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 ?

Step-by-step solution

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 C₀x^n + ^ n C₂x^ n-2 y^2 + ) , we have: u+f+v = 2(¹⁰C₀( 2 )¹⁰ + ¹⁰C₂( 2 )^8 + + ¹⁰C₁₀) 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 di