that

|z_1-z_0|\,|z_2-z_0|=1

. If

z_0

,

z_1

and

z_2

are collinear, then the smaller value of

|z_2|^2

is equal to

(A) $\dfrac{13}{2}$
(B) $\dfrac52$
(C) $\dfrac32$
(D) $\dfrac72$

Q66·MathematicsNumericalJEE Main 2023

Let

S=\left\{z\in\mathbb{C}-\{i,2i\}:\dfrac{z^2+8iz-15}{z^2-3iz-2}\in\mathbb{R}\right\}

. If

\alpha-\dfrac{13}{11}i\in S

,

\alpha\in\mathbb{R}-\{0\}

, then

242\alpha^2

is equal to

Q67·MathematicsSingle correctJEE Main 2023

For

a\in\mathbb{C}

, let

A=\{z\in\mathbb{C}:\operatorname{Re}(a+\bar{z})>\operatorname{Im}(\bar{a}+z)\}

and

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)Only (S1) is true
(B)Both are false
(C)Only (S2) is true
(D)Both are true

Q68·MathematicsSingle correctJEE Main 2023

Let

w_1

be the point obtained by the rotation of

z_1 = 5 + 4i

about the origin through a right angle in the anticlockwise direction, and

w_2

be the point obtained by the rotation of

z_2 = 3 + 5i

about the origin through a right angle in the clockwise direction. Then the principal argument of

w_1 - w_2

is equal to

(A) $-\pi + \tan^{-1} \frac{33}{5}$
(B) $-\pi - \tan^{-1} \frac{33}{5}$
(C) $-\pi + \tan^{-1} \frac{8}{9}$
(D) $\pi - \tan^{-1} \frac{8}{9}$

Q69·MathematicsSingle correctJEE Main 2023

Let

S=\left\{z=x+iy:\frac{2z-3i}{4z+2i}\text{ is a real number}\right\}

. Then which of the following is NOT correct?

(A) $y+x^2+y^2\ne-\frac{1}{4}$
(B) $x=0$
(C) $(x,y)=\left(0,-\frac{1}{2}\right)$
(D) $y\in\left(-\infty,-\frac{1}{2}\right)\cup\left(-\frac{1}{2},\infty\right)$

Q70·MathematicsSingle correctJEE Main 2023

Let the complex number

z=x+iy

be such that

\dfrac{2z-3i}{2z+i}

is purely imaginary. If

x+y^2=0

, then

y^4+y^2-y

is equal to:

(A) $\dfrac32$
(B) $\dfrac43$
(C) $\dfrac23$
(D) $\dfrac34$

Q71·MathematicsSingle correctJEE Main 2023

Let

A=\left\{\theta\in(0,2\pi):\dfrac{1+2i\sin\theta}{1-i\sin\theta}\text{ is purely imaginary}\right\}

. Then the sum of the elements in A is

(A) $\pi$
(B) $2\pi$
(C) $4\pi$
(D) $3\pi$

Q72·MathematicsSingle correctJEE 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

(A) $x^{2}+7x+12=0$
(B) $x^{2}+3x-4=0$
(C) $x^{2}+2x-3=0$
(D) $x^{2}+x-12=0$

Q73·MathematicsNumericalJEE Main 2023

For

\alpha,\beta,z\in \mathbb{C}

and

\lambda>1

, if

\sqrt{\lambda-1}

is the radius of the circle

|z-\alpha|^2+|z-\beta|^2=2\lambda

, then

|\alpha-\beta|

is equal to

Q74·MathematicsSingle correctJEE Main 2023

Let

a\neq b

be two non-zero real numbers. Then the number of elements in the set

X=\{z\in \mathbb{C}:\operatorname{Re}(az^2+bz)=a \text{ and } \operatorname{Re}(bz^2+az)=b\}

is equal to

(A)1
(B)3
(C)0
(D)2

Q75·MathematicsSingle correctJEE Main 2023

If the center and radius of the circle

\left|\frac{z-2}{z-3}\right|=2

are respectively

(\alpha,\beta)

and

\gamma

, then

3(\alpha+\beta+\gamma)

is equal to

(A) $11$
(B) $12$
(C) $9$
(D) $10$

Q76·MathematicsSingle correctJEE Main 2023

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

C_2

. Then :

(A)the curve $C_1$ lies inside $C_2$
(B)the curve $C_2$ lies inside $C_1$
(C)the curves $C_1$ and $C_2$ intersect at 4 points
(D)the curves $C_1$ and $C_2$ intersect at 2 points

Q77·MathematicsSingle correctJEE Main 2023

The complex number

z=\dfrac{i-1}{\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}}

is equal to:

(A) $\sqrt2 i\left(\cos\frac{5\pi}{12}-i\sin\frac{5\pi}{12}\right)$
(B) $\sqrt2\left(\cos\frac{\pi}{12}+i\sin\frac{\pi}{12}\right)$
(C) $\sqrt2\left(\cos\frac{5\pi}{12}+i\sin\frac{5\pi}{12}\right)$
(D) $\cos\frac{\pi}{12}-i\sin\frac{\pi}{12}$

Q78·MathematicsNumericalJEE Main 2023

Let

z=1+i

and

z_1=\dfrac{1+i\bar z}{\bar z(1-z)+\dfrac1z}

. Then

\dfrac{12}{\pi}\arg(z_1)

is equal to

Q79·MathematicsNumericalJEE Main 2023

Let

\alpha=8-14i,\,A=\left\{z\in\mathbb{C}:\dfrac{\alpha z-\bar\alpha\bar z}{z^{2}-(\bar z)^{2}-112i}=1\right\}

and

B=\{z\in\mathbb{C}:\,|z+3i|=4\}

. Then

\displaystyle\sum_{z\in A\cap B}(\operatorname{Re}z-\operatorname{Im}z)

is equal to _____.

Q80·MathematicsSingle correctJEE Main 2023

For two non-zero complex numbers

z_1

and

z_2

, if

\text{Re}(z_1 \bar{z_2}) = 0

and

\text{Re}(z_1 + z_2) = 0

, then which of the following are possible? A.

\text{Im}(z_1) > 0

and

\text{Im}(z_2) > 0

B.

\text{Im}(z_1) < 0

and

\text{Im}(z_2) > 0

C.

\text{Im}(z_1) > 0

and

\text{Im}(z_2) < 0

D.

