Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 58

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If $| z - 1 | \leq 2$ and $|ω z - 1 - ω^{2}| = a$ (where $ω$ is a cube root of unity), then complete set of values of $a$ 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 59

If

|z^{2} - 3| = 3 |z|

, then the maximum value of

|z|

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 60

If

| 2 z - 1 | = | z - 2 |

and

z_{1}, z_{2}

are complex numbers such that

|z_{1} - α| < α, |z_{2} - β| < β,

then

|\frac{z_{1} + z_{2}}{α + β}|

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 61

If

z_{1}

is a root of the equation

a_{0} z^{n} + a_{1} z^{n - 1} + . . . + a_{n - 1} z + a_{n} = 3,

where

|a_{i}| < 2

for

i = 0, 1, . . ., n,

then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 62

If

|z| < \sqrt{2} - 1,

then

|z^{2} + 2 z \cos α|

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 63

Let

|z_{r} - r| \leq r, \forall r = 1, 2, 3,

....

n .

Then,

|\sum_{r = 1}^{n} z_{r}|

is less than

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 64

All the roots of the equation

11 z^{10} + 10 i z^{9} + 10 i z - 11 = 0

lie

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 65

Let

λ \in R,

the origin and the non-real roots of

2 z^{2} + 2 z + λ = 0

form the three vertices of an equilateral triangle in the Argand plane, then

λ

is

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 66

The roots of the equation

r^{3} + 3 a t^{2} + 3 b t + c = 0

are

z_{1}, z_{2}, z_{3}

which represent the vertices of an equilateral triangle. Then