Sets, Relations & Functions
Functions
star_batch_jee_advanced_2025
Grade 11
Question:
Compute: $\left[\frac{2103^3}{2104 \times 2105} - \frac{2104^3}{2105 \times 2106}\right]$. $[x]$ denotes greatest integer $\leq x$.
Step-by-Step Solution
Key Concept: Algebraic simplification of symmetric expressions reduces a complex fraction to a form where the integer and fractional parts are easily identified.
Let $t = 2105$. The given expression simplifies to $\frac{(t+1)^3 + (t-1)^3}{(t-1)t(t+1)} = 8 + \frac{16t}{t^3-t} = 8 + \frac{16}{t^2-1}$. Since $\frac{16}{t^2-1} < 1$, the integral part is $8$.
Correct Answer: 8