\text{Im}(z_1) < 0

and

\text{Im}(z_2) < 0

. Choose the correct answer from the options given below:

(A)B and D
(B)A and B
(C)B and C
(D)A and C

Q81·MathematicsNumericalJEE Main 2023

Let

\alpha_1,\alpha_2,\dots,\alpha_7

be the roots of the equation

x^7+3x^5-13x^3-15x=0

and

|\alpha_1|\ge|\alpha_2|\ge\dots\ge|\alpha_7|

. Then

\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6

is equal to _____.

Q82·MathematicsSingle correctJEE Main 2023

Let

z_1=2+3i

and

z_2=3+4i

. The set

S=\{z\in\mathbb{C}:|z-z_1|^2-|z-z_2|^2=|z_1-z_2|^2\}

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 $14$

Q83·MathematicsSingle correctJEE Main 2023

Let

z

be a complex number such that

\left|\dfrac{z-2i}{z+i}\right|=2

,

z\neq -i

. Then

z

lies on the circle of radius

2

and centre

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

Q84·MathematicsSingle correctJEE Main 2023

The value of

\left(\dfrac{1+\sin\frac{2\pi}{9}+i\cos\frac{2\pi}{9}}{1+\sin\frac{2\pi}{9}-i\cos\frac{2\pi}{9}}\right)^3

is

(A) $-\dfrac12(\sqrt3-i)$
(B) $-\dfrac12(1-i\sqrt3)$
(C) $\dfrac12(1-i\sqrt3)$
(D) $\dfrac12(\sqrt3+i)$

Q85·MathematicsSingle correctJEE Main 2023

Let

p,q\in\mathbb{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

(A) $x^2-4x-1=0$
(B) $x^2-4x+1=0$
(C) $x^2+4x-1=0$
(D) $x^2+4x+1=0$

Q86·MathematicsMultiple correctJEE Advanced 2022

Let z̄ denote the complex conjugate of a complex number z. If z is a non-zero complex number for which both real and imaginary parts of

\left(\bar{z}\right)^{2}

+

\frac{1}{z^{2}}

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

(A) $\left(\frac{43+3\sqrt{205}}{2}\right)^{\frac{1}{4}}$
(B) $\left(\frac{7+\sqrt{33}}{4}\right)^{\frac{1}{4}}$
(C) $\left(\frac{9+\sqrt{65}}{4}\right)^{\frac{1}{4}}$
(D) $\left(\frac{7+\sqrt{13}}{6}\right)^{\frac{1}{4}}$

Q87·MathematicsNumericalJEE Advanced 2022

Let

\bar{z}

denote the complex conjugate of a complex number z and let

i = \sqrt{-1}

. In the set of complex numbers, the number of distinct roots of the equation

\bar{z} - z^{2} = i\left(\bar{z} + z^{2}\right)

is __________.

Q88·MathematicsNumericalJEE Advanced 2022

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

\frac{2+3z+4z^{2}}{2-3z+4z^{2}}

is a real number, then the value of

|z|^{2}

is __________.

Q89·MathematicsSingle correctJEE Main 2022

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) $\sqrt{2} + 1$

Q90·MathematicsSingle correctJEE Main 2022

If

z = 2 + 3i

, then

z^{5} + (\bar{z})^{5}

is equal to :

(A)244
(B)224
(C)245
(D)265

Q91·MathematicsSingle correctJEE Main 2022

Let

S = \{ z = x + iy : |z - 1 + i| \ge |z|, |z| < 2, |z + i| = |z - 1| \}

. Then the set of all values of x, for which

w = 2x + iy \in 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) $\left(-\frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}\right]$

Q92·MathematicsSingle correctJEE Main 2022

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 :

(A)0
(B) $\frac{1}{2}$
(C) $\frac{3}{2}$
(D) $\frac{5}{2}$

Q93·MathematicsNumericalJEE Main 2022

Let

z = a + ib

,

b \neq 0

be complex numbers satisfying

z^{2}=\bar{z}\cdot 2^{1-|z|}

. Then the least value of

n \in N

, such that

z^{n} = (z+1)^{n}

, is equal to _____

Q94·MathematicsSingle correctJEE Main 2022

Let the minimum value

v_0

of

v = |z|^2 + |z-3|^2 + |z-6i|^2

,

z \in \mathbb{C}

is attained at

z = z_0

. Then

\left| 2z_0^2 - \bar{z}_0^3 + 3 \right|^2 + v_0^2

is equal to

(A)1000
(B)1024
(C)1105
(D)1196

Q95·MathematicsNumericalJEE Main 2022

Let

S=\left\{z\in\mathbb{C}:z^{2}+\bar{z}=0\right\}

. Then

\sum_{z\in S}\left(\mathrm{Re}\left(z\right)+\mathrm{Im}\left(z\right)\right)

is equal to ______.

Q96·MathematicsSingle correctJEE Main 2022

Let S be the set of all

(\alpha, \beta)

,

\pi < \alpha, \beta < 2\pi

, for which the complex number

\frac{1-i\sin\alpha}{1+2i\sin\alpha}

is purely imaginary and

\frac{1+i\cos\beta}{1-2i\cos\beta}

is purely real. Let

Z_{\alpha\beta} = \sin 2\alpha + i\cos 2\beta, (\alpha,\beta) \in S

. Then

\sum_{(\alpha,\beta)\in S}\left(i Z_{\alpha\beta} + \frac{1}{i\,\overline{Z}_{\alpha\beta}}\right)

is equal to :

(A) $3$
(B) $3i$
(C) $1$
(D) $2-i$

Q97·MathematicsSingle correctJEE Main 2022

If

z = x + iy

satisfies

|z| - 2 = 0

and

|z-i|-|z+5i|=0

, then

(A) $x + 2y - 4 = 0$
(B) $x^{2} + y - 4 = 0$
(C) $x + 2y + 4 = 0$
(D) $x^{2}- y + 3 = 0$

Q98·MathematicsSingle correctJEE Main 2022

Let O be the origin and A be the point z

_{1}

= 1 + 2i. If B is the point z

_{2}

, Re(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 $_{2}$ = π – tan $^{-1}$ 3
(B)arg (z $_{1}$ – 2z $_{2}$ ) = – $\tan^{-1}\dfrac{4}{3}$
(C) $\left|z_{2}\right| = \sqrt{10}$
(D) $\left|2z_{1} - z_{2}\right| = 5$

Q99·MathematicsSingle correctJEE 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 .

(A) $-4$
(B) $-1$
(C) $1$
(D) $4$

Q100·MathematicsNumericalJEE Main 2022

Let

S = \{z \in C : |z - 2| \leq 1, z(1 + i) + \bar{z}(1 - i) \leq 2\}

. Let

|z - 4i|

attains minimum and maximum values, respectively, at

z_1 \in S

and

z_2 \in S

. If

5(|z_1|^2 + |z_2|^2) = \alpha + \beta\sqrt{5}

, where α and β are integers, then the value of α + β is equal to _____.

Q101·MathematicsSingle correctJEE Main 2022

Let α and β be the roots of the equation

x^{2} + (2i - 1) = 0

. Then, the value of

|\alpha^{8} + \beta^{8}|

is equal to :

(A)50
(B)250
(C)1250
(D)1500

Q102·MathematicsSingle correctJEE Main 2022

Let arg (z) represent the principal argument of the complex number z. The,

\left| z \right| = 3

and

\arg (z - 1) - \arg (z + 1) = \frac{\pi}{4}

intersect:

(A)Exactly at one point
(B)Exactly at two points
(C)Nowhere
(D)At infinitely many points.

Q103·MathematicsSingle correctJEE 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:

(A) $1$
(B) $\alpha$
(C) $1 + \alpha$
(D) $1 + 2\alpha$

Q104·MathematicsNumericalJEE Main 2022

The number of elements in the set

\{z = a + ib \in \mathbb{C} : a, b \in \mathbb{Z} \text{ and } 1 < |z - 3 + 2i| < 4\}

is ______.

Q105·MathematicsNumericalJEE Main 2022

Sum of squares of modulus of all the complex numbers

z

satisfying

\bar{z} = iz^2 + z^2 - z

is equal to

Q106·MathematicsSingle correctJEE Main 2022

The area of the polygon, whose vertices are the non-real roots of the equation

\overline{z} = iz^{2}

is :

(A) $\frac{3\sqrt{3}}{4}$
(B) $\frac{3\sqrt{3}}{2}$
(C) $\frac{3}{2}$
(D) $\frac{3}{4}$

Q107·MathematicsSingle correctJEE Main 2022

Let for some real numbers

\alpha

and

\beta

,

a=\alpha-i\beta

. If the system of equations

4ix+(1+i)y=0

and

8\left(\cos\dfrac{2\pi}{3}+i\sin\dfrac{2\pi}{3}\right)x+\bar{a}y=0

has more than one solution then

\dfrac{\alpha}{\beta}

is equal to :

(A) $-2+\sqrt{3}$
(B) $2-\sqrt{3}$
(C) $2+\sqrt{3}$
(D) $-2-\sqrt{3}$

Q108·MathematicsSingle correctJEE Main 2022

The number of points of intersection of

\left|z-(4+3i)\right|=2

and

\left|z\right|+\left|z-4\right|=6

,

z \in \mathbb{C}

is :

(A) $0$
(B) $1$
(C) $2$
(D) $3$

Q109·MathematicsNumericalJEE Main 2022

If

z^{2} + z + 1 = 0

,

z \in \mathbb{C}

, then

\left|\sum_{n=1}^{15}\left(Z^{n} + \left(-1\right)^{n}\frac{1}{Z^{n}}\right)^{2}\right|

is equal to ______ .

Q110·MathematicsSingle correctJEE Main 2022

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 :

(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

Q111·MathematicsSingle correctJEE Main 2022

Let

z_1

and

z_2

be two complex numbers such that

\overline{z}_1 = i\overline{z}_2

and

\arg\left(\frac{z_1}{\overline{z}_2}\right) = \pi

. Then

(A) $\arg z_2 = \frac{\pi}{4}$
(B) $\arg z_2 = -\frac{3\pi}{4}$
(C) $\arg z_1 = \frac{\pi}{4}$
(D) $\arg z_1 = -\frac{3\pi}{4}$

Q112·MathematicsNumericalJEE Main 2022

Let

S = \left\{z \in \mathbb{C} : |z-3| \le 1 \text{ and } z(4+3i) + \bar{z}(4-3i) \le 24\right\}

. If

\alpha + i\beta

is the point in S which is closest to

4i

, then

25(\alpha + \beta)

is equal to ______.

Q113·MathematicsSingle correctJEE Main 2022

Let

A=\{z\in\mathbb{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

Q114·MathematicsSingle correctJEE Advanced 2021

Let

\theta_1, \theta_2, ..., \theta_{10}

be positive valued angles (in radian) such that

\theta_1 + \theta_2 + ... + \theta_{10} = 2\pi

. Define the complex numbers

z_1 = e^{i\theta_1}

,

z_k = z_{k-1}e^{i\theta_k}

for

k = 2, 3, ..., 10

, where

i = \sqrt{-1}

. Consider the statements P and Q given below : P :

\left|z_2 - z_1\right| + \left|z_3 - z_2\right| + ... + \left|z_{10} - z_9\right| + \left|z_1 - z_{10}\right| \le 2\pi

Q :

\left|z_2^2 - z_1^2\right| + \left|z_3^2 - z_2^2\right| + .... + \left|z_{10}^2 - z_9^2\right| + \left|z_1^2 - z_{10}^2\right| \le 4\pi

Then,

(A)P is TRUE and Q is FALSE
(B)Q is TRUE and P is FALSE
(C)both P and Q are TRUE
(D)both P and Q are FALSE

Q115·MathematicsMultiple correctJEE Advanced 2021

For any complex number

w = c + id

, let

\arg(w) \in (-\pi, \pi]

, where

i = \sqrt{-1}

. Let

\alpha

and

\beta

be real numbers such that for all complex numbers

z = x + iy

satisfying

\arg\left(\frac{z + \alpha}{z + \beta}\right) = \frac{\pi}{4}

, the ordered pair

(x, y)

lies on the circle

x^2 + y^2 + 5x - 3y + 4 = 0

. Then which of the following statements is (are) TRUE ?

(A) $\alpha = -1$
(B) $\alpha\beta = 4$
(C) $\alpha\beta = -4$
(D) $\beta = 4$

Q116·MathematicsNumericalJEE Main 2021

If for the complex numbers z satisfying |z − 2 − 2i| ≤ 1, the maximum value of |3iz + 6| is attained at a + ib, then a + b is equal to ________ .

