Question : If $a^{3}+\frac{1}{a^{3}}=2$, then the value of $\frac{a^{2}+1}{a}$ is ($a$ is a positive number):
Option 1: 1
Option 2: 2
Option 3: 3
Option 4: 4
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Correct Answer: 2
Solution :
Given: $a^{3}+\frac{1}{a^{3}}=2$
⇒ $a^{3}+\frac{1}{a^{3}}=2$
⇒ $a^{6}+1 =2a^3$
⇒ $a^{6}-2a^3+1=0$
⇒ $(a^3-1)^2=0$
$a = 1$, the above equation is satisfied.
Now putting $a = 1$, in the equation, we get
$\frac{a^{2}+1}{a} = \frac{1^{2}+1}{1} = 2$
Hence, the correct answer is 2.
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