MCQ
If ${a^2} + {b^2} + {c^2} + ab + bc + ca \leq 0\,\forall a,\,b,\,c\, \in \,R$ , then the value of determinant $\left| {\begin{array}{*{20}{c}}
  {{{(a + b + c)}^2}}&{{a^2} + {b^2}}&1 \\ 
  1&{{{(b + c + 2)}^2}}&{{b^2} + {c^2}} \\ 
  {{c^2} + {a^2}}&1&{{{(c + a + 2)}^2}} 
\end{array}} \right|$ 
  • $65$
  • B
    $a^2+b^2+c^2+31$
  • C
    $4(a^2+b^2+c^2)$
  • D
    $0$

Answer

Correct option: A.
$65$
a
$\therefore \sum \mathrm{a}^{2}+\sum \mathrm{ab} \leq 0 $

$\Rightarrow(\mathrm{a}+\mathrm{b})^{2}+(\mathrm{b}+\mathrm{c})^{2}+(\mathrm{c}+\mathrm{a})^{2} \leq 0$

$\Rightarrow \mathrm{a}+\mathrm{b}=0, \mathrm{b}+\mathrm{c}=0, \mathrm{c}+\mathrm{a}=0$

$\Rightarrow \mathrm{a}=\mathrm{b}=\mathrm{c}=0$

$\therefore \left|\begin{array}{lll}{4} & {0} & {1} \\ {1} & {4} & {0} \\ {0} & {1} & {4}\end{array}\right|=65$

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