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 uv ?

Step-by-step solution

Given ( 2 +1 )¹⁰ = u+f and ( 2 -1 )¹⁰ = v . Multiplying these two equations, we get: (u+f)v = ( 2 +1 )¹⁰ ( 2 -1 )¹⁰ (u+f)v = (( 2 )^2 - 1^2 )¹⁰ = 1¹⁰ = 1 uv + fv = 1 uv = 1 - fv Since f is the fractional part, 0 Also, v = ( 2 -1 )¹⁰ , and since 0 Because both f and v lie strictly between 0 and 1 , their product fv also lies strictly between 0 and 1 , i.e., 0 Therefore, uv = 1 - fv must satisfy: 0 Answer: 0