Core Syllabus Module

Complex Numbers

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Main 2024 (01 Feb Shift 1)

LEVELJEE Advanced

Let $P = {z \in C : ∣ z + 2 - 3 i ∣ \leq 1}$ and $Q = {z \in C : z (1 + i) + \overset{z}{ˉ} (1 - i) \leq - 8}$ . Let in $P \cap Q$ , $∣ z - 3 + 2 i ∣$ be maximum and minimum at $z_{1}$ and $z_{2}$ respectively. If $∣ z_{1} ∣^{2} + 2∣ z_{2} ∣^{2} = α + β 2$ , where $α, β$ are integers, then $α + β$ equals

JEE(ADVANCED)-201

LEVELJEE Advanced

For a non-zero complex number $z$ , let $ar g (z)$ denote the principal argument with $- π < ar g (z) \leq π$ . Then, which of the following statement(s) is (are) FALSE ?

* Multiple Correct Options

(A)

ar g (- 1 - i) = \frac{π}{4}

, where

i = - 1

(B)

The function

f : R \to (- π, π]

, defined by

f (t) = ar g (- 1 + i t)

for all

t \in R

, is continuous at all points of

R

, where

i = - 1

(C)

For any two non-zero complex numbers

z_{1}

and

z_{2}

ar g \frac{z _{1}}{z _{2}} - ar g (z_{1}) + ar g (z_{2})

is an integer multiple of

2 π

(D)

For any three given distinct complex numbers

z_{1}, z_{2}

and

z_{3}

, the locus of the point

z

satisfying the condition

ar g (\frac{( z - z _{1} ) ( z _{2} - z _{3} )}{( z - z _{3} ) ( z _{2} - z _{1} )}) = π

, lies on a straight line

JEE Advanced 2008

LEVELJEE Advanced

Comprehension Passage

Let

A, B, C

be three sets of complex numbers as defined below

A = {z : I m z \geq 1}

B = {z : ∣ z - 2 - i ∣ = 3}

C = {z : R e ((1 - i) z) = 2}

Question 1:

The number of elements in the set $A \cap B \cap C$ is

\infty

Question 2:

Let $z$ be any point in $A \cap B \cap C$ . Then, $∣ z + 1 - i ∣^{2} + ∣ z - 5 - i ∣^{2}$ lies between

Let $z$ be any point $A \cap B \cap C$ and let $w$ be any point satisfying $∣ w - 2 i ∣ < 3$ . Then, $∣ z ∣ - ∣ w ∣ + 3$ lies between

Match the statements given in Column-I with the values given in Column-II.

List-I

(P)

a = \hat{j} + 3 \hat{k}

b = - \hat{j} + 3 \hat{k}

and

c = 23 \hat{k}

form a triangle, then the internal angle of the triangle between

a

and

b

(Q)

\int_{a}^{b} (f (x) - 3 x) d x = a^{2} - b^{2}

, then the value of

f (\frac{π}{6})

(R)

The value of

\frac{π ^{2}}{l n 3} \int_{7/6}^{5/6} sec (π x) d x

(S)

The maximum value of

Arg (\frac{1}{1 - z})

for

∣ z ∣ = 1, z \neq = 1

is given by

List-II

(1)

\frac{π}{6}

(2)

\frac{2 π}{3}

(3)

\frac{π}{3}

(4)

π

(5)

\frac{π}{2}

JEE Advanced 1990

LEVELJEE Advanced

Let $z_{1} = 10 + 6 i$ and $z_{2} = 4 + 6 i$ . If $z$ is any complex number such that the argument of $\frac{z - z _{1}}{z - z _{2}}$ is $\frac{π}{4}$ , then prove that $∣ z - 7 - 9 i ∣ = 32$ .

