Question : If $\mathrm{y}+\frac{1}{\mathrm{y}}=3$, then what is the value of $\frac{1}{\mathrm{y}^3}+\mathrm{y}^3+2$ ?
Option 1: 24
Option 2: 18
Option 3: 20
Option 4: 29
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Correct Answer: 20
Solution :
Given, $\mathrm{y}+\frac{1}{\mathrm{y}}=3$
Cubing both sides, we get
⇒ $(\mathrm{y}+\frac{1}{\mathrm{y}})^3=3^3$
⇒ $\mathrm{y^3}+\frac{1}{\mathrm{y^3}}+3\times\mathrm{y}\times\frac{1}{\mathrm{y}}\times(\mathrm{y}+\frac{1}{\mathrm{y}})=27$
⇒ $\mathrm{y^3}+\frac{1}{\mathrm{y^3}}+3(3)=27$
⇒ $\mathrm{y^3}+\frac{1}{\mathrm{y^3}}+9=27$
⇒ $\mathrm{y^3}+\frac{1}{\mathrm{y^3}}+2=27-7$
⇒ $\mathrm{y^3}+\frac{1}{\mathrm{y^3}}+2=20$
Hence, the correct answer is 20.
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