Question : If $\left ( a+\frac{1}{a} \right )^{2}=3$, then the value of $\left ( a^{2}+\frac{1}{a^{2}} \right )$ will be:
Option 1: 0
Option 2: 1
Option 3: 2
Option 4: 3
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Correct Answer: 1
Solution :
Given:
$(a+\frac{1}{a})^2=3$
Expanding the left side of the equation:
⇒ $a^2 + 2(a)(\frac{1}{a}) + \left(\frac{1}{a}\right)^2 = 3$
⇒ $a^2 + 2 + \frac{1}{a^2} = 3$
⇒ $a^2 + \frac{1}{a^2} = 3 - 2$
⇒ $a^2 + \frac{1}{a^2} = 1$
Therefore, the value of $a^2 + \frac{1}{a^2}$ is 1.
Hence, the correct answer is 1.
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