JEE Advanced 2023

LEVELJEE Advanced

Let $A_{1}, A_{2}, A_{3}, \dots, A_{8}$ 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 $P A_{i}$ denote the distance between the points $P$ and $A_{i}$ for $i = 1, 2, \dots, 8$ . If $P$ varies over the circle, then the maximum value of the product $P A_{1} \cdot P A_{2} \cdot \dots \cdot P A_{8}$ , is

JEE Main 2022 (27 June Shift 2)

LEVELJEE Advanced

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

JEE Main 2022 (29 June Shift 1)

LEVELJEE Advanced

Let $S = {z \in C : ∣ z - 2∣ \leq 1, z (1 + i) + \overset{z}{ˉ} (1 - i) \leq 2}$ . Let $∣ z - 4 i ∣$ attains minimum and maximum values, respectively, at $z_{1} \in S$ and $z_{2} \in S$ . If $5 (∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2}) = α + β 5$ , where $α$ and $β$ are integers, then the value of $α + β$ is equal to \_\_\_\_\_.

JEE Main 2025 (January)

LEVELJEE Advanced

Let integers $a, b \in [- 3, 3]$ be such that $a + b \neq = 0$ . Then the number of all possible ordered pairs $(a, b)$ , for which $∣ \frac{z - a}{z + b} ∣ = 1$ and $∣ z + 1 ω ω^{2} ω z + ω^{2} 1 ω^{2} 1 z + ω ∣ = 1, z \in C$ , where $ω$ and $ω^{2}$ are the roots of $x^{2} + x + 1 = 0$ is equal to

JEE Advanced 2019

LEVELJEE Advanced

Let $S$ be the set of all complex numbers $z$ satisfying $∣ z - 2 + i ∣ \geq 5$ . If the complex number $z_{0}$ is such that $\frac{1}{∣ z _{0} - 1∣}$ is the maximum of the set ${\frac{1}{∣ z - 1∣} : z \in S}$ , then the principal argument of $\frac{4 - z _{0} - z ˉ _{0}}{z _{0} - z ˉ _{0} + 2 i}$ is

(A)

\frac{π}{4}

(B)

- \frac{π}{2}

(C)

\frac{3 π}{4}

(D)

\frac{π}{2}

JEE Advanced 1993

LEVELJEE Advanced

$A B C D$ is a rhombus. Its diagonals $A C$ and $B D$ intersect at the point $M$ and satisfy $B D = 2 A C$ . If the points $D$ and $M$ represent the complex numbers $1 + i$ and $2 - i$ respectively, then $A$ represents the complex number $\dots$ or $\dots$

JEE Main 2021 (27 July Shift 2)

LEVELJEE Advanced

Let $C$ be the set of all complex numbers. Let $S_{1} = {z \in C : ∣ z - 2∣ \leq 1}$ and $S_{2} = {z \in C : z (1 + i) + \overset{z}{ˉ} (1 - i) \geq 4}$ . Then, the maximum value of $∣ z - \frac{5}{2} ∣^{2}$ for $z \in S_{1} \cap S_{2}$ is equal to :

(A)

\frac{3 + 2 2}{4}

(B)

\frac{5 + 2 2}{2}

(C)

\frac{3 + 2 2}{2}

(D)

\frac{5 + 2 2}{4}

JEE Advanced 2013

LEVELJEE Advanced

Comprehension Passage

Let

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

, where

S_{1} = {z \in C : ∣ z ∣ < 4}

S_{2} = {z \in C : I m [\frac{z - 1 + 3 i}{1 - 3 i}] > 0}

and

S_{3} = {z \in C : R ez > 0}

Question 1:

Area of $S =$

(A)

\frac{10 π}{3}

(B)

\frac{20 π}{3}

(C)

\frac{16 π}{3}

(D)

\frac{32 π}{3}

Question 2:

$min_{z \in S} ∣1 - 3 i - z ∣ =$

(A)

\frac{2 - 3}{2}

(B)

\frac{2 + 3}{2}

(C)

\frac{3 - 3}{2}

(D)

\frac{3 + 3}{2}

JEE Advanced 2021

LEVELJEE Advanced

For any complex number $w = c + i d$ , let $ar g (w) \in (- π, π]$ , where $i = - 1$ . Let $α$ and $β$ be real numbers such that for all complex numbers $z = x + i y$ satisfying $ar g (\frac{z + α}{z + β}) = \frac{π}{4}$ , the ordered pair $(x, y)$ lies on the circle $x^{2} + y^{2} + 5 x - 3 y + 4 = 0$ . Then which of the following statements is (are) TRUE ?

