Mathematics

Mathematics

Problem

Let f:[0,∞)→Rf: [0, \infty) \to \mathbb{R} be a continuous function such that f(x)=1−2x+∫0xex−tf(t)dtf(x) = 1 - 2x + \int_{0}^{x} e^{x-t} f(t) dt for all x∈[0,∞)x \in [0, \infty). Then, which of the following statement(s) is (are) TRUE? [JEE(Advanced) 2018] (A) The curve y=f(x)y = f(x) passes through the point (1,2)(1, 2) (B) The curve y=f(x)y = f(x) passes through the point (2,−1)(2, -1) (C) The area of the region {(x,y)∈[0,1]×R:f(x)≤y≤1−x2}\left\{(x, y) \in [0, 1] \times \mathbb{R} : f(x) \le y \le \sqrt{1-x^2}\right\} is π−24\frac{\pi-2}{4} (D) The area of the region {(x,y)∈[0,1]×R:f(x)≤y≤1−x2}\left\{(x, y) \in [0, 1] \times \mathbb{R} : f(x) \le y \le \sqrt{1-x^2}\right\} is π−14\frac{\pi-1}{4}

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