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
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
•
Total views: 363.9k
•
Views today: 6.63k
Recently Updated Pages
What do you mean by public facilities

Paragraph on Friendship

Slogan on Noise Pollution

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

10 Slogans on Save the Tiger

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

One cusec is equal to how many liters class 8 maths CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE
