Complex Numbers | MCQs

Explanation

To find the sum, we need to calculate each term:

i^101 = i (since i^1 = i and the powers repeat every 4 terms)

i^102 = i^2 = -1

i^103 = i^3 = -i

i^104 = i^4 = 1

Now, let's add them up:

i + (-1) + (-i) + 1 = 0

Explanation

Given the product:

(5i) × (-2i)

= -10i²

Since i² = -1:

= -10(-1)

= 10

Explanation

√-1 is represented as "i" (the imaginary unit) in complex numbers.

√-1 × √-1 = i × i = i².

By definition, i² = -1.

Explanation

The cube roots of unity are 1, ω, and ω², where ω = -1 + √3i / 2 and ω² = -1 - √3i / 2.
Their product is 1 × ω × ω² = 1.

Explanation

-9×-9=81

So B is correct

Explanation

Given:

Z1 = 2 - i

Z2 = 3 + i

Product of Z1 and Z2:

Z1 * Z2 = (2 - i) * (3 + i)

= 6 + 2i - 3i - i²

Since i² = -1,

= 6 - i + 1

= 7 - i

Explanation

To find z1z2:

z1 = 6 + 3i

z2 = 3 - 5i

z1z2 = (6 + 3i)(3 - 5i)

= 18 - 30i + 9i - 15i^2

= 18 - 21i + 15 (since i^2 = -1)

= 33 - 21i

Explanation

The square root of any real number is never negative, so √x = -10 has no real solution.

Hence, the solution set is empty, written as { }.

Explanation

The calculation confirms that:

3i(2 + 5i) = 6i + 15i^2

= 6i - 15

Comparing this to x + 6i:

x = -15

Explanation

Using the identity:

(1 - ω + ω²)ⁿ = (-1)ⁿ ω²ⁿ

Substituting n = 7:

(1 - ω + ω²)⁷ = (-1)⁷ ω²⁷

= -1 × ω²⁷ (since (-1)⁷ = -1)

= -128ω² (since ω²⁷ = (ω²)⁷ × ω⁰ = (ω²)⁷ = 2⁷ω² = 128ω², and multiplying by -1)

The correct answer is:

-128ω²