Question

If o is a complex cube root of unity, then the
value of 099 +0100 +0101 is:
(A)/-1
(B) 1
(C) 0
(D) 3
​

Answer 1

Answer:

What are Cube Roots of Unity?

There are a total of three cube roots of unity which are as follows:

Cube Root of Unity Value Nature of Cube Root

1 Real

−½ + i √(3/ 2) Complex

−½ – i √(3/ 2) Complex

Here, a = 1 is the real cube root of unity while a = – ½ + i √(3/ 2) and a = – ½ – i √(3/ 2) are the imaginary or complex cube roots of unity.

How to Find Cube Root of Unity Values (Derivation)?

Suppose the cube root of 1 is “a”,

i .e 3√1 = a.

According to the general cube roots definition, a3 = 1 or a3 – 1 = 0

(a3 – b3) = (a – b) ( a2 + ab + b2)

Now, (a3 – 13) = 0

or (a – 1)( a2 + a + 1) = 0

Therefore, a = 1

or

( a2 + a + 1) = 0

By using quadratic roots formula for the above equation, we get;

a = [(-1) ± √(12-4.1.1)]/2

= [-1 ± √-3]/2

= -1/2 ± i√(3)/2

Therefore, the three cube roots of unity are:

1, -1/2+i√(3)/2, -1/2 – i√(3)/2

Answer 2

Answer:

(d) 3 will be the answer of the question