Answer
455.7k+ views
Hint: - Use $\alpha = \omega ,{\text{ }}\beta = {\omega ^2}$
As we know if $\alpha $and $\beta $are the complex cube roots of unity therefore
$1 + \alpha + \beta = 0................\left( 1 \right)$
As we know cube roots of unity are $1,\omega ,{\omega ^2}$
Where $\omega $and${\omega ^2}$are non-real complex cube roots of unity therefore
${\omega ^3} = 1........\left( 2 \right),{\text{ }}1 + \omega + {\omega ^2} = 0...............\left( 3 \right)$
So, from equations (1) and (3)
$\alpha = \omega ,{\text{ }}\beta = {\omega ^2}$
Now given equation is ${\alpha ^4} + {\beta ^4} + {\alpha ^{ - 1}}{\beta ^{ - 1}}$
\[
\Rightarrow {\omega ^4} + {\left( {{\omega ^2}} \right)^4} + {\omega ^{ - 1}}{\left( {{\omega ^2}} \right)^{ - 1}} \\
\Rightarrow {\omega ^3}.\omega + {\omega ^8} + {\omega ^{ - 1}}\left( {{\omega ^{ - 2}}} \right) \\
\Rightarrow {\omega ^3}.\omega + {\left( {{\omega ^3}} \right)^2}{\omega ^2} + \dfrac{1}{{{\omega ^3}}} \\
\]
From equation (2)
$
{\omega ^3} = 1 \\
\Rightarrow 1.\omega + {\left( 1 \right)^2}{\omega ^2} + 1 \\
\Rightarrow 1 + \omega + {\omega ^2} \\
$
From equation (3)
$
1 + \omega + {\omega ^2} = 0 \\
\Rightarrow {\alpha ^4} + {\beta ^4} + {\alpha ^{ - 1}}{\beta ^{ - 1}} = 1 + \omega + {\omega ^2} = 0 \\
$
Hence, option (d) is correct.
Note: - Whenever we face such types of problems the key concept is that always remember the condition of cube roots of unity which is stated above, then substitute the values in the given equation then simplify we will get the required answer.
As we know if $\alpha $and $\beta $are the complex cube roots of unity therefore
$1 + \alpha + \beta = 0................\left( 1 \right)$
As we know cube roots of unity are $1,\omega ,{\omega ^2}$
Where $\omega $and${\omega ^2}$are non-real complex cube roots of unity therefore
${\omega ^3} = 1........\left( 2 \right),{\text{ }}1 + \omega + {\omega ^2} = 0...............\left( 3 \right)$
So, from equations (1) and (3)
$\alpha = \omega ,{\text{ }}\beta = {\omega ^2}$
Now given equation is ${\alpha ^4} + {\beta ^4} + {\alpha ^{ - 1}}{\beta ^{ - 1}}$
\[
\Rightarrow {\omega ^4} + {\left( {{\omega ^2}} \right)^4} + {\omega ^{ - 1}}{\left( {{\omega ^2}} \right)^{ - 1}} \\
\Rightarrow {\omega ^3}.\omega + {\omega ^8} + {\omega ^{ - 1}}\left( {{\omega ^{ - 2}}} \right) \\
\Rightarrow {\omega ^3}.\omega + {\left( {{\omega ^3}} \right)^2}{\omega ^2} + \dfrac{1}{{{\omega ^3}}} \\
\]
From equation (2)
$
{\omega ^3} = 1 \\
\Rightarrow 1.\omega + {\left( 1 \right)^2}{\omega ^2} + 1 \\
\Rightarrow 1 + \omega + {\omega ^2} \\
$
From equation (3)
$
1 + \omega + {\omega ^2} = 0 \\
\Rightarrow {\alpha ^4} + {\beta ^4} + {\alpha ^{ - 1}}{\beta ^{ - 1}} = 1 + \omega + {\omega ^2} = 0 \\
$
Hence, option (d) is correct.
Note: - Whenever we face such types of problems the key concept is that always remember the condition of cube roots of unity which is stated above, then substitute the values in the given equation then simplify we will get the required answer.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 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)
Give 10 examples for herbs , shrubs , climbers , creepers
![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)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english 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)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)