1.

If P = \(\frac{1+2x}{1-2x}\) and Q = \(\frac{1-2x}{1+2x}\) then \(\frac{P-Q}{P+Q}\) = ..................A) \(-\frac{4x}{1+4x^2}\)B) \(\frac{1+4x^2}{4x}\)C) \(\frac{-(1+4x^2)}{4x}\)D) \(\frac{4x}{1+4x^2}\)

Answer»

Correct option is (D) \(\frac{4x}{1+4x^2}\)

We have P = \(\frac{1+2x}{1-2x}\) and Q = \(\frac{1-2x}{1+2x}\)

\(\therefore\) \(\frac{P-Q}{P+Q}\) \(=\cfrac{\frac{1+2x}{1-2x}-\frac{1-2x}{1+2x}}{\frac{1+2x}{1-2x}+\frac{1-2x}{1+2x}}\) \(=\cfrac{\frac{(1+2x)^2-(1-2x)^2}{(1-2x)(1+2x)}}{\frac{(1+2x)^2+(1-2x)^2}{(1-2x)(1+2x)}}\)

\(=\frac{(1+2x)^2-(1-2x)^2}{(1+2x)^2+(1-2x)^2}\) \(=\frac{(1+4x+4x^2)-(1-4x+4x^2)}{(1+4x+4x^2)+(1-4x+4x^2)}\)

\(=\frac{8x}{2(1+4x^2)}\) \(=\frac{4x}{1+4x^2}\)

Correct option is A) \(-\frac{4x}{1+4x^2}\)



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