Consider the following statements : I. (u + v + f) is an integer. II. (f + v) is an integer. Which of the statements given above is/are correct ?
For the next five (05) items that follow : Let u be a positive integer and f be a real number lying between 0 and 1. Further, \( (\sqrt{2} + 1)^{10} = u + f \) and \( (\sqrt{2} - 1)^{10} = v \)
- A. I only
- B. II only
- C. Both I and II ✓
- D. Neither I nor II
Correct Answer: C. Both I and II
Explanation
\( u + f + v = (\sqrt{2} + 1)^{10} + (\sqrt{2} - 1)^{10} \) which equals an even integer. Thus I is true. Since u is an integer, f + v must also be an integer. Thus II is true.
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