Answer
Verified
481.5k+ 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
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE