JEE Class main Answered
mth root of unity is written as
where n = 0, 1, 2 ..........(m-1)
Cube roots of unity are
cos(0) , cos((2π)/3) + i sin((2π)/3) and cos((4π)/3) + i sin((4π)/3)
If they are 1, ω and ω2 respectively , then
1 = cos(0)
ω = cos((2π)/3) + i sin((2π)/3)
and ω2 = cos((4π)/3) + i sin((4π)/3)
1 + ω + ω2 = [ 1 + cos((2π)/3)+ cos((4π)/3) ] +i [ sin((2π)/3)+ sin((4π)/3) ]
1 + ω + ω2 = 0 + i 0 = 0
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we have the matric equation as
Let us do elementary operation R1 = ( R1+R2+R3 ) in first matrix and second matrix
Second matrix vanishes because all elements of first row are 1 + ω + ω2 = 0
Hence we get
x × 3 ( ω3 - 1 ) = 3
It is not possible to get value of x from above expression because ( ω3 - 1 ) = 0 .
Hence the answer is (D) None of these