Question

For the given example choose the correct alternative and fill in the blanks :
$(2012{)}^3+(2013{)}^3+(2014{)}^3-3\times 2012\times 2013\times 2014$
$=(.........)\times \left\{\begin{array}{c}(2012{)}^2+(2013{)}^2+(2014{)}^2-2012\times 2013-2013\times 2014-2014\times 2012\end{array}\right\}$

• A. 6036
• B. 6039
• C. 6042
• D. 6048

Answer 1

The correct option is B 6039

We have the formula,

$a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)+3abc$

$⟹a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)$

Let $a=2012,b=2013,c=2014$

$⟹(2012{)}^3+(2013{)}^3+(2014{)}^3-3\times 2012\times 2013\times 2014$

$=(2012+2013+2014)\left(\right(2012{)}^2+(2013{)}^2+(2014{)}^2-2012\times 2013-2013\times 2014-2014\times 2012)$

$=\left(6039\right)\left(\right(2012{)}^2+(2013{)}^2+(2014{)}^2-2012\times 2013-2013\times 2014-2014\times 2012)$.

Hence $6039$ will replace the blank.