* Multiple Correct Options

(A)

α = - 1

(B)

α β = 4

(C)

α β = - 4

(D)

β = 4

JEE Advanced 2021

LEVELJEE Advanced

Let $θ_{1}, θ_{2}, \dots, θ_{10}$ be positive valued angles (in radian) such that $θ_{1} + θ_{2} + \dots + θ_{10} = 2 π$ . Define the complex numbers $z_{1} = e^{i θ_{1}}$ , $z_{k} = z_{k - 1} e^{i θ_{k}}$ for $k = 2, 3, \dots, 10$ , where $i = - 1$ . Consider the statements P and Q given below : P : $∣ z_{2} - z_{1} ∣ + ∣ z_{3} - z_{2} ∣ + \dots + ∣ z_{10} - z_{9} ∣ + ∣ z_{1} - z_{10} ∣ \leq 2 π$ Q : $∣ z_{2}^{2} - z_{1}^{2} ∣ + ∣ z_{3}^{2} - z_{2}^{2} ∣ + \dots + ∣ z_{10}^{2} - z_{9}^{2} ∣ + ∣ z_{1}^{2} - z_{10}^{2} ∣ \leq 4 π$ 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

JEE Advanced 2010

LEVELJEE Advanced

Match the statements in Column I with those in Column II. [Note : Here $z$ takes values in the complex plane and $I m z$ and $R ez$ denote , respectively, the imaginary part and the real part of $z$ . ]

List-I

(P)

The set of points

z

satisfying

∣ z - i ∣ z ∣∣ = ∣ z + i ∣ z ∣∣

is contained in or equal to

(Q)

The set of points

z

satisfying

∣ z + 4∣ + ∣ z - 4∣ = 10

is contained in or equal to

(R)

∣ w ∣ = 2

, then the set of points

z = w - \frac{1}{w}

is contained in or equal to

(S)

∣ w ∣ = 1

, then the set of points

z = w + \frac{1}{w}

is contained in or equal to

List-II

(1)

an ellipse with eccentricity

\frac{4}{5}

(2)

the set of points

z

satisfying

I m z = 0

(3)

the set of points

z

satisfying

∣ I m z ∣ \leq 1

(4)

the set of points

z

satisfying

∣ R ez ∣ < 2

(5)

the set of points

z

satisfying

∣ z ∣ \leq 3

JEE Advanced 2024

LEVELJEE Advanced

Let $S = {a + b 2 : a, b \in Z}$ , $T_{1} = {(- 1 + 2)^{n} : n \in Z}$ , and $T_{2} = {(1 + 2)^{n} : n \in N}$ . Then which of the following statements is(are) TRUE ?

* Multiple Correct Options

(A)

Z \cup T_{1} \cup T_{2} \subset S

(B)

T_{1} \cap (0, \frac{1}{2024}) = ϕ

, where

ϕ

denotes the empty set

(C)

T_{2} \cap (2024, \infty) \neq = ϕ

(D)

For any given

a, b \in Z

cos (π (a + b 2)) + i sin (π (a + b 2)) \in Z

if and only if

b = 0

, where

i = - 1

JEE Advanced 2014

LEVELJEE Advanced

Let $z_{k} = cos (\frac{2 k π}{10}) + i sin (\frac{2 k π}{10}); k = 1, 2, \dots, 9$ . Match the statements in List-I with those in List-II.

List-I

(P)

For each

z_{k}

there exists a

z_{j}

such that

z_{k} \cdot z_{j} = 1

(Q)

There exists a

k \in {1, 2, \dots, 9}

such that

z_{1} \cdot z = z_{k}

has no solution

z

in the set of complex numbers

(R)

\frac{∣1 - z _{1} ∣∣1 - z _{2} ∣ \dots ∣1 - z _{9} ∣}{10}

equals

(S)

1 - \sum_{k = 1}^{9} cos (\frac{2 k π}{10})

equals

List-II

JEE Main 2020 - 4 Sep (Morning)

LEVELJEE Advanced

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

(A)

1/2

(B)

3/2

(C)

2

(D)

4

JEE Main 2026 (22 January Shift 1)

LEVELJEE Advanced

Let $α = \frac{- 1 + i 3}{2}$ and $β = \frac{- 1 - i 3}{2}, i = - 1$ . If $(7 - 7 α + 9 β)^{20} + (9 + 7 α - 7 β)^{20} + (- 7 + 9 α + 7 β)^{20} + (14 + 7 α + 7 β)^{20} = m^{10}$ , then $m$ is .........

