If $a = 2012, b = 2011, c = 2010$ then the value of $a^{2} + b^{2} + c^{2} - a b - b c - c a$ is

0

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2012

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3

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4024

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Solution

The correct option is B $3$
$L e t$
$a = 2012, b = 2011, c = 2010$
$x = a^{2} + b^{2} + c^{2} - a b - b c - c a$
$2 x = 2 (a^{2} + b^{2} + c^{2} - a b - b c - c a)$
$\Rightarrow 2 x = 2 a^{2} + 2 b^{2} + 2 c^{2} - 2 a b - 2 b c - 2 c a$
$\Rightarrow 2 x = a^{2} - 2 a b + b^{2} + b^{2} - 2 b c + c^{2} + c^{2} - 2 c a + a^{2}$
$\Rightarrow 2 x = {(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2}$
$\Rightarrow 2 x = {(2012 - 2011)}^{2} + {(2011 - 2010)}^{2} + {(2010 - 2012)}^{2}$
$\Rightarrow 2 x = 1 + 1 + 4$
$\Rightarrow x = 3$

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