Q117·MathematicsSingle correctJEE Main 2021

If

a_{r} = \cos\frac{2r\pi}{9} + i \sin\frac{2r\pi}{9}

,

r = 1, 2, 3, \ldots

,

i = \sqrt{-1}

, then the determinant

\begin{vmatrix} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{vmatrix}

is equal to :

(A) $a_{2}a_{6} - a_{4}a_{8}$
(B) $a_{9}$
(C) $a_{1}a_{9} - a_{3}a_{7}$
(D) $a_{5}$

Q118·MathematicsNumericalJEE 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

\left| z - 9\sqrt{2} - 2i \right|^{2}

is equal to ______.

Q119·MathematicsSingle correctJEE Main 2021

If

z

is a complex number such that

\frac{z - i}{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) $2\sqrt{2}$

Q120·MathematicsNumericalJEE Main 2021

Let

z_{1}

and

z_{2}

be two complex numbers such that

\arg\,(z_{1} - z_{2}) = \frac{\pi}{4}

and

z_{1}

,

z_{2}

satisfy the equation

|z - 3| = \mathrm{Re}(z)

. Then the imaginary part of

z_{1} + z_{2}

is equal to ___________ .

Q121·MathematicsSingle correctJEE Main 2021

If

S = \left\{ z \in \mathbb{C} : \frac{z-i}{z+2i} \in \mathbb{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

Q122·MathematicsSingle correctJEE Main 2021

The equation

\arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{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 at $(0,0)$ and radius $\sqrt{2}$
(D)centre at $(0,1)$ and radius $2$

Q123·MathematicsNumericalJEE Main 2021

The least positive integer n such that

\frac{(2i)^{n}}{(1-i)^{n-2}}, i = \sqrt{-1}

is a positive integer, is ________.

Q124·MathematicsNumericalJEE 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} + ... + \left(z^{21} + \frac{1}{z^{21}}\right)^{3}

is ________.

Q125·MathematicsSingle correctJEE Main 2021

If

\left(\sqrt{3} + i\right)^{100} = 2^{99}\left(p + iq\right)

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

Q126·MathematicsSingle correctJEE Main 2021

Let c be the set of all complex numbers. Let S

_1

=

\left\{ z \in C : |z - 2| \le 1 \right\}

and S

_2

=

\left\{ z \in C : z(1+i) + \bar{z}(1-i) \ge 4 \right\}

. Then, the maximum value of

\left| z - \frac{5}{2} \right|^2

for

z \in S_1 \cap S_2

is equal to :

(A) $\frac{5 + 2\sqrt{2}}{4}$
(B) $\frac{5 + 2\sqrt{2}}{2}$
(C) $\frac{3 + 2\sqrt{2}}{4}$
(D) $\frac{3 + 2\sqrt{2}}{2}$

Q127·MathematicsNumericalJEE Main 2021

If the real part of the complex number z =

\frac{3+2i\cos\theta}{1-3i\cos\theta}

, θ ∈

\left(0,\frac{\pi}{2}\right)

is zero, then the value of

\sin^{2}3\theta

+

\cos^{2}\theta

is equal to………

Q128·MathematicsSingle correctJEE Main 2021

Let C be the set of all complex numbers. Let

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

,

S_2 = \left\{ z \in C \,\middle|\, \operatorname{Re}(z) \ge 5 \right\}

and

S_3 = \left\{ z \in C \,\middle|\, \left| z - \bar{z} \right| \ge 8 \right\}

. Then the number of element in

S_1 \cap S_2 \cap S_3

is equal to:

(A)0
(B)2
(C)1
(D)infinite

Q129·MathematicsNumericalJEE Main 2021

The equation of a circle is

Re\left(z^{2}\right) + 2\left(Im(z)\right)^{2} + 2Re(z) = 0

, where

z = x + iy

. A line which passes through the center of the given circle and the vertex of the parabola,

x^{2} - 6x - y + 13 = 0

, has y-intercept equal to.........

Q130·MathematicsSingle correctJEE Main 2021

Let n denote the number of solutions of the equation

z^2 + 3\bar{z} = 0

, where z is a complex number. Then the value of

\sum_{k=0}^{\infty} \frac{1}{n^k}

is equal to :

(A)1
(B)2
(C) $\frac{3}{2}$
(D) $\frac{4}{3}$

Q131·MathematicsSingle correctJEE Main 2021

If the real part of the complex number

\left(1-\cos\theta+2i\sin\theta\right)^{-1}

is

\frac{1}{5}

for

\theta\in(0,\pi)

, then the value of the integral

\int_{0}^{\theta}\sin x\,dx

is equal to :

(A) $-1$
(B) $0$
(C) $2$
(D) $1$

Q132·MathematicsSingle correctJEE Main 2021

If z and ω are two complex numbers such that

|z\omega| = 1

and

\arg(z) - \arg(w) = \frac{3\pi}{2}

, then

\arg\left(\frac{1 - 2\bar{z}\omega}{1 + 3\bar{z}\omega}\right)

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

(A) $\frac{\pi}{4}$
(B) $-\frac{3\pi}{4}$
(C) $-\frac{\pi}{4}$
(D) $\frac{3\pi}{4}$

Q133·MathematicsNumericalJEE Main 2021

If

f(x)

and g(x) are two polynomials such that the polynomial

P(x) = f(x^{3}) + xg(x^{3})

is divisible by

x^{2} + x + 1

, then P(1) is equal to ________ .

