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

HARD

JEE Advanced

IMPORTANT

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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:

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

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

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Important Questions on Complex Numbers

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}

is

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 ?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 4 - Complex Numbers>JEE Advanced Paper 1 - 2013>Q 16

Let complex numbers

α

and

\frac{1}{\bar{α}}

lie on circles

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} = r^{2}

, and

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} = 4 r^{2}

, respectively. If

z_{0} = x_{0} + i y_{0}

satisfies the equation

2 {|z_{0}|}^{2} = r^{2} + 2

, then

|α| =