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
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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