JEE Advanced2018MathematicsDifferential EquationsActual
Let f : 0 , ∞ → R be a continuous function such that f x = 1 - 2 x + ∫ 0 x e x - t f t d t for all x ∈ 0 , ∞ . Then, which of the following statement(s) is (are) TRUE?
Options
- AThe curve y = f x pass through the point 1 , 2
- BThe curve y = f x passes through the point 2 , - 1
- CThe area of the region x , y ∈ 0,1 × R : f x ≤ y ≤ 1 - x 2   i s π - 2 4
- DThe area of the region x , y ∈ 0,1 × R : f x ≤ y ≤ 1 - x 2   i s π - 1 4
Correct answer
B. The curve y = f x passes through the point 2 , - 1
Step-by-step solution
f x = 1 - 2 x + ∫ 0 x e x - t f t d t ⇒ e - x f x = e - x 1 - 2 x + ∫ 0 x e - t f t d t Differentiate w.r.t. x e - x f x + e + x f ′ x = - e - x 1 - 2 x + e - x - 2 + e - x f x ⇒ - f x + f ′ x = - 1 - 2 x - 2 + f x ⇒ f ′ x - 2 f x = 2 x - 3 Integrating factor = e - 2 x f x . e - 2 x = ∫ e - 2 x 2 x - 3 d x = 2 x - 3 ∫ e - 2 x d x - ∫ 2 ∫ e - 2 x d x d x = 2 x - 3 e - 2 x - 2 - e - 2 x 2 + c f x = 2 x - 3 - 2 - 1 2 + c e 2 x f 0 = 3 2 - 1 2 + c = 1 ⇒ c = 0 ∴ f x = 1 - x Area = π 4 - 1 2 = π - 2 4