Question : Which of the following statements is correct?
I. If $K+\frac{1}{K}=12$, then ${K}^2+\frac{1}{K^2}=142$
II. The value of $({k}^2+\frac{1}{{k}^2})({k}-\frac{1}{{k}})({k}^4+\frac{1}{{k}^4})({k}+\frac{1}{{k}})$ is ${k}^{16}-\frac{1}{{k}^{16}}$
Option 1: Only I
Option 2: Neither I nor II
Option 3: Both I and II
Option 4: Only II
Correct Answer: Only I
Solution :
I. $K+\frac{1}{K}=12$
Squaring, $(K+\frac{1}{K})^2=12^2$
⇒ ${K}^2+\frac{1}{K^2}+2=144$
⇒ ${K}^2+\frac{1}{K^2}=142$
So the first statement is correct.
II. $({k}^2+\frac{1}{{k}^2})({k}-\frac{1}{{k}})({k}^4+\frac{1}{{k}^4})({k}+\frac{1}{{k}})$
= $({k}^2+\frac{1}{{k}^2})({k}^2-\frac{1}{{k}^2})({k}^4+\frac{1}{{k}^4})$
= $({k}^4-\frac{1}{{k}^4})({k}^4+\frac{1}{{k}^4})$
= $({k}^8-\frac{1}{{k}^8})$
So, the second statement is not correct.
Hence, the correct answer is only I.
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