Mathematics Crash Course JEE Main>Chapter 5 - Complex Numbers>Level 3>Q 10

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IMPORTANT

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Number of ordered pairs $(z, ω)$ of the complex numbers $z$ and $ω$ , satisfying the system of equations, $z^{3} + (\bar{ω})^{7} = 0$ and $z^{5} \cdot ω^{11} = 1$ is

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Important Questions on Complex Numbers

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 5 - Complex Numbers>Level 3>Q 11

Let

P

denote a complex number

Z = r (\cos θ + i \sin θ)

on the Argand's plane and

Q

denote a complex number

\sqrt{2 {|Z|}^{2}} (\cos (θ + \frac{π}{4}) + i \sin (θ + \frac{π}{4}))

. If

O

is the origin, then

△ O P Q

is

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 5 - Complex Numbers>Level 3>Q 12

Let

A (2, 0)

and

B (z)

are two points on the circle

|z| = 2 . M (z^{'})

is a point on

A B

. If the point

{\bar{z}}^{'}

lies on the median of the triangle

O A B

, where

O

is origin, then

\arg (z^{'})

is

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 5 - Complex Numbers>Level 3>Q 13

If a variable circle

S

touches

S_{1} : |z - z_{1}| = r_{1}

internally and

S_{2} : |z - z_{2}| = r_{2}

externally while the curves

S_{1} &

S_{2}

touch internally to each other. Then the eccentricity of the locus of the centre of the curve

S

is equal to

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 5 - Complex Numbers>Level 3>Q 14

The eccentricity of conic represented by

|z - 2| + |z - 2 + 17 i| = 19

is

e

, then

{(19 e)}^{2}

is equal to

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 5 - Complex Numbers>Level 3>Q 15

A, B

and

C

are points represented by complex numbers

z_{1}, z_{2}

and

z_{3}

. If the circumcentre of the

Δ A B C

is at the origin and the altitude

A D

of the triangle meets the circumcircle again at

P

, then the complex number representing point

P

is

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JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 5 - Complex Numbers>Level 3>Q 16

Let

P (z = x + i y)

lie on the curve

| z | = 2

in Argand plane and a point

Q

representing

z + \frac{1}{z}

moves on an ellipse having length of semi-major and semi-minor axes as

a and b

respectively. The value of

a^{2} - b^{2}

is