Q134·MathematicsNumericalJEE Main 2021

Let

z_{1}, z_{2}

be the roots of the equation

z^{2} + az + 12 = 0

and

z_{1}, z_{2}

form an equilateral triangle with origin. Then, the value of

|a|

is

Q135·MathematicsSingle correctJEE Main 2021

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) =

\frac{\pi}{2}

. Then the area of the triangle with vertices origin, z and w is equal to :

(A)4
(B) $\frac{1}{2}$
(C) $\frac{1}{4}$
(D)2

Q136·MathematicsSingle correctJEE Main 2021

If the equation

a\left|z\right|^{2} + \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) $\left|\alpha\right|^{2} - ad \neq 0$
(B) $|\alpha|^{2} - ad > 0$ and $a \in R - \{0\}$
(C) $|\alpha|^{2} - ad \geq 0$ and $a \in R$
(D) $\alpha = 0$ , $a, d \in R^{+}$

Q137·MathematicsSingle correctJEE Main 2021

The area of the triangle with vertices

A(z)

,

B(iz)

and

C(z + iz)

is :

(A)1
(B) $\frac{1}{2}|z|^2$
(C) $\frac{1}{2}$
(D) $\frac{1}{2}|z + iz|^2$

Q138·MathematicsSingle correctJEE Main 2021

Let

S_{1}

,

S_{2}

and

S_{3}

be three sets defined as

S_{1} = \left\{z \in \mathbb{C} : |z-1| \le \sqrt{2}\right\}

S_{2} = \left\{z \in \mathbb{C} : \mathrm{Re}\left((1-i)z\right) \ge 1\right\}

S_{3} = \left\{z \in \mathbb{C} : \mathrm{Im}(z) \le 1\right\}

Then the set

S_{1} \cap S_{2} \cap S_{3}

(A)is a singleton
(B)has exactly two elements
(C)has infinitely many elements
(D)has exactly three elements

Q139·MathematicsNumericalJEE Main 2021

Let z and w be two complex numbers such that

w = z\bar{z} - 2z + 2

,

\left|\frac{z+i}{z-3i}\right| = 1

and Re(w) has minimum value. Then, the minimum value of n ∈ ℕ for which

w^{n}

is real, is equal to _____.

Q140·MathematicsSingle correctJEE Main 2021

The least value of

|z|

where z is complex number which satisfies the inequality

\exp\left(\frac{\left(|z|+3\right)\left(|z|-1\right)}{\left||z|+1\right|}\log_{e}2\right) \ge \log_{\sqrt{2}}\left|5\sqrt{7}+9i\right|,

i=\sqrt{-1}

, is equal to :

(A) $3$
(B) $\sqrt{5}$
(C) $2$
(D) $8$

Q141·MathematicsSingle correctJEE Main 2021

Let a complex number z, |z| ≠ 1, satisfy

\log_{\frac{1}{\sqrt{2}}}\left(\frac{|z| + 11}{(|z| - 1)^{2}}\right) \le 2

. Then, the largest value of |z| is equal to _____ .

(A)8
(B)7
(C)6
(D)5

Q142·MathematicsNumericalJEE Main 2021

Let z be those complex number which satisfy

|z+5| \leq 4

and

z(1+i) + \bar{z}\left(1-i\right) \geq -10, i = \sqrt{-1}

. If the maximum value of

|z+1|^{2}

is

\alpha + \beta\sqrt{2}

, then the value of

\left(\alpha + \beta\right)

is ________________.

Q143·MathematicsNumericalJEE Main 2021

The sum of

162^{\text{th}}

power of the roots of the equation

x^{3}-2x^{2}+2x-1=0

is _______.

Q144·MathematicsSingle correctJEE Main 2021

Let the lines

(2 - i)z = (2 + i)\bar{z}

and

(2 + i)z + (i - 2)\bar{z} - 4i = 0

, (here

i^{2} = -1

) be normal to a circle C. If the line

iz + \bar{z} + 1 + i = 0

is tangent to this circle C, then its radius is :

(A) $\frac{3}{\sqrt{2}}$
(B) $3\sqrt{2}$
(C) $\frac{3}{2\sqrt{2}}$
(D) $\frac{1}{2\sqrt{2}}$

Q145·MathematicsSingle correctJEE Main 2021

If α, β∈ R are such that 1 – 2i (here i

^{2}

= –1) is a root of z

^{2}

+ αz + β = 0, then (α – β) is equal to:

(A)7
(B)–3
(C)3
(D)–7

Q146·MathematicsNumericalJEE Main 2021

If the least and the largest real values of

\alpha

, for which the equation

z + \alpha|z - 1| + 2i = 0

\left(z \in C \text{ and } i = \sqrt{-1}\right)

has a solution, are p and q respectively; then

4(p^2 + q^2)

is equal to ____

Q147·MathematicsNumericalJEE Main 2021

Let

i=\sqrt{-1}

. If

\frac{(-1+i\sqrt{3})^{21}}{(1-i)^{24}}+\frac{(1+i\sqrt{3})^{21}}{(1+i)^{24}}=k,

and

\mathrm{n}=\left[\left|k\right|\right]

be the greatest integral part of

|k|

. Then

\sum_{j=0}^{n+5}(j+5)^{2}-\sum_{j=0}^{n+5}(j+5)

is equal to________.

Q148·MathematicsIntegerJEE Advanced 2020

For a complex number

z

, let

\mathrm{Re}(z)

denote the real part of

z

. Let

S

be the set of all complex numbers

z

satisfying

z^{4} - |z|^{4} = 4iz^{2}

, where

i = \sqrt{-1}

. Then the minimum possible value of

|z_{1} - z_{2}|^{2}

, where

z_{1}, z_{2} \in S

with

\mathrm{Re}(z_{1}) > 0

and

\mathrm{Re}(z_{2}) < 0

, is ________

Q149·MathematicsMultiple correctJEE Advanced 2020

Let S be the set of all complex numbers z satisfying

|z^{2} + z + 1| = 1

. Then which of the following statements is/are TRUE ?

(A) $\left|z + \frac{1}{2}\right| \leq \frac{1}{2}$ for all $z \in S$
(B) $|z| \leq 2$ for all $z \in S$
(C) $\left|z + \frac{1}{2}\right| \geq \frac{1}{2}$ for all $z \in S$
(D)The set S has exactly four elements

Q150·MathematicsSingle correctJEE 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:

(A)line, $y = -x$
(B)imaginary axis
(C)line, $y = x$
(D)real axis

Q151·MathematicsSingle correctJEE Main 2020

The region represented by

\{z = x+iy \in C : |z| - \mathrm{Re}(z) \le 1\}

is also given by the inequality:

(A) $y^2 \ge 2(x+1)$
(B) $y^2 \le 2\left(x+\dfrac{1}{2}\right)$
(C) $y^2 \le x + \dfrac{1}{2}$
(D) $y^2 \ge x+1$

Q152·MathematicsSingle correctJEE Main 2020

The value of

\left(\dfrac{-1+i\sqrt{3}}{1-i}\right)^{30}

is:

