If `a^(2)+b^(2)+c^(2)+ab+bc+ca le 0 AA a,b,c in R`, then value of the determinant `|{:((a+b+2)^(2),a^(2)+b^(2),1),(1,(b+c+2)^(2),b^(2)+c^(2)),(c^(2)+a^(2),1,(c+a+2)^(2)):}|` equals

Options

A. 65
B. `a^(2) + b^(2) + c^(2) + 31`
C. `4(a^(2) + b^(2) + c^(2))`
D. 0

Answer

A

Explanation

We have,
`a^(2) + b^(2) + c^(3) + ab + bc + ca le 0`
`rArr 2a^(2) + 2b^(2) + 2c^(2) + 2ab + 2bc + 2ca le 0`
`rArr (a +b)^(2) + (b +c)^(2) + (c +a)^(2) le 0`
`rArr a + b = 0, b +c = 0, c + a= 0`
`rArr (a +b) + (b+c) + (c +a) = 0 rArr a + b + c = 0`
Thus, we have `a + b = 0, b + c = 0, c + a = 0 and a + b + c = 0`
`rArr a = b = c = 0`
`:. [((a + b +2)^(2),a^(2) + b^(2),1),(1,(b + c =2)^(2),b^(2) + c^(2)),(c^(2) + a^(2),1,(c + a + 2)^(2))] = [(4,0,1),(1,4,0),(0,1,4)] = 65`