Complex Numbers And Quadratic Equations

Let assume :




Therefore,

So, the correct option is (C) : unit

Given Determinant:

Using the identities: , , , , and using , we rewrite the matrix as:

This can be expressed as the square of a 3x3 determinant:

This is a Vandermonde determinant, and its value is:

So,

… (i)

Now consider the quadratic equation:

Its roots are and . So,

Then,

Also, note from (i):

Substitute the values:

Thus, comparing with the form , we get:

Final Answer: Option (A):