Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 10

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Let $|z| = 2$ and $w = \frac{z + 1}{z - 1},$ where $z, w \in C$ (where $C$ is the set of complex numbers), then the product of the least and greatest value of the modulus of $w$ is ___.

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Important Questions on Complex Numbers

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 11

If

z

is a complex number satisfying

z^{4} + z^{3} + 2 z^{2} + z + 1 = 0,

then

|z|

is equal to _______.

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 12

Let

1, ω, ω^{2}

be the cube roots of unity. The least possible degree of a polynomial with real coefficients having roots

2 ω, (2 + 3 ω), (2 + 3 ω^{2}), (2 - ω - ω^{2})

is_______.

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 13

If

ω

is the imaginary cube root of unity, then the number of pairs of integers

(a, b)

such that

|a ω + b| = 1

is ________.

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 14

Suppose that

z

is a complex number that satisfies

|z - 2 - 2 i| \leq 1 .

The maximum value of

|2 i z + 4|

is equal to_________.

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 15

If

|z + 2 - i| = 5

and maximum value of

|3 z + 9 - 7 i|

is

M

, then the value of

M

is_______.,

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 16

Let

A = {a \in R |

the equation

(1 + 2 i) x^{3} - 2 (3 + i) x^{2} + (5 - 4 i) x + 2 a^{2} = 0}

has at least one real root. Then the value of

\sum a^{2}

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 17

The minimum value of the expression

E = {|z|}^{2} + {|z - 3|}^{2} + {|z - 6 i|}^{2}

is

m

, then the value of

m

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 18

If

z_{1}

lies on

|z - 3| + |z + 3| = 8

such that

\arg z_{1} = \frac{π}{6}

, then

37 {|z_{1}|}^{2} =

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