(A) $2^{15}i$
(B) $-2^{15}$
(C) $6^5$
(D) $-2^{15}i$

Q153·MathematicsSingle correctJEE Main 2020

If the four complex numbers,

z, \bar{z}, \bar{z} - 2\text{Re}(\bar{z})

and

z - 2\text{Re}(z)

represent the vertices of a square of side 4 units in the Argand plane, then

|z|

is equal to:

(A) $2\sqrt{2}$
(B)2
(C) $4\sqrt{2}$
(D)4

Q154·MathematicsSingle correctJEE 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:

(A)9
(B)33
(C)57
(D)24

Q155·MathematicsSingle correctJEE Main 2020

Let

u = \frac{2z + i}{z - ki}

,

z = x + iy

and

k > 0

. If the curve represented by

\mathrm{Re}(u) + \mathrm{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) $\frac{1}{2}$
(C) $\frac{3}{2}$
(D) $4$

Q156·MathematicsNumericalJEE Main 2020

If

\left(\frac{1+i}{1-i}\right)^{m/2} = \left(\frac{1+i}{i-1}\right)^{n/3} = 1

,

(m, n \in N)

Then the greatest common divisor of the least values of m and n is __________.

Q157·MathematicsSingle correctJEE Main 2020

If

z_1, z_2

are complex numbers such that

\mathrm{Re}(z_1) = |z_1 - 1|, \mathrm{Re}(z_2) = |z_2 - 1|

and

\arg(z_1 - z_2) = \frac{\pi}{6}

, then

\mathrm{Im}(z_1 + z_2)

is equal to:

(A) $2\sqrt{3}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\frac{1}{\sqrt{3}}$
(D) $\frac{2}{\sqrt{3}}$

Q158·MathematicsSingle correctJEE Main 2020

If z be a complex number satisfying

\left|Re(z)\right| + \left|Im(z)\right| = 4

, then

\left|z\right|

cannot be:

(A) $\sqrt{\dfrac{17}{2}}$
(B) $\sqrt{10}$
(C) $\sqrt{8}$
(D) $\sqrt{7}$

Q159·MathematicsSingle correctJEE Main 2020

Let z be a complex number such that

\left|\dfrac{z-i}{z+2i}\right|=1

and

|z|=\dfrac{5}{2}

. Then the value of

|z+3i|

is:

(A) $\dfrac{15}{4}$
(B) $2\sqrt{3}$
(C) $\sqrt{10}$
(D) $\dfrac{7}{2}$

Q160·MathematicsSingle correctJEE Main 2020

If the equation,

x^{2}+bx+45=0\ (b\in R)

has conjugate complex roots and they satisfy

|z+1|=2\sqrt{10}

, then:

(A) $b^{2}+b=12$
(B) $b^{2}-b=42$
(C) $b^{2}-b=30$
(D) $b^{2}+b=72$

Q161·MathematicsSingle correctJEE Main 2020

Let

\alpha=\dfrac{-1+i\sqrt{3}}{2}

. If

a=(1+\alpha)\displaystyle\sum_{k=0}^{100}\alpha^{2k}

and

b=\displaystyle\sum_{k=0}^{100}\alpha^{3k}

, then a and b are the roots of the quadratic equation:

(A) $x^{2}-101x+100=0$
(B) $x^{2}-102x+101=0$
(C) $x^{2}+102x+101=0$
(D) $x^{2}+101x+100=0$

Q162·MathematicsSingle correctJEE Main 2020

If

\dfrac{3+i\sin\theta}{4-i\cos\theta}, \theta \in [0,2\pi]

, is a real number, then an argument of

\sin\theta+i\cos\theta

is:

(A) $-\tan^{-1}\left(\dfrac{3}{4}\right)$
(B) $\pi-\tan^{-1}\left(\dfrac{4}{3}\right)$
(C) $\pi-\tan^{-1}\left(\dfrac{3}{4}\right)$
(D) $\tan^{-1}\left(\dfrac{4}{3}\right)$

Q163·MathematicsSingle correctJEE Main 2020

If

Re\left(\frac{z-1}{2z+i}\right)=1

, where

z=x+iy

, then the point (x, y) lies on a:

(A)straight line whose slope is $\frac{3}{2}$
(B)circle whose diameter is $\frac{\sqrt{5}}{2}$
(C)straight line whose slope is $-\frac{2}{3}$
(D)circle whose centre is at $\left(-\frac{1}{2},-\frac{3}{2}\right)$

Q164·MathematicsNumericalJEE Advanced 2019

Let

\omega \ne 1

be a cube root of unity. Then the minimum of the set

\{|a + b\omega + c\omega^2|^2 : a, b, c

distinct non-zero integers

\}

equals ____

Q165·MathematicsSingle correctJEE Advanced 2019

Let S be the set of all complex numbers z satisfying

|z - 2 + i| \ge \sqrt{5}

. If the complex number

z_0

is such that

\frac{1}{|z_0 - 1|}

is the maximum of the set

\left\{ \frac{1}{|z - 1|} : z \in S \right\}

, then the principal argument of

\frac{4 - z_0 - \overline{z_0}}{z_0 - \overline{z_0} + 2i}

is

(A) $\frac{3\pi}{4}$
(B) $\frac{\pi}{4}$
(C) $-\frac{\pi}{2}$
(D) $\frac{\pi}{2}$

Q166·MathematicsSingle correctJEE Main 2019

Let

z \in C

with

\text{Im}(z) = 10

and it satisfies

\dfrac{2z-n}{2z+n} = 2i - 1

for some natural number n. Then :

(A)n = 40 and Re(z) = 10
(B)n = 20 and Re(z) = 10
(C)n = 40 and Re(z) = $-10$
(D)n = 20 and Re(z) = $-10$

Q167·MathematicsSingle correctJEE Main 2019

The equation

|z-i|=|z-1|, i=\sqrt{-1}

, represents :

(A)a circle of radius $\frac{1}{2}$
(B)the line through the origin with slope 1
(C)a circle of radius 1
(D)the line through the origin with slope $-1$

Q168·MathematicsSingle correctJEE 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:

(A) $-\frac{3}{5}-\frac{1}{5}i$
(B) $-\frac{1}{5}-\frac{3}{5}i$
(C) $-\frac{1}{5}+\frac{3}{5}i$
(D) $\frac{1}{5}-\frac{3}{5}i$

