Question
Let f : - π 2 , π 2 → ℝ be a continuous function such that f 0 = 1 and ∫ 0 π 3 f t d t = 0 Then which of the following statements is (are) TRUE?
JEE Advanced
Mathematics
Definite Integration
2021
JEE Advanced 2021 (Paper 2)
JEE Advanced Mathematics Question (2021) — Solution
Options
- A. The equation f x - 3 cos 3 x = 0 has at least one solution in 0 , π 3
- B. The equation f x - 3 sin 3 x = - 6 π has at least one solution in 0 , π 3
- C. lim x → 0 x ∫ 0 x f t d t 1 - e x 2 = - 1
- D. lim x → 0 sin x ∫ 0 x f t d t x 2 = - 1
Answer
C. lim x → 0 x ∫ 0 x f t d t 1 - e x 2 = - 1
Step-by-step solution
(A) Let g x = f x - 3 cos 3 x Now ∫ 0 π 3 g x d x = ∫ 0 π 3 f x d x - 3 ∫ 0 π 3 cos 3 x d x = 0 - sin 3 x 0 π 3 = 0 Hence g x = 0 has a root in 0 , π 3 (B) Let h x = f x - 3 sin 3 x + 6 π Now ∫ 0 π 3 h x d x = ∫ 0 π 3 f x d x - 3 ∫ 0 π 3 sin 3 x d x + ∫ 0 π 3 6 π d x = 0 - - cos 3 x 0 π 3 + 6 x π 0 π 3 = 0 - 2 + 2 = 0 Hence h x = 0 has a root in 0 , π 3 (C) lim x → 0 x ∫ 0 x f ( t ) d t 1 - e x 2 = lim x → 0 x 2 1 - e x 2 × ⏟ - 1 ∫ 0 x f ( t ) d t x ⏟ Apply   L ' hospital   rule = - 1 lim x → 0 f x 1 = - 1 (D) lim x → 0 ( sin x ) ∫ 0 x f t d t x 2 = lim x → 0 sin x x ⏟ 1 × ∫ 0 x f ( t ) d t x ⏟ Apply   L ' hospital   rule   = 1 lim x → 0 f x 1 = 1
Related: Mathematics — Definite Integration · All PYQ Banks