# For the given example choose the correct alternative and fill in the blanks:

$$\eqalign{

& {\left( {2012} \right)^3} + {\left( {2013} \right)^3} + {\left( {2014} \right)^3} - 3 \times 2012 \times 2013 \times 2014 \cr

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

A).6036

B).6039

C).6042

D).6048

Answer

Verified

363.9k+ views

Hint: We are going to solve this problem by using formula of ${a^3} + {b^3} + {c^3}$

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

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

Let a=2012, b=2013, and c=2014, Taking L.H.S from the given equation

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

It is in the form of ${a^3} + {b^3} + {c^3} - 3abc$

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

$\therefore $6039 is the number required in the given blank.

Note:

Here we solved the given problem using basic algebraic formula ${a^3} + {b^3} + {c^3} = (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ac) + 3abc$

We compared the given problem with this formula and simplified the expression to get the required value.

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

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

Let a=2012, b=2013, and c=2014, Taking L.H.S from the given equation

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

It is in the form of ${a^3} + {b^3} + {c^3} - 3abc$

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

$\therefore $6039 is the number required in the given blank.

Note:

Here we solved the given problem using basic algebraic formula ${a^3} + {b^3} + {c^3} = (a + b + c)({a^2} + {b^2} + {c^2} - ab - bc - ac) + 3abc$

We compared the given problem with this formula and simplified the expression to get the required value.

Last updated date: 28th Sep 2023

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