Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 5

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Let $a, b$ and $c$ be distinct non-zero real numbers satisfying $a^{3} + b^{3} + 6 a b c = 8 c^{3}$ . Then, which of the following can be correct? $($ $ω$ is complex cube root of unity $)$

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Important Questions on Complex Numbers

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 6

Let

z_{1}

and

z_{2}

be two non-zero complex numbers such that

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

. Then

\frac{z_{1}}{z_{2}}

can be equal to (

ω

is imaginary cube root of unity)

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 7

If

p = a + b ω + c ω^{2}

,

q = b + c ω + a ω^{2}

, and

r = c + a ω + b ω^{2}

, where

a, b, c \neq 0

and

ω

is the complex cube root of unity, then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 8

Let

P (x)

and

Q (x)

be two polynomials. Suppose that

f (x) = P (x^{3}) + x Q (x^{3})

is divisible by

x^{2} + x + 1

, then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 9

If

α

is a complex constant such that

α z^{2} + z + α = 0

has a real root, then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 10

If

z^{3} + (3 + 2 i) z + (- 1 + i a) = 0

has one real root, then the value of

a

lies in the interval

(a \in R)

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 11

Given that the complex numbers which satisfy the equation

|z z^{3}| + |z z^{3}| = 350

form a rectangle in the Argand plane with the length of its diagonal having an integral number of units, then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 12

If the points

A (z), B (- z)

and

C (1, - z)

are the vertices of an equilateral triangle

A B C

, then

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 13

If

|z - 3| = \min \{|z - 1|, |z - 5|\}

, then

Re (z)

equals to