Let u be a positive integer and f be a real number lying between 0 and 1. Further, (√2 + 1)^10 = u + f and (√2 − 1)^10 = v. 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?
A.I only
B.II only
C.Both I and II✓ Correct
D.Neither I nor II
Explanation
When we expand (√2 + 1)^10 + (√2 − 1)^10 using binomial theorem, the irrational terms cancel and we get an integer (since the exponent 10 is even, the result is 2 times the sum of even-indexed terms). So u + f + v = integer, making statement I correct. Since u is an integer, this means f + v is also an integer. Both statements I and II are correct.
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