Let `omega` be the complex number `cos((2pi)/3)+isin((2pi)/3)`. Then the number of distinct complex cos numbers z satisfying `Delta=|(z+1,omega,omega^2),(omega,z+omega^2,1),(omega^2,1,z+omega)|=0` is

Options

A. 1
B. 0
C. 2
D. 3

Answer

A

Explanation

Clearly, `omega` is a complete cube root of unity.
`therefore 1+omega+omega^(2)=0`
We have, `|{:(z+1,omega,omega^(2)),(omega, z+omega^(2),1),(omega^(2),1,z+omega)|=0`
`rArr |{:(z+1+omega+omega^(2),omega,omega^(2)),(z+1+omega+omega^(2),1),(z+1+omega+omega^(2),1,z+omega):}|=0`
Applying
`C_(1) to C_(1)+C_(2)+C_(3)`
`rArr z|{:(1,omega,omega^(2)),(1,z+omega^(2),1),(1,1,z+omega)|=0`
`rArr z|{:(1,omega,omega^(2)),(0,z+omega^(2)-omega,1-omega^(2)):}|=0` Applying `R_(2) to R_(2) -R_(1), R_(3) to R_(3)-R_(1)`
`rArr z^(3)=0 rArr z=0`