Question : If $\frac{1}{N}=\frac{(\sqrt6+\sqrt5)}{(\sqrt6-\sqrt5)}$, what is the value of $N$?
Option 1: $6-\sqrt{30}$
Option 2: $6+\sqrt{30}$
Option 3: $11-2\sqrt{30}$
Option 4: $11+2\sqrt{5}$
Correct Answer: $11-2\sqrt{30}$
Solution :
Given: $\frac{1}{N}=\frac{(\sqrt6+\sqrt5)}{(\sqrt6-\sqrt5)}$
⇒ $N=\frac{(\sqrt6-\sqrt5)}{(\sqrt6+\sqrt5)}$
Multiplying numerator and denominator by $(\sqrt6-\sqrt5)$,
$N=\frac{(\sqrt6-\sqrt5)\times (\sqrt6-\sqrt5)}{(\sqrt6+\sqrt5)\times (\sqrt6-\sqrt5)}$
Using identities: $(a^2-b^2)=(a-b)(a+b)$ and $(a-b)^2=a^2+b^2-2ab$,
$N=\frac{(6+5-2\times \sqrt6\times \sqrt5)}{(6-5)}$
⇒ $N=11-2\sqrt{30}$
Hence, the correct answer is $11-2\sqrt{30}$.
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