Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 14

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The possible values of the real parameter $m$ for which $|z - (m^{2} - 7 m + 11 + i)| = 1$ and $\arg (z) \geq \frac{π}{2}$ is satisfied by at least one $z$ is/are

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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 15

If

β = e^{i \frac{2 π}{7}}

, and

|\sum_{r = 0}^{3 n - 1} β^{2^{r}}| = 4 \sqrt{2}

then

n

is equal to

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

_{If area of the region on complex plane by complex number $z$ , such that $\frac{π}{4} \leq \arg (\frac{z - 2}{z + 2}) \leq \frac{3 π}{4}$ is $a π + b$ then $(b - a)$ equals}

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 1

A man walks a distance of

3

units from the origin towards the North-East (

N

45 °

E

) direction. From there, he walks a distance of

4

units towards the North-West (

N

45 °

W

) direction to reach a point

P

. Then, the position of

P

in the argand plane is

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 2

If

|z| = 1

and

z \neq 1,

then all the values of

\frac{z}{1 - z^{2}}

lie on

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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})

at

θ = 2^{°}

is

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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}

is