If o is a complex cube root of unity, then the value of 099 +0100 +010

1.	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» 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

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»

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

( 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