Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 13

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The equation of the radical axis of the two circles is represented by the equations, $|z - 2| = 3$ and $|z - 2 - 3 i| = 4$ on the complex plane is

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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 14

If

\log_{\frac{1}{2}} (\frac{|z - 1| + 4}{3 |z - 1| - 2}) > 1,

then the locus of

z

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 15

Points

z_{1}

and

z_{2}

are adjacent vertices of a regular octagon. The vertex

z_{3}

adjacent to

z_{2} (z_{3} \neq z_{1})

is represented by :

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 16

If

p = a + b ω + c ω^{2}; q = b + c ω + a ω^{2}

and

r = c + a ω + b ω^{2}

where

a, b, c \neq 0

and

ω

is the non-real complex cube root of unity, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 17

The points

z_{1} = 3 + \sqrt{3} i

and

z_{2} = 2 \sqrt{3} + 6 i

are given on a complex plane. The complex number lying on the bisector of the angle formed by the vectors

z_{1}

and

z_{2}

is :

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 18

Let

ω

be the non real cube root of unity which satisfy the equation

h (x) = 0

where

h (x) = x f (x^{3}) + x^{2} g (x^{3})

If

h (x)

is polynomial with real coefficient then which statement is incorrect.

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Alpha Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>Exercise-2>Q 19

If

1, α_{1}, α_{2}, α_{3}, \dots \dots, α_{n - 1}

be the

n^{th}

roots of unity, then the value of

\sin \frac{π}{n} \cdot \sin \frac{2 π}{n} \cdot \sin \frac{3 π}{n} \dots \dots . . \sin \frac{(n - 1) π}{n}

equals

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

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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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).