EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Main - 27 July 2022 Shift 1>Q 97

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JEE Main

IMPORTANT

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

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Important Questions on Complex Numbers

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Main - 27 July 2022 Shift 1>Q 98

Let

S = \{z \in ℂ : z^{2} + \bar{z} = 0\}

. Then

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

is equal to _______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Main - 27 July 2022 Shift 2>Q 99

Let

S

be the set of all

(α, β), π < α, β < 2 π

, for which the complex number

\frac{1 - i \sin α}{1 + 2 i \sin α}

is purely imaginary and

\frac{1 + i \cos β}{1 - 2 i \cos β}

is purely real. Let

Z_{α β} = \sin 2 α + i \cos 2 β, (α, β) \in S

.
Then

\sum_{(α, β) \in S} (i Z_{α β} + \frac{1}{i {\bar{Z}}_{α β}})

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Main - 28 July 2022 Shift 1>Q 100

Let

S_{1} = \{z_{1} \in C : |z_{1} - 3| = \frac{1}{2}\}

and

S_{2} = \{z_{2} \in C : |z_{2} - |z_{2} + 1|| = |z_{2} + |z_{2} - 1||\} .

Then, for

z_{1} \in S_{1}

and

z_{2} \in S_{2}

, the least value of

|z_{2} - z_{1}|

is

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JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Main - 28 July 2022 Shift 2>Q 101

Let

z = a + i b, 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 _____ .

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Main - 29 July 2022 Shift 1>Q 102

If

z = 2 + 3 i

, then

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

is equal to:

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Main - 29 July 2022 Shift 2>Q 103

If

z \neq 0

be a complex number such that

|z - \frac{1}{z}| = 2

, then the maximum value of

|z|

is

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JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Main - 29 July 2022 Shift 2>Q 104

Let

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 S

for some

y \in ℝ

, is

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Main - 24 January 2023 Shift 1>Q 105

Let

p, q \in ℝ

and

(1 - \sqrt{3} i)^{200} = 2^{199} (p + i q)

,

i = \sqrt{- 1}

. Then,

p + q + q^{2}

and

p - q + q^{2}

are roots of the equation.