Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 32

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The area of the region bounded by $|z| \leq 4 & - \frac{π}{2} \leq a r g z \leq \frac{π}{3}$ on the Argand diagram is expressed in the form of $\frac{a π}{b} .$ Then find the value of $a b$ (where $a & b$ are co-prime natural number)

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Important Questions on Complex Numbers

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 34

If

ω

and

ω^{2}

are the non-real cube roots of unity and

a, b, c \in R

such that

\frac{1}{a + ω} + \frac{1}{b + ω} + \frac{1}{c + ω} = 2 ω^{2}

and

\frac{1}{a + ω^{2}} + \frac{1}{b + ω^{2}} + \frac{1}{c + ω^{2}} = 2 ω .

If

\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c + 1} = λ

, then determine

λ^{4}

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 38

If the biquadratic

x^{4} + a x^{3} + b x^{2} + c x + d = 0 (a, b, c, d \in R)

has

4

non-real roots, two with sum

3 + 4 i

and the other two with product

13 + i .

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 39

The quadratic equation

z^{2} + (p + i p^{'}) z + q + i q^{'} = 0;

where

p, p^{'}, q, q^{'}

are all real.

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 40

The value of

i^{n} + i^{- n},

for

i = \sqrt{- 1}

and

n \in I

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 41

If

amp (z_{1} z_{2}) = 0

and

|z_{1}| = |z_{2}| = 1

, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 42

Let $z_{1}$ and $z_{2}$ are two complex numbers such that $(1 - i) z_{1} = 2 z_{2}$ and arg $(z_{1} z_{2}) = \frac{π}{2},$ then arg $(z_{2})$ is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 43

If

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

(where

z_{1}

and

z_{2}

are non-zero complex numbers), then

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 44

a, b, c

are real numbers in the polynomial,

P (z) = 2 z^{4} + a z^{3} + b z^{2} + c z + 3 .

If two roots of the equation

P (z) = 0

are

2

and

i

. Then which of the following are true.