Question : If $x=a^{\frac{1}{2}}+a^{-\frac{1}{2}}$, $y=a^{\frac{1}{2}}- a^{-\frac{1}{2}}$, then the value of $(x^{4}-x^{2}y^{2}-1)+(y^{4}-x^{2}y^{2}+1)$ is:
Option 1: 16
Option 2: 13
Option 3: 12
Option 4: 14
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Correct Answer: 16
Solution : Given: $x=a^{\frac{1}{2}}+a^{\frac{-1}{2}}$ and $y=a^{\frac{1}{2}}-a^{\frac{-1}{2}}$ ⇒ $x=\sqrt{a}+{\frac{1}{\sqrt{a}}}$ and $y=\sqrt{a}-{\frac{1}{\sqrt{a}}}$ ⇒ $x+y=2\sqrt{a}$ and $x-y=\frac{2}{\sqrt{a}}$ So, $x^{4}-x^{2}y^{2}-1+y^{4}-x^{2}y^{2}+1$ = $x^{4}+y^{4}-2x^{2}y^{2}$ = $(x^{2}-y^{2})^{2}$ = $[(x-y)(x+y)]^{2}$ = $[(2\sqrt{a})(\frac{2}{\sqrt{a}})]^{2}$ = $16$ Hence, the correct answer is 16.
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