Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 3 - Complex Numbers>DPP>Q 10

HARD

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IMPORTANT

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Complex numbers whose real and imaginary parts $x$ and $y$ are integers and satisfy the equation $3 x^{2} - | x y | - 2 y^{2} + 7 = 0$ .

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Important Questions on Complex Numbers

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IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 3 - Complex Numbers>DPP>Q 11

For a complex number

z

, if all the three roots of

a z^{3} + b z^{2} + c z + d = 0

have negative real parts

(a, b, c \in R)

, then

HARD

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IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 3 - Complex Numbers>DPP>Q 12

Suppose three real numbers

a, b, c

are in

G . P .

and

z = \frac{a + i b}{c - i b}

. Then,

EASY

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IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 3 - Complex Numbers>DPP>Q 1

The modulus and amplitude of

\frac{1 + 2 i}{1 - (1 - i)^{2}}

are

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IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 3 - Complex Numbers>DPP>Q 2

The argument of

(z - a) (\bar{z} - b)

is equal to that of

\frac{(\sqrt{3} + i) (1 + \sqrt{3} i)}{1 + i}

where

a, b

are real numbers. If the locus of

z

in the Argand diagram is a circle with its centre at

\frac{(3 + i)}{2}

, then the ordered pair

(a, b)

is

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Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 3 - Complex Numbers>DPP>Q 3

If a complex number

z

satisfies

{|z|}^{2} + \frac{4}{| z |^{2}} - 2 (\frac{z}{\bar{z}} + \frac{\bar{z}}{z}) - 16 = 0

, then the maximum value of

| z |

is

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IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 3 - Complex Numbers>DPP>Q 4

If

cosα + cosβ + cosγ = 0 = sinα + sinβ + sinγ,

then

\frac{\sin 3 α + \sin 3 β + \sin 3 γ}{\sin (α + β + γ)}

is equal to

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IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 3 - Complex Numbers>DPP>Q 5

The least value of

{|z - 3 - 4 i|}^{2} + {|z + 2 - 7 i|}^{2} + {|z - 5 + 2 i|}^{2}

occurs when

z

is

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IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Algebra JEE Main & Advanced>Chapter 3 - Complex Numbers>DPP>Q 6

The roots of the equation

x^{4} - 2 x^{2} + 4 = 0

are the vertices of a