If 1 - i z z - i = 1 , then z lies on

a line not passing through origin

The equation z 2 = z - has

An infinite number of solutions

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Which sentence structure is used in the following sentence: 'After the game, the players were tired but happy.'

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Important Questions on Complex Numbers and Quadratic Equations

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Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

Arrange the following compounds in order of decreasing acidity :
Question Image

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Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

For all complex numbers

z

of the form

1 + i α, α \in R,

z^{2} = x + i y,

then

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Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The value of sum

\sum_{n = 1}^{13} (i^{n} + i^{n + 1}), where i = \sqrt{- 1},

equals

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

z

is a complex number such that

Re (z) = I m (z)

, then

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The maximum value of

|z|

where

z

satisfies the condition

|z + \frac{2}{z}| = 2

, is

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Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The equation, whose roots are

i, \frac{1 + i}{\sqrt{2}}, \frac{i - 1}{\sqrt{2}},

their conjugates,

- 1 and 1

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The value of

\sum_{n = 0}^{100} i^{n!}

equals (where

i = \sqrt{- 1}

)

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

(3 + i) z = (3 - i) \bar{z}

, then the complex number

z

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Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The relation between the real numbers

a

and

b

, which satisfy the equation

\frac{1 - i x}{1 + i x} = a - i b

, for some real value of

x

, is

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

Let

z

be a complex number such that the imaginary part of

z

is nonzero and

a = z^{2} + z + 1

is real. Then

a

cannot take the value

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

If $a = \cos α + i \sin α, b = \cos β + i \sin β$ , $c = \cos γ + i \sin γ$ and $\frac{b}{c} + \frac{c}{a} + \frac{a}{b} = 1$ , then $\cos (β - γ) + \cos (γ - α) + \cos (α - β)$ is equal to $(where i = \sqrt{- 1})$

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

Evaluate

{(- \sqrt{- 1})}^{4 n + 3}, n \in N

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

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

then

z

lies on

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The value of sum

\sum_{n = 1}^{13} (i^{n} + i^{n + 1})

, where

i = \sqrt{- 1}

, equals

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The complex number which satisfy the equation

z + \sqrt{2} |z + 1| + i = 0

MEDIUM

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

The equation

z^{2} = \bar{z}

has

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

Dividing

f (z)

z - i

, we obtain the remainder

i

and dividing it by

z + i

, we get the remainder

1 + i

. The remainder upon the division of

f (z)

z^{2} + 1

EASY

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

Δ = |\begin{array}{l} \sqrt{6} & 2 i & 3 + \sqrt{6} \\ \sqrt{12} & \sqrt{3} + \sqrt{8} i & 3 \sqrt{2} + \sqrt{6} i \\ \sqrt{18} & \sqrt{2} + \sqrt{12} i & \sqrt{27} + 2 i \end{array}|

, then

Δ

HARD

Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

If the complex number

z

satisfy the equation

(i - z) (1 + 2 i) + (1 - i z) (3 - 4 i) = 1 + 7 i,

then

z + z + z z

is equal to -

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Mathematics>Algebra>Complex Numbers and Quadratic Equations>Complex Numbers

z = x - i y

and

z^{\frac{1}{3}} = p + i q

, then

\frac{\frac{x}{p} + \frac{y}{q}}{(p^{2} + q^{2})}

is equal to: (where

i = \sqrt{- 1}

)

Which sentence structure is used in the following sentence: 'After the game, the players were tired but happy.'

Important Questions on Complex Numbers and Quadratic Equations

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