If 1, , ^2 are the cube roots of unity, k is positive integer and (1- + ^2 )^ 3 k + (1- ^2+ )^ 3 k = (1- + ^2 )^ 3 k +1 + (1+ - ^2 )^ 3 k +1 , then k =

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Correct answer: A. r , r , N

aligned & (1- + ^2 )^ 3 k + (1- ^2+ )^ 3 k & = (1- + ^2 )^ 3 k+1 + (1+ - ^2 )^ 3 k+1 & (- - )^ 3 k + (- ^2- ^2 )^ 3 k & =(- - )^ 3 k+1 + (- ^2- ^2 )^ 3 k+1 & (-2)^ 3 k ( )^3+(-2)^ 3 k ( ^2 )^3 & =(-2 ) (-2)^ 3 k ( )^ 3 k + (-2 ^2 ) (-2)^ 3 k ^ 3 k & 1+1=-2 -2 ^2 & 2=-2 ( + ^2 ) & 1=-1(-1) 1=1 aligned The given statement is true for all k N k=r , where r N

If 1, , ^2 are the cube roots of unity, k is positive integer and (1- + ^2 )^ 3 k + (1- ^2+ )^ 3 k = (1- + ^2 )^ 3 k +1 + (1+ - ^2 )^ 3 k +1 , then k =

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