Passage: 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. Question: 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 ?
C. Both I and II
Given ( 2 +1)^ 10 = u+f and ( 2 -1)^ 10 = v. Adding both equations, we get: u+f+v = ( 2 +1)^ 10 + ( 2 -1)^ 10 Using the binomial expansion (x+y)^n + (x-y)^n = 2(^ n C_ 0 x^n + ^ n C_ 2 x^ n-2 y^2 + ), we have: u+f+v = 2(^ 10 C_ 0 ( 2 )^ 10 + ^ 10 C_ 2 ( 2 )^8 + + ^ 10 C_ 10 ) Since all powers of 2 are even, the right-hand side is an even integer. Let this integer be I. Therefore, u+f+v = I, which means (u+v+f) is an integer. Thus, Statement I is correct. Since u is given as a positive integer and u+f+v = I, it follows that f+v = I - u. The difference of two integers is an integer, so (f+v) is an integer. Thus, Statement II is also correct. Answer: Both I and II
Related: Mathematics — Binomial Theorem · All PYQ Banks