Question : If $y+\frac{1}{y}=11$, then the value of $y^3-\frac{1}{y^3}$ is:
Option 1: $345 \sqrt{13}$
Option 2: $360 \sqrt{13}$
Option 3: $352 \sqrt{13}$
Option 4: $368 \sqrt{13}$
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Correct Answer: $360 \sqrt{13}$
Solution :
Given, $y+\frac{1}{y}=11$
Squaring both sides, we get:
$⇒(y+\frac{1}{y})^2=11^2$
$⇒(y-\frac{1}{y})^2+4(y)(\frac{1}y)=121$
$⇒(y-\frac{1}{y})^2+4=121$
$⇒(y-\frac{1}{y})^2=121-4$
$⇒(y-\frac{1}{y})=3\sqrt{17}$
Cubing both sides, we get:
$⇒(y-\frac{1}{y})^3=(3\sqrt{17})^3$
$⇒y^3-\frac{1}{y^3}-3(y)(\frac{1}{y})(y-\frac{1}{y})=351\sqrt{13}$
$⇒y^3-\frac{1}{y^3}-3(3\sqrt{17})=351\sqrt{13}$
$⇒y^3-\frac{1}{y^3}=351\sqrt{13}+9\sqrt{13}$
$\therefore y^3-\frac{1}{y^3}=360\sqrt{13}$
Hence, the correct answer is $360\sqrt{13}$.
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