Definite Integration
Definite Integration
nta_pyq_2025_jan
Grade 12
Question:
Let f : (0, \infty) \to R be a twice differentiable function. If for some a̸ = 0, \int 1 f (\lambdax)d\lambda = 0 af (x), f (1) = 1 and f (16) = 18 , then 16 - f ′ ( 16 1 ) is equal to _______.
Step-by-Step Solution
Key Concept: Apply the core result for definite integral properties and simplify using the given constraints.
Given, \int 1 f (\lambdax)d\lambda = af (x) 0 (112) Let \lambdax = u 1 d\lambda = du x \therefore From (1) 1 x \int x 0 f (u)du = af (x) x \Rightarrow \int f (u)du = axf(x) 0 Differentiate both sides ′ f (x) = a (xf (x) + f (x)) ′ \Rightarrow f (x) