If `a^(2) + b^(2) + c^(3) + ab + bc + ca le 0` for all, `a, b, c in R`, then the 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))|`, is equal toA. 65B. `a^(2) + b^(2) + c^(2) + 31`C. `4(a^(2) + b^(2) + c^(2))`D. 0

If `a^(2) + b^(2) + c^(3) + ab + bc + ca le 0` for all, `a, b, c in R`

1.	If `a^(2) + b^(2) + c^(3) + ab + bc + ca le 0` for all, `a, b, c in R`, then the 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))\|`, is equal toA. 65B. `a^(2) + b^(2) + c^(2) + 31`C. `4(a^(2) + b^(2) + c^(2))`D. 0
Answer» Correct Answer - A 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`