Let a , b ∈ R and a 2 + b 2 ≠ 0 . Suppose S = z ∈ C : z = 1 a + i b t , t ∈ R , t ≠ 0 , where i = - 1 . If z = x + i y and z ∈ S , then x , y lies on

The y - axis for a = 0 , b ≠ 0

The circle with radius - 1 2 a and centre - 1 2 a , 0 for a < 0 , b ≠ 0

Let a , b ∈ R and a 2 + b 2 ≠ 0 . Suppose S = z ∈ C : z = 1 a + i b t , t ∈ R , t ≠ 0 , where i = - 1 . If z = x + i y and z ∈ S , then x , y lies on

The y - axis for a = 0 , b ≠ 0

For a non-zero complex number z , let arg ( z ) denote the principal argument with - π < a r g z ≤ π then, which of the following statement(s) is (are) FALSE?

For any three given distinct complex numbers z 1 , z 2 & z 3 , the locus of the point z satisfying the condition arg z - z 1 z 2 - z 3 z - z 3 z 2 - z 1 = π lies on a straight line

For any two non-zero complex numbers z 1 & z 2 , a r g z 1 z 2 - a r g z 1 + a r g z 2 is an integer multiple of 2 π

Let s , t , r be non-zero complex numbers and L be the set of solutions z = x + i y x , y ∈ ℝ , i = - 1 of the equation s z + t z ¯ + r = 0 , where z ¯ = x - i y . Then, which of the following statement(s) is (are) TRUE?

If L has more than one element, then L has infinitely many elements

If s = t , then L has infinitely many elements

For a non-zero complex number z , let arg ( z ) denote the principal argument with - π < a r g z ≤ π then, which of the following statement(s) is (are) FALSE?

For any three given distinct complex numbers z 1 , z 2 & z 3 , the locus of the point z satisfying the condition arg z - z 1 z 2 - z 3 z - z 3 z 2 - z 1 = π lies on a straight line

For any two non-zero complex numbers z 1 & z 2 , a r g z 1 z 2 - a r g z 1 + a r g z 2 is an integer multiple of 2 π

HARD

JEE Advanced

IMPORTANT

Earn 100

Let $a, b \in R$ and $a^{2} + b^{2} \neq 0$ . Suppose $S = \{z \in C : z = \frac{1}{a + i b t}, t \in R, t \neq 0\},$ where $i = \sqrt{- 1} .$ If $z = x + i y$ and $z \in S,$ then $(x, y)$ lies on

7.69% studentsanswered this correctly

Important Questions on Complex Numbers

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Advanced Paper 2 - 2013>Q 6

Let

ω = \frac{\sqrt{3} + i}{2}

and

P = \{ω^{n} : n = 1, 2, 3 \dots\}

. Further

H_{1} = \{z \in C : R e (z) > \frac{1}{2}\}

and

H_{2} = \{z \in C : R e (z) < \frac{- 1}{2}\}

, where

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

∠ z_{1} O z_{2} =

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Advanced Paper 1 - 2021>Q 8

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 $i = \sqrt{- 1}$ . Consider the statements $P$ and $Q$ given below:

$P : |z_{2} - z_{1}| + |z_{3} - z_{2}| + \dots + |z_{10} - z_{9}| + |z_{1} - z_{10}| \leq 2 π$

$Q : |z_{2}^{2} - z_{1}^{2}| + |z_{3}^{2} - z_{2}^{2}| + \dots + |z_{10}^{2} - z_{9}^{2}| + |z_{1}^{2} - z_{10}^{2}| \leq 4 π$

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Advanced Paper 1 - 2021>Q 9

For any complex number

w = c + i d

, let

\arg (w) \in (- π, π]

, where

i = \sqrt{- 1}

. Let

α

and

β

be real numbers such that for all complex numbers

z = x + i y

satisfying

\arg (\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?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Advanced Paper 1 - 2020>Q 10

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?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Advanced Paper 1 - 2019>Q 11

Let

S

be the set of all complex numbers

z

satisfying

|z - 2 + i| \geq \sqrt{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} - \bar{z_{0}}}{z_{0} - \bar{z_{0}} + 2 i}

MEDIUM

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Advanced Paper 1 - 2019>Q 12

Let

ω \neq 1

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

{{|a + b ω + c ω^{2}|}^{2} : a, b, c

are distinct non-zero integers

}

equals ________

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Advanced Paper 1 - 2018>Q 13

For a non-zero complex number

z

, let

\arg (z)

denote the principal argument with

- π < a r g (z) \leq π

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

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Advanced Paper 1 - 2017>Q 14

Let

a, b, x

and

y

be real numbers such that

a - b = 1

and

y \neq 0

. If the complex number

z = x + i y

satisfies

Im (\frac{a z + b}{z + 1}) = y,

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

x ?

Let a, b ∈R and a2+b2≠0 . Suppose S=z∈C :z=1a+ibt, t ∈R, t ≠0, where i= -1. If z=x+iy and z∈S, then x, y lies on

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Let $a, b \in R$ and $a^{2} + b^{2} \neq 0$ . Suppose $S = \{z \in C : z = \frac{1}{a + i b t}, t \in R, t \neq 0\},$ where $i = \sqrt{- 1} .$ If $z = x + i y$ and $z \in S,$ then $(x, y)$ lies on