Question : If $\frac{x}{3}+\frac{3}{x}=1$, then the value of $x^3$ is:
Option 1: 1
Option 2: 27
Option 3: 0
Option 4: –27
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Correct Answer: –27
Solution :
Given:
$\frac{x}{3}+\frac{3}{x}=1$
⇒ $\frac{x^2+9}{3x}=1$
⇒ $x^2+9=3x$
⇒ $x^2-3x+9=0$
⇒ $(x+3)(x^2-3x+9)=(x+3)×0$ [Multiplying both sides by $(x+3)$]
⇒ $x^3+3^3=0$ (Applying formula)
⇒ $x^3=-(3)^3$
⇒ $x^3=-27$
Hence, the correct answer is –27.
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