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Q.

If the vectors (a, b, c), (b, c, a) and (c, a, b) are linealy dependent then

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a

$a^{3} + b^{3} + c^{3} = 3 abc$

b

$a^{2} + b^{2} + c^{2} = 2 abc$

c

$a^{3} + b^{3} + c^{3} + 3 abc = 0$

d

$a^{2} + b^{2} + c^{2} + 2 abc = 0$

answer is A.

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Detailed Solution

$\Rightarrow \begin{array}{rcl} |\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}| \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} 0 \end{array} \begin{array}{rcl} \Rightarrow & a (bc - a^{2}) - b (b^{2} - ac) + (ab - c^{2}) = 0 \end{array} \begin{array}{rcl} a \end{array} \begin{array}{rcl} \end{array}$

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