Complex Number | 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 of two complex numbers (x + iy) and (x' + iy') is zero:

(x + iy)(x' + iy') = 0

This implies either (x + iy) = 0 or (x' + iy') = 0.

Therefore, at least one of the complex numbers must be zero.

Explanation

The argument of the product of two complex numbers is the sum of their arguments.

This property comes from the multiplication of complex numbers in polar form.

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

Z1Z2 = (2 - i)(3 - i)

= 6 - 2i - 3i + i²

= 6 - 5i - 1 (since i² = -1)

= 5 - 5i

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