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

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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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Important Questions on Complex Numbers

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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.

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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