CAT2009-1

Question

If $mx^{m}-nx^{n}=0$, then what is the value of $\frac{1}{x^{m}+x^{n}}+\frac{1}{x^{m}-x^{n}}$ in terms of $x^{n}$?

1. $\frac{2mn}{x^{n}(n^{2}-m^{2})}$
2. $\frac{2mn}{x^{n}(n^{2}+m^{2})}$
3. $\frac{2mn}{x^{n}(m^{2}-n^{2})}$
4. $\frac{2mn}{x^{n}(m^{2}+n^{2})}$

Answer 1

$mx^m - nx^n =0$

So, $\frac{x^m}{x^n} =\frac{n}{m}$

$\frac{1}{x^m+x^n} +\frac{1}{x^m-x^n} = \frac{1}{x^n\left(\frac{x^m}{x^n}+1 \right )} + \frac{1}{x^n\left(\frac{x^m}{x^n}-1 \right )}$

$= \frac{1}{x^n}\left(\frac{1}{\frac{n}{m}+1} +\frac{1}{\frac{n}{m}-1} \right)$

$= \frac{m}{x^n}\left(\frac{1}{n+m} +\frac{1}{n-m} \right)$

$= \frac{2mn}{x^n(n^2-m^2)}$