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If a^(2) + b^(2) + c^(3) + ab + bc + ca ...

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 to

A

`65`

B

`a^(2)+b^(2)+c^(2)+31`

C

`4(a^(2)+b^(2)+c^(2))`

D

`0`

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` We have `a^(2)+b^(6)+c^(2)+ab+bc+ca le 0`
`:.(a+b)^(2)+(b+c)^(2)+(c+a)^(2)le0`
`:.a+b=0`, `b+c=0` and `c+a=0`
`:.a=b=c=0`
`implies|{:((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`
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