Q169·MathematicsSingle correctJEE Main 2019

If z and w are two complex numbers such that

|zw| = 1

and

\arg(z) - \arg(w) = \dfrac{\pi}{2}

, then

(A) $\bar{z}w = i$
(B) $z\bar{w} = \dfrac{-1 + i}{\sqrt{2}}$
(C) $z\bar{w} = \dfrac{1 - i}{\sqrt{2}}$
(D) $\bar{z}w = -i$

Q170·MathematicsSingle correctJEE Main 2019

All the points in the set

S = \left\{\dfrac{\alpha+i}{\alpha-i} : \alpha \in R\right\}

(i=\sqrt{-1})

lie on a:

(A)straight line whose slope is 1
(B)circle whose radius is $\sqrt{2}$
(C)straight line whose slope is $-1$
(D)circle whose radius is 1

Q171·MathematicsSingle correctJEE Main 2019

Let

z \in C

be such that

|z|<1

. If

w=\dfrac{5+3z}{5(1-z)}

, then

(A) $5\,\text{Im}(w)<1$
(B) $4\,\text{Im}(w)>5$
(C) $5\,\text{Re}(w)>1$
(D) $5\,\text{Re}(w)>4$

Q172·MathematicsSingle correctJEE Main 2019

If

z=\frac{\sqrt{3}}{2}+\frac{i}{2}\left(i+\sqrt{-1}\right)

, then

\left(1+iz+z^{5}+iz^{8}\right)^{9}

is equal to:

(A)-1
(B)1
(C) $(-1+2i)^{9}$
(D)0

Q173·MathematicsSingle correctJEE Main 2019

If

\alpha

and

\beta

be the roots of the equation

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

, then the least value of n for which

\left(\frac{\alpha}{\beta}\right)^{n}=1

is:

(A) $4$
(B) $2$
(C) $5$
(D) $3$

Q174·MathematicsSingle correctJEE Main 2019

Let

z_{1}

and

z_{2}

be two complex numbers satisfying

|z_{1}| = 9

and

|z_{2} - 3 - 4i| = 4

. Then the minimum value of

|z_{1} - z_{2}|

is :

(A) $0$
(B) $\sqrt{2}$
(C) $1$
(D) $2$

Q175·MathematicsSingle correctJEE Main 2019

If

\frac{z-\alpha}{z+\alpha}(\alpha \in R)

is a purely imaginary number and

|z|=2

, then a value of

\alpha

is:

(A)2
(B)1
(C) $\frac{1}{2}$
(D) $\sqrt{2}$

Q176·MathematicsSingle correctJEE Main 2019

Let z be a complex number such that

|z|+z=3+i

(where

i=\sqrt{-1}

). Then

|z|

is equal to:

(A) $\dfrac{\sqrt{34}}{3}$
(B) $\dfrac{5}{3}$
(C) $\dfrac{\sqrt{41}}{4}$
(D) $\dfrac{5}{4}$

Q177·MathematicsSingle correctJEE Main 2019

Let

\left(-2-\dfrac{1}{3}i\right)^{3}=\dfrac{x+iy}{27}

(i=\sqrt{-1})

, where x and y are real numbers, then

y-x

equals:

(A)91
(B)-85
(C)85
(D)-91

Q178·MathematicsSingle correctJEE Main 2019

Let

A=\left\{\theta\in\left(-\dfrac{\pi}{2},\pi\right):\dfrac{3+2i\sin\theta}{1-2i\sin\theta}\ \text{purely imaginary}\right\}

. Then the sum of the elements in A is:

(A) $\dfrac{5\pi}{6}$
(B) $\pi$
(C) $\dfrac{3\pi}{4}$
(D) $\dfrac{2\pi}{3}$

Q179·MathematicsSingle correctJEE Main 2019

Let

\alpha

and

\beta

be two roots of the equation

x^{2}+2x+2=0

, then

\alpha^{15}+\beta^{15}

is equal to:

(A) $-256$
(B) $512$
(C) $-512$
(D) $256$

Q180·MathematicsSingle correctJEE Main 2019

Let

z_{0}

be a root of the quadratic equation,

x^{2}+x+1=0

. If

z=3+6iz_{0}^{81}-3iz_{0}^{93}

, then

\arg z

is equal to

(A) $\frac{\pi}{4}$
(B) $\frac{\pi}{3}$
(C)0
(D) $\frac{\pi}{6}$

Q181·MathematicsMultiple correctJEE Advanced 2018

Let

s

,

t

,

r

be non-zero complex numbers and L be the set of solutions

z = x + iy

(

x

,

y \in R

,

i = \sqrt{-1}

) of the equation

sz + t\bar{z} + r = 0

, where

\bar{z} = x - iy

. Then, which of the following statement(s) is (are) TRUE ?

(A)If L has exactly one element, then $|s| \neq |t|$
(B)If $|s| = |t|$ , then L has infinitely many elements
(C)The number of elements in $L \cap \{z : |z - 1 + i| = 5\}$ is at most 2
(D)If L has more than one element, then L has infinitely many elements

Q182·MathematicsMultiple correctJEE Advanced 2018

For a non-zero complex number z, let arg(z) denote the principal argument with

-\pi < \arg(z) \le \pi

. Then, which of the following statement(s) is (are) FALSE ?

(A) $\arg(-1-i) = \frac{\pi}{4}$ , where $i = \sqrt{-1}$
(B)The function $f : R \to (-\pi, \pi]$ , defined by $f(t) = \arg(-1+it)$ for all $t \in R$ , is continuous at all points of R, where $i = \sqrt{-1}$
(C)For any two non-zero complex numbers $z_{1}$ and $z_{2}$ , $\arg\left(\frac{z_{1}}{z_{2}}\right) - \arg(z_{1}) + \arg(z_{2})$ is an integer multiple of $2\pi$
(D)For any three given distinct complex numbers $z_{1}$ , $z_{2}$ and $z_{3}$ , the locus of the point z satisfying the condition $\arg\left(\frac{(z-z_{1})(z_{2}-z_{3})}{(z-z_{3})(z_{2}-z_{1})}\right) = \pi$ , lies on a straight line

Q183·MathematicsMultiple correctJEE Advanced 2017

Let

a

,

b

,

x

and

y

be real numbers such that

a - b = 1

and

y \neq 0

. If the complex number

z = x + iy

satisfies

\mathrm{Im}\left(\frac{az + b}{z + 1}\right) = y

, then which of the following is(are) possible value(s) of

x

?

