Question
The number of elements in the set S = \ x: x [0,100] . and . _ 0 ^ x t^ 2 (x-t) d t=x^ 2 \ is \_\_\_\_.
JEE Main
Mathematics
Definite Integration
2026
JEE Main 2026 (23 January Shift 2)
JEE Main Mathematics Question (2026) — Solution
Options
- A. A
- B. B
- C. C
- D. D
Answer
A. A
Step-by-step solution
Using u = x - t: _0^x t^2 (x-t)\,dt = _0^x (x-u)^2 u\,du. Expanding and integrating term by term: = x^2(1- x) - 2x(-x x + x) + (-x^2 x + 2x x + 2 x - 2) = x^2 + 2 x - 2. Setting equal to x^2: 2 x - 2 = 0 x = 1 x = 2n . In [0, 100]: 2n 100 n 50 15.9. So n = 0, 1, 2, , 15, giving 16 elements.
Related: Mathematics — Definite Integration · All PYQ Banks