If z = 1 and z ≠ 1 , then all the values of z 1 - z 2 lie on

a line not passing through the origin

Let a , b ∈ ℝ and a 2 + b 2 ≠ 0 . Suppose S = z ∈ ℂ : Z = 1 a + i b t , t ∈ ℝ , t ≠ 0 , where i = − 1 . If z = x + i y and z ∈ S , then x , y lies on

The y -axis for a = 0 , b ≠ 0

The circle with radius - 1 2 a and centre - 1 2 a , 0 for a < 0 , b ≠ 0

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

If L has more than one element, then L has infinitely many elements

If s = t , then L has infinitely many elements

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If $|z| = 1$ and $z \neq 1,$ then all the values of $\frac{z}{1 - z^{2}}$ lie on

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Important Questions on Complex Numbers

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 3

A particle

P

starts from the point

z_{0} = 1 + 2 i,

where

i = \sqrt{- 1} .

It moves first horizontally away from origin by

5 units

and then vertically away from origin by

3 units

to reach a point

z_{1} .

From

z_{2}

the particle moves

\sqrt{2}

units in the direction of the vector

\hat{i} + \hat{j}

and then it moves through an angle

\frac{π}{2}

in anticlockwise direction on a circle with centre at origin, to reach a point

z_{2} .

The point

z_{2}

is given by

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 4

Let

z = \cos θ + i \sin θ

. Then the value of

\sum_{m = 1}^{15} Im (z^{2 m - 1})

θ = 2^{°}

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 5

Let complex numbers

α

and

\frac{1}{\bar{α}}

lies on circles

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} = r^{2}

and

{(x - x_{0})}^{2} + {(y - y_{0})}^{2} = 4 r^{2}

, respectively. If

z_{0} = x_{0} + i y_{0}

satisfies the equation

2 {|z_{0}|}^{2} = r^{2} + 2

, then

| α | =

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 6

Let

S

be the set of all comlex numbers

z

satisfying

| z - 2 + i | \geq \sqrt{5}

. if the complex number

z_{0}

is such that

\frac{1}{|z_{0} - 1|}

is the maximum of the set

\{\frac{1}{| z - 1 |} : z \in S\}

, then the principle argument of

\frac{4 - z_{0} - \bar{z_{0}}}{z_{0} - \bar{z_{0}} + 2 i}

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 7

Let

z_{1}

and

z_{2}

be two distinct complex numbers and let

z = (1 - t) z_{1} + t z_{2}

for some real number

t

with

0 < t < 1

. If

\arg (w)

denotes the principal argument of a non-zero complex number

w

, then

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 8

Let

ω

be a complex cube root of unity with

ω \neq 1

and

P = [p_{i j}]

be a

n \times n

matrix with

p_{i j} = ω^{i + j} .

Then

P^{2} \neq 0,

when

n =

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 9

Let

W = \frac{\sqrt{3} + i}{2}

and

P = \{W^{n} : n = 1, 2, 3, \dots\}

. Further

H_{1} = \{z \in C : Re (z) > \frac{1}{2}\}

and

H_{2} = [z \in C : Re (z) < \frac{- 1}{2}]

, where

C

is the set of all complex numbers. If

z_{1} \in P \cap H_{1}

z_{2} \in P \cap H_{2}

and

O

represents the origin, then

∠ z_{1} O z_{2}

is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 10

Let

a, b \in ℝ

and

a^{2} + b^{2} \neq 0

. Suppose

S = \{z \in ℂ : Z = \frac{1}{a + i b t}, t \in ℝ, t \neq 0\}

, where

i = \sqrt{- 1}

. If

z = x + i y

and

z \in S

, then

(x, y)

lies on

If z=1 and z≠1, then all the values of z1-z2 lie on

Important Questions on Complex Numbers

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If $|z| = 1$ and $z \neq 1,$ then all the values of $\frac{z}{1 - z^{2}}$ lie on