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
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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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