Question : If $\left(3 y+\frac{3}{y}\right)=8$, then find the value of $\left(y^2+\frac{1}{y^2}\right)$.
Option 1: $5\frac{1}{9}$
Option 2: $4\frac{5}{6}$
Option 3: $7\frac{1}{9}$
Option 4: $9\frac{1}{9}$
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Correct Answer: $5\frac{1}{9}$
Solution :
Given: $\left(3 y+\frac{3}{y}\right)=8$
$⇒(y+\frac{1}{y})=\frac{8}{3}$
Squaring both sides,
$⇒(y^2+\frac{1}{y^2}+2)=\frac{64}{9}$
$⇒(y^2+\frac{1}{y^2})=\frac{64}{9} - 2$
$⇒(y^2+\frac{1}{y^2})=\frac{64-18}{9} $
$⇒(y^2+\frac{1}{y^2})=\frac{46}{9}=5\frac{1}{9}$
Hence, the correct answer is $5\frac{1}{9}$.
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