Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 18

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

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Important Questions on Complex Numbers

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 19

If

|z - \frac{4}{z}| = 2

, then the maximum value of

| z |

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 20

If

α

and

β

are the roots of the equation

x^{2} - x + 1 = 0

, then

α^{2009} + β^{2009} =

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 21

If

ω (\neq 1)

is a cube root of unity and

(1 + ω)^{7} = A + B ω .

Then,

(A, B)

equals

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 22

Let

α, β

be real and

z

be a complex number. If

z^{2} + α z + β = 0

has two distinct roots on the line

Re z = 1,

then it is necessary that

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 23

If

z

is a complex number of unit modulus and argument

θ,

then

a r g (\frac{1 + z}{1 + \bar{z}})

equals

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 24

If $z$ a complex number such that $|z| \geq 2$ , then the minimum value of $|z + \frac{1}{2}|$

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 25

A complex number

z

is said to be unimodular, if

|z| = 1

Suppose

z_{1}

and

z_{2}

are complex numbers such that

\frac{z_{1} - 2 z_{2}}{2 - z_{1} {\bar{z}}_{2}}

is unimodular and

z_{2}

is not unimodular. Then, the point

z_{1}

lies on a

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-3>Q 26

A value of

θ

for which

\frac{2 + 3 isinθ}{1 - 2 isinθ}

is purely imaginary, is