Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 20

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If $a$ and $b$ are positive integers such that $N = {(a + i b)}^{3} - 107 i$ is a positive integer, then find the value of $\frac{N}{2}$

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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 22

If

x = 9^{\frac{1}{3}} 9^{\frac{1}{9}} 9^{\frac{1}{27}} \dots \infty, y = 4^{\frac{1}{3}} 4^{- \frac{1}{9}} 4^{\frac{1}{27}} \dots \infty

and

z = \sum_{r = 1}^{\infty} {(1 + i)}^{- r}

and principal argument of

P = (x + y z)

is

- \tan^{- 1} (\frac{\sqrt{a}}{b})

, then determine

a^{2} + b^{2}

(where

a & b

are co-prime natural numbers).

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 23

z_{1}, z_{2} \in c

and

z_{1}^{2} + z_{2}^{2} \in R

,

z_{1} (z_{1}^{2} - 3 z_{2}^{2}) = 2, z_{2} (3 z_{1}^{2} - z_{2}^{2}) = 11

. If

z_{1}^{2} + z_{2}^{2} = λ

, then determine

λ^{2}

.

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 31

Given that,

|z - 1| = 1,

where

{}^{'}z^{'}

is a point on the Argand plane.

\frac{z - 2}{2 z} = α i \tan (a r g z) .

. Then determine

\frac{1}{α^{4}}

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

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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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