(A) $-1 - \sqrt{1 - y^{2}}$
(B) $1 + \sqrt{1 + y^{2}}$
(C) $1 - \sqrt{1 + y^{2}}$
(D) $-1 + \sqrt{1 - y^{2}}$

Q184·MathematicsIntegerJEE Advanced 2016

Let

z = \frac{-1 + \sqrt{3}i}{2}

, where

i = \sqrt{-1}

, and

r, s \in \{1, 2, 3\}

. Let

P = \begin{bmatrix} (-z)^{r} & z^{2s} \\ z^{2s} & z^{r} \end{bmatrix}

and

I

be the identity matrix of order 2. Then the total number of ordered pairs

(r, s)

for which

P^{2} = -I

is

Q185·MathematicsMultiple correctJEE Advanced 2016

Let

a, b \in \mathbb{R}

and

a^{2} + b^{2} \neq 0

. Suppose

S = \left\{ z \in \mathbb{C} : z = \frac{1}{a + ibt}, t \in \mathbb{R}, t \neq 0 \right\}

, where

i = \sqrt{-1}

. If

z = x + iy

and

z \in S

, then (x, y) lies on

(A)the circle with radius $\frac{1}{2a}$ and centre $\left( \frac{1}{2a}, 0 \right)$ for $a > 0, b \neq 0$
(B)the circle with radius $-\frac{1}{2a}$ and centre $\left( -\frac{1}{2a}, 0 \right)$ for $a < 0, b \neq 0$
(C)the x-axis for $a \neq 0, b = 0$
(D)the y-axis for $a = 0, b \neq 0$

Q186·MathematicsIntegerJEE Advanced 2015

For any integer

k

, let

\alpha_{k} = \cos\left(\dfrac{k\pi}{7}\right) + i\sin\left(\dfrac{k\pi}{7}\right)

, where

i = \sqrt{-1}

. The value of the expression

\dfrac{\displaystyle\sum_{k=1}^{12}\left|\alpha_{k+1} - \alpha_{k}\right|}{\displaystyle\sum_{k=1}^{3}\left|\alpha_{4k-1} - \alpha_{4k-2}\right|}

is

Q187·MathematicsSingle correctJEE Advanced 2014

Let

z_{k} = \cos\left(\frac{2k\pi}{10}\right) + i\sin\left(\frac{2k\pi}{10}\right)

;

k = 1, 2, \ldots, 9

.

List – I	List – II
P.For each $z_{k}$ there exists a $z_{j}$ such that $z_{k} \cdot z_{j} = 1$	1.True
Q.There exists a $k \in \{1, 2, \ldots, 9\}$ such that $z_{1} \cdot z = z_{k}$ has no solution $z$ in the set of complex numbers	2.False
R. $\frac{\|1-z_{1}\|\|1-z_{2}\| \ldots \|1-z_{9}\|}{10}$ equals	3.1
S. $1 - \sum_{k=1}^{9} \cos\left(\frac{2k\pi}{10}\right)$ equals	4.2

(A)P-1, Q-2, R-4, S-3
(B)P-2, Q-1, R-3, S-4
(C)P-1, Q-2, R-3, S-4
(D)P-2, Q-1, R-4, S-3

Q188·MathematicsMultiple correctJEE Advanced 2013

Let

w = \frac{\sqrt{3} + i}{2}

and

P = \{w^{n} : n = 1, 2, 3, \ldots\}

. Further

H_{1} = \left\{z \in \mathbb{C} : \operatorname{Re} z > \frac{1}{2}\right\}

and

H_{2} = \left\{z \in \mathbb{C} : \operatorname{Re} z < \frac{-1}{2}\right\}

, where

\mathbb{C}

is the set of all complex numbers. If

z_{1} \in P \cap H_{1}

,

z_{2} \in P \cap H_{2}

and O represents the origin, then

\angle z_{1} O z_{2} =

(A) $\frac{\pi}{2}$
(B) $\frac{\pi}{6}$
(C) $\frac{2\pi}{3}$
(D) $\frac{5\pi}{6}$

Q189·MathematicsSingle correctJEE Advanced 2013

Let

S = S_{1} \cap S_{2} \cap S_{3}

, where

S_{1} = \{z \in \mathbb{C} : |z| < 4\}

,

S_{2} = \left\{z \in \mathbb{C} : \operatorname{Im}\left[\frac{z - 1 + \sqrt{3}i}{1 - \sqrt{3}i}\right] > 0\right\}

and

S_{3} = \{z \in \mathbb{C} : \operatorname{Re} z > 0\}

. Area of

S =

(A) $\frac{10\pi}{3}$
(B) $\frac{20\pi}{3}$
(C) $\frac{16\pi}{3}$
(D) $\frac{32\pi}{3}$

Q190·MathematicsSingle correctJEE Advanced 2013

Let complex numbers

\alpha

and

\frac{1}{\bar{\alpha}}

lie on circles

(x-x_0)^2+(y-y_0)^2=r^2

and

(x-x_0)^2+(y-y_0)^2=4r^2

, respectively. If

z_0=x_0+iy_0

satisfies the equation

2|z_0|^2=r^2+2

, then

|\alpha|=

(A) $\frac{1}{\sqrt{2}}$
(B) $\frac{1}{2}$
(C) $\frac{1}{\sqrt{7}}$
(D) $\frac{1}{3}$

Q191·MathematicsSingle correctJEE Advanced 2013

Let

S = S_{1} \cap S_{2} \cap S_{3}

, where

S_{1} = \{z \in \mathbb{C} : |z| < 4\}

,

S_{2} = \left\{z \in \mathbb{C} : \operatorname{Im}\left[\frac{z - 1 + \sqrt{3}i}{1 - \sqrt{3}i}\right] > 0\right\}

and

S_{3} = \{z \in \mathbb{C} : \operatorname{Re} z > 0\}

.

\min_{z \in S} |1 - 3i - z| =

(A) $\frac{2 - \sqrt{3}}{2}$
(B) $\frac{2 + \sqrt{3}}{2}$
(C) $\frac{3 - \sqrt{3}}{2}$
(D) $\frac{3 + \sqrt{3}}{2}$

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