Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 32

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If $z = (3 + 7 i) (a + i b)$ where $a, b \in Z - \{0\},$ is purely imaginary, then the minimum value of ${|z|}^{2}$ is

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Important Questions on Complex Numbers

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 33

If

(\cos θ + i \sin θ) (\cos 2 θ + i \sin 2 θ) . . . (\cos n θ + i \sin n θ) = 1,

then the value of

θ

is,

m \in Z

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 34

Given

z = {(1 + i \sqrt{3})}^{100},

then

[\frac{Re (z)}{Im (z)}]

equals

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 35

The expression

{[\frac{1 + \sin \frac{π}{8} + i \cos \frac{π}{8}}{1 + \sin \frac{π}{8} - i \cos \frac{π}{8}}]}^{8} =

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 36

The number of complex numbers

z

satisfying

|z - 3 - i| = |z - 9 - i|

and

|z - 3 + 3 i| = 3

are

MEDIUM

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 37

P (z)

be a variable point in the Argand plane such that

|z| = minimum \{|z - 1|, |z + 1|\},

then

z + z

will be equal to

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Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 38

If

|z^{2} - 1| = {|z|}^{2} + 1,

then

z

lies on

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 39

If

z = x + i y (x, y \in R, x \neq - \frac{1}{2})

, the number of values of

z

satisfying

{|z|}^{n} = z^{2} {|z|}^{n - 2} + z {|z|}^{n - 2} + 1 . (n \in N, n > 1)

is

HARD

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IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 3 - Complex Numbers>Exercises>Q 40

The number of solutions of the equation

z^{3} + \frac{3 {(\bar{z})}^{2}}{|z|} = 0

(where,

z

is a complex number) are