Question
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 ?
NDA
Mathematics
Binomial Theorem
2026
NDA 2026 (Phase 1)
NDA Mathematics Question (2026) — Solution
Options
- A. 1
- B. 2
- C. 0<uv<1
- D. 1<uv<2
Answer
C. 0<uv<1
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 = 1^ 10 = 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<uv<1
Related: Mathematics — Binomial Theorem · All PYQ Banks