The value of a for which system of equations , `a^3x+(a+1)^3y+(a+2)^3z=0, ax+(a+1)y+(a+2)z=0, x + y + z = 0 ,` has a non-zero solution is:

Options

A. 0
B. `-1`
C. 1
D. none of these

Answer

B

Explanation

The given system of equations will have a non-zero solution, if `|(a^(3),(a +1)^(3),(a +2)^(3)),(a,(a +1),(a +2)),(1,1,1)|=0`
`rArr |(a^(3) - (a +2)^(3),(a +1)^(3) - (a +2)^(3),(a + 2)^(3)),(-2,-1,a +2),(0,0,1)| = 0 " " [("Applying " C_(1) rarr C_(1) - C_(3)),(and C_(2) rarr C_(2) - C_(3))]`
`rArr -a^(3) + (a +2)^(3) + 2(a +1)^(3) - 2(a +2)^(3) = 0`
`rArr (a +2)^(3) -2( a +1)^(3) + a^(3) = 0`
`rArr (6a^(2) + 12a + 8) - 2 (3a^(2) + 3a +1) = 0`
`rArr 6a + 6 = 0`
`rArr a = -1`