Micro-Topic Strategy Matrix

Algebraic Operations on Complex Numbers

Conjugate and Modulus

Argand Plane and Polar Representation

Euler's Form and De Moivre's Theorem

Cube Roots and nth Roots of Unity

Geometrical Applications of Complex Numbers

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For a non-zero complex number z, let arg(z) denote the principal argument with −π<arg(z)≤π. Then, which of the following statement(s) is (are) FALSE ?

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of elements in the set A∩B∩C is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z be any point in A∩B∩C. Then, ∣z+1−i∣2+∣z−5−i∣2 lies between

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z be any point A∩B∩C and let w be any point satisfying ∣w−2i∣<3. Then, ∣z∣−∣w∣+3 lies between

Match the statements given in Column-I with the values given in Column-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z1​=10+6i and z2​=4+6i. If z is any complex number such that the argument of z−z2​z−z1​​ is 4π​, then prove that ∣z−7−9i∣=32​.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of points of intersection of ∣z−(4+3i)∣=2 and ∣z∣+∣z−4∣=6,z∈C is :

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Area of S=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } minz∈S​∣1−3i−z∣=

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={a+b2​:a,b∈Z}, T1​={(−1+2​)n:n∈Z}, and T2​={(1+2​)n:n∈N}. Then which of the following statements is(are) TRUE ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let zk​=cos(102kπ​)+isin(102kπ​);k=1,2,…,9. Match the statements in List-I with those in List-II.

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let u=z−ki2z+i​, z=x+iy and k>0. If the curve represented by Re(u)+Im(u)=1 intersects the y-axis at the point P and Q where PQ=5, then the value of K is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α=2−1+i3​​ and β=2−1−i3​​,i=−1​. If (7−7α+9β)20+(9+7α−7β)20+(−7+9α+7β)20+(14+7α+7β)20=m10, then m is .........

For a non-zero complex number $z$ , let $ar g (z)$ denote the principal argument with $- π < ar g (z) \leq π$ . Then, which of the following statement(s) is (are) FALSE ?

The number of elements in the set $A \cap B \cap C$ is

Let $z$ be any point in $A \cap B \cap C$ . Then, $∣ z + 1 - i ∣^{2} + ∣ z - 5 - i ∣^{2}$ lies between

Let $z$ be any point $A \cap B \cap C$ and let $w$ be any point satisfying $∣ w - 2 i ∣ < 3$ . Then, $∣ z ∣ - ∣ w ∣ + 3$ lies between

Let $z_{1} = 10 + 6 i$ and $z_{2} = 4 + 6 i$ . If $z$ is any complex number such that the argument of $\frac{z - z _{1}}{z - z _{2}}$ is $\frac{π}{4}$ , then prove that $∣ z - 7 - 9 i ∣ = 32$ .

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

Area of $S =$

$min_{z \in S} ∣1 - 3 i - z ∣ =$

Let $S = {a + b 2 : a, b \in Z}$ , $T_{1} = {(- 1 + 2)^{n} : n \in Z}$ , and $T_{2} = {(1 + 2)^{n} : n \in N}$ . Then which of the following statements is(are) TRUE ?

Let $z_{k} = cos (\frac{2 k π}{10}) + i sin (\frac{2 k π}{10}); k = 1, 2, \dots, 9$ . Match the statements in List-I with those in List-II.

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

Let $α = \frac{- 1 + i 3}{2}$ and $β = \frac{- 1 - i 3}{2}, i = - 1$ . If $(7 - 7 α + 9 β)^{20} + (9 + 7 α - 7 β)^{20} + (- 7 + 9 α + 7 β)^{20} + (14 + 7 α + 7 β)^{20} = m^{10}$ , then $m$ is .........