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

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If both the roots of the quadratic equation $a x^{2} + b x + c = 0; a, b, c \in R - {0}$ are nonreal and if ' $θ$ ' denotes the principle argument of the complex number $Z = a + i c,$ then which of the following options is necessarily true?

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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 7

Consider two complex numbers

z_{1}

and

z_{2}

such that

|z_{1} + z_{2}| = |z_{1}| - |z_{2}|,

then the complex number

z = \frac{z_{1}}{z_{2}}

will always lie on a

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

The sum of the real roots of polynomial

72 - 36 z - 26 z^{2} + 13 z^{3} + 2 z^{4} - z^{5} = 0

will be

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

If

f (x) = 7 x^{5} + 3 x^{4} + a_{1} x^{3} + a_{2} x^{2} + a_{3} x + 1

such that for any

z = x + i y

where

x y \neq 0

and

x, y \in R, f (\bar{z}) = \bar{f (z)}

then the product of

a_{1} a_{2} a_{3}

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

a, b, c

are real numbers in the polynomial

p (z) = 2 z^{4} + a z^{3} + b z^{2} + c z + 5 .

If two roots of the equation

p (z) = 0

are

2

and

i (= \sqrt{- 1}),

then the value of

a

is

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 11

Given that the equation

z^{2} + (a + i b) z + c + i d = 0

where

a, b, c, d

are non-zero, has a real root then

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 13

If

x = - 5 + 2 \sqrt{- 4}

, then the value of the expression

x^{4} + 9 x^{3} + 35 x^{2} - x + 4

is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 5 - Complex Numbers>Assignment>Q 15

It is given that the equation

| z |^{2} - 2 i z + 2 α (λ + i) = 0

possesses a solution for all

α \in R,

then the number of integral value(s) of '

λ

' for which it is true is

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

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.