A) \[2+\frac{3{{x}^{2}}}{4{{a}^{2}}}+....\]
B) \[1+\frac{3{{x}^{2}}}{8{{a}^{2}}}+....\]
C) \[2+\frac{x}{a}+\frac{3{{x}^{2}}}{4{{a}^{2}}}+....\]
D) \[2-\frac{x}{a}+\frac{3{{x}^{2}}}{4{{a}^{2}}}\]+......
Correct Answer: A
Solution :
\[{{\left( \frac{a+x}{a} \right)}^{-1/2}}+{{\left( \frac{a-x}{a} \right)}^{-1/2}}={{\left( 1+\frac{x}{a} \right)}^{-1/2}}+{{\left( 1-\frac{x}{a} \right)}^{-1/2}}\] \[=\left[ 1+\left( -\frac{1}{2} \right)\,\left( \frac{x}{a} \right)+\frac{\left( -\frac{1}{2} \right)\,\left( -\frac{3}{2} \right)}{2.1}{{\left( \frac{x}{a} \right)}^{2}}+.... \right]\]\[+\left[ 1+\left( -\frac{1}{2} \right)\,\left( -\frac{x}{a} \right)+\frac{\left( -\frac{1}{2} \right)\,\left( -\frac{3}{2} \right)}{2.1}{{\left( -\frac{x}{a} \right)}^{2}}+.... \right]\] \[=2+\frac{3{{x}^{2}}}{4{{a}^{2}}}+\]......... Here odd terms cancel each other.
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