Answer
468k+ views
Hint : (Use the given equation to get the value of asked equation. Start solving from first equation and use that in second equation for final answer)
The given equations are
$x = 2 + {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}$ ……(i)
${x^3} - 6{x^2} + 6x\,$……(ii)
We have to find the value of second equation
We will do operation in first equation to get the value of second
As we know ${(a - b)^3} = {a^3} - {b^3} + 3a{b^2} - 3{a^2}b$
On considering equation one and solving
$
x = 2 + {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}} \\
x - 2 = {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}} \\
\\
$
Cubing both sides we get,
$
{(x - 2)^3} = {({2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}})^3} \\
{x^3} - 8 - 6{x^2} + 12x = {2^{\dfrac{{2(3)}}{3}}} + {2^{\dfrac{{1(3)}}{3}}} + {3.2^{\dfrac{{2(2)}}{3}}}{.2^{\dfrac{1}{3}}} + {3.2^{\dfrac{{1(2)}}{3}}}{.2^{\dfrac{2}{3}}} \\
$
On subtracting $6x$ from both sides we get,
$
{x^3} - 6{x^2} + 12x - 6x = 8 + 6 + 3({2^{\dfrac{5}{3}}} + {2^{\dfrac{4}{3}}}) - 6x \\
{x^3} - 6{x^2} + 6x = 14 + 3({2^{\dfrac{5}{3}}} + {2^{\dfrac{4}{3}}}) - 6(2 + {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}) \\
{x^3} - 6{x^2} + 6x = 2 + 3({2^{\dfrac{5}{3}}}) + 3({2^{\dfrac{4}{3}}}) - 6({2^{\dfrac{2}{3}}}) - 6({2^{\dfrac{1}{3}}}) \\
{x^3} - 6{x^2} + 6x = 2 + 3(2)({2^{\dfrac{2}{3}}}) + 3(2)({2^{\dfrac{1}{3}}}) - 6({2^{\dfrac{2}{3}}}) - 6({2^{\dfrac{1}{3}}}) \\
{x^3} - 6{x^2} + 6x = 2 \\
$
Hence the value of the second equation is 2 .
Therefore the correct option is B.
Note :- In this question we have used the formula of ${(a - b)^3} = {a^3} - {b^3} + 3a{b^2} - 3{a^2}b $in equation (i) above. We then got the value of equation (ii) by simplifying the equation (i). We have also used the concept of adding of the power when the base is same during obtaining the value of asked equation. There is nothing needed other than this to solve this question. Never try to put the values directly, it will make equation complex and the solution will not look good.
The given equations are
$x = 2 + {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}$ ……(i)
${x^3} - 6{x^2} + 6x\,$……(ii)
We have to find the value of second equation
We will do operation in first equation to get the value of second
As we know ${(a - b)^3} = {a^3} - {b^3} + 3a{b^2} - 3{a^2}b$
On considering equation one and solving
$
x = 2 + {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}} \\
x - 2 = {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}} \\
\\
$
Cubing both sides we get,
$
{(x - 2)^3} = {({2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}})^3} \\
{x^3} - 8 - 6{x^2} + 12x = {2^{\dfrac{{2(3)}}{3}}} + {2^{\dfrac{{1(3)}}{3}}} + {3.2^{\dfrac{{2(2)}}{3}}}{.2^{\dfrac{1}{3}}} + {3.2^{\dfrac{{1(2)}}{3}}}{.2^{\dfrac{2}{3}}} \\
$
On subtracting $6x$ from both sides we get,
$
{x^3} - 6{x^2} + 12x - 6x = 8 + 6 + 3({2^{\dfrac{5}{3}}} + {2^{\dfrac{4}{3}}}) - 6x \\
{x^3} - 6{x^2} + 6x = 14 + 3({2^{\dfrac{5}{3}}} + {2^{\dfrac{4}{3}}}) - 6(2 + {2^{\dfrac{2}{3}}} + {2^{\dfrac{1}{3}}}) \\
{x^3} - 6{x^2} + 6x = 2 + 3({2^{\dfrac{5}{3}}}) + 3({2^{\dfrac{4}{3}}}) - 6({2^{\dfrac{2}{3}}}) - 6({2^{\dfrac{1}{3}}}) \\
{x^3} - 6{x^2} + 6x = 2 + 3(2)({2^{\dfrac{2}{3}}}) + 3(2)({2^{\dfrac{1}{3}}}) - 6({2^{\dfrac{2}{3}}}) - 6({2^{\dfrac{1}{3}}}) \\
{x^3} - 6{x^2} + 6x = 2 \\
$
Hence the value of the second equation is 2 .
Therefore the correct option is B.
Note :- In this question we have used the formula of ${(a - b)^3} = {a^3} - {b^3} + 3a{b^2} - 3{a^2}b $in equation (i) above. We then got the value of equation (ii) by simplifying the equation (i). We have also used the concept of adding of the power when the base is same during obtaining the value of asked equation. There is nothing needed other than this to solve this question. Never try to put the values directly, it will make equation complex and the solution will not look good.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)