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: What is the value of u ?

Step-by-step solution

Given ( 2 +1)¹⁰ = u + f and ( 2 -1)¹⁰ = v . Adding the two equations: u + f + v = ( 2 +1)¹⁰ + ( 2 -1)¹⁰ Using the binomial expansion (x+y)^n + (x-y)^n = 2 [^ n C₀ x^n + ^ n C₂ x^ n-2 y^2 + ] : u + f + v = 2 [¹⁰C₀( 2 )¹⁰ + ¹⁰C₂( 2 )^8 + ¹⁰C₄( 2 )^6 + ¹⁰C₆( 2 )^4 + ¹⁰C₈( 2 )^2 + ¹⁰C₁₀] u + f + v = 2 [1(32) + 45(16) + 210(8) + 210(4) + 45(2) + 1(1)] u + f + v = 2 [32 + 720 + 1680 + 840 + 90 + 1] u + f + v = 2 [3363] = 6726 Since 0 We are given 0 Since u and 6726 are integers, f + v must be an integer. The only integer