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

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If $a, b, c$ are real numbers and $z$ is a complex number such that, $a^{2} + b^{2} + c^{2} = 1$ and $b + i c = (1 + a) z$ then $\frac{1 + i z}{1 - i z}$ equals.

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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 2

If

z

satisfies

| z - 1 | < | z + 3 |,

then

ω = 2 z + 3 - i

satisfies

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

If

| z + ω |^{2} = | z |^{2} + | ω |^{2}

, where

z

and

ω

are complex numbers, then

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

Let

z_{1}, z_{2}

be two complex numbers represented by points on the circle

|z| = 1

and

|z| = 2

respectively, then

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

Let complex number

z

satisfy

|z - \frac{2}{z}| = 1,

then

| z |

can take all values except

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

If

z_{1} = a + i b

and

z_{2} = c + i d

are complex numbers such that

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

and

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

, then the pair of complex numbers

ω_{1} = a + i c

and

ω_{2} = b + i d

satisfies

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

If

z_{1}, z_{2}

be two complex numbers satisfying the equation

|\frac{z_{1} + z_{2}}{z_{1} - z_{2}}| = 1

, then

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

If

Z_{1}, Z_{2}, Z_{3} \in C

such that

|Z_{1}| = |Z_{2}| = |Z_{3}| = 1

and

\frac{Z_{1}^{2}}{Z_{2} Z_{3}} + \frac{Z_{2}^{2}}{Z_{1} Z_{3}} + \frac{Z_{3}^{2}}{Z_{1} Z_{2}} + 1 = 0,

then the integral values that

|Z_{1} + Z_{2} + Z_{3}|

can attain, is

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

If

|\frac{2 i z_{1} - z_{1} - z_{2}}{2 i z_{1} + z_{1} + z_{2}}| = |\frac{\cos 2 ϕ + i \sin 2 ϕ}{\cos 2 ϕ - i \sin 2 ϕ}|

and

z_{1}, z_{2}

are non-zero complex numbers, then