Question : If, $\frac{r}{13}+\frac{13}{r}=1$, then the value of $r^3$ is:
Option 1: –2157
Option 2: 2197
Option 3: 2157
Option 4: –2197
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Correct Answer: –2197
Solution : $\frac{r}{13}+\frac{13}{r}=1$ We know that if $x+\frac{1}{x} = 1$, then $x^3 = -1$ Let $\frac{r}{13} = x$ The equation becomes, $x+\frac{1}{x} = 1$ ⇒ $x^2-x+1=0$ $x^3+1^3=(x+1) (x^2-x+1)$ ⇒ $x^3+1^3 =0$ ⇒ $x^3 = -1$ ⇒ $(\frac{r}{13})^3 = -1$ ⇒ $r^3 = -2197$ Hence, the correct answer is –2197.
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