Question : If $\left(a+\frac{1}{a}\right)=6$, then what is the value of $\frac{3}{4}\left(a^2+\frac{1}{a^2}\right)$?
Option 1: 22.5
Option 2: 34
Option 3: 25.5
Option 4: 36
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Correct Answer: 25.5
Solution :
Given that $\left(a+\frac{1}{a}\right)=6$,
Squaring both sides, we get,
$⇒\left(a+\frac{1}{a}\right)^2 = 6^2$
$⇒a^2 + 2 + \frac{1}{a^2} = 36$
$⇒a^2 + \frac{1}{a^2} = 36 - 2 = 34$
$ \therefore\frac{3}{4}\left(a^2+\frac{1}{a^2}\right) = \frac{3}{4} \times 34 = 25.5$
Hence, the correct answer is 25.5.
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