Area Under the Curve
Area Under Curves
nta_pyq_2025_jan
Grade 12
Question:
Let f : R \to R be a twice differentiable function such that f (x + y) = f (x)f (y) for all x, y \in R. If f (0) = 4a ′ and f satisfies f (x) - 3af (x) - f (x) = 0, a > 0, then the area of the region ′′ ′ R = {(x, y) ∣ 0 \le y \le f (ax), 0 \le x \le 2} is:
e - 1 2
e + 1 2
e + 1 4
e - 1 4
Step-by-Step Solution
Key Concept: Apply the core result for area bounded by curves and simplify using the given constraints.
f (x + y) = f (x) ⋅ f (y) \lambdax ′ (1) \Rightarrow f (x) = e f (0) = 4a ′ \lambdax \Rightarrow f (x) = \lambdae \Rightarrow \lambda = 4a So, f (x) = e 4x ′′ ′ f (x) - 3af (x) - f (x) = 0 2 \Rightarrow \lambda - 3a\lambda - 1 = 0 1 2 2 2 \Rightarrow 16a - 12a - 1 = 0 \Rightarrow 4a = 1 \Rightarrow a = 2 2x F(x) = e 2 x 2 Area = \int e dx = e - 1 0
Correct Answer: 1