Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 9 - Complex Numbers>Exercise>Q 6

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IMPORTANT

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Which of the following is equal to $\sqrt[3]{- 1}$ ?

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Important Questions on Complex Numbers

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 9 - Complex Numbers>Exercise>Q 7

If

| z - 1 | \leq 2

and

|ω z - 1 - ω^{2}| = a

(where

ω

is a cube root of unity), then complete set of values of

a

is

HARD

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 9 - Complex Numbers>Exercise>Q 8

Let

|z_{r} - r| \leq r, \forall r = 1, 2, 3,

....

n .

Then,

|\sum_{r = 1}^{n} z_{r}|

is less than

HARD

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IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 9 - Complex Numbers>Exercise>Q 9

If

z

is a complex number such that

0 \leq \arg z \leq \frac{π}{2},

then which of the following inequality is true?

HARD

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IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 9 - Complex Numbers>Exercise>Q 10

If

' z'

is complex number then the locus of

' z'

satisfying the condition

| 2 z - 1 | = | z - 1 |

is

HARD

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IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 9 - Complex Numbers>Exercise>Q 11

If

t

and

c

are two complex numbers such that

| t | \neq | c |, | t | = 1

and

z = \frac{a t + b}{t - c}, z = x + i y .

Locus of

z

is (where

a, b

are complex numbers)

HARD

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IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 9 - Complex Numbers>Exercise>Q 12

Let

a

be a complex number such that

| a | < 1

and

z_{1}, z_{2}, z_{3}, \dots

be the vertices of a polygon such that

z_{k} = 1 +

a + a^{2} + \dots \dots . + a^{k - 1}

for all

k = 1, 2, 3, \dots

, then

z_{1}, z_{2}, \dots . .

lie within the circle

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 9 - Complex Numbers>Exercise>Q 13

Let

λ \in R,

the origin and the non-real roots of

2 z^{2} + 2 z + λ = 0

form the three vertices of an equilateral triangle in the Argand plane, then

λ

is

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IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 9 - Complex Numbers>Exercise>Q 14

Let

z = 1 - t + i \sqrt{t^{2} + t + 2},

where

t

is a real parameter. The locus of

z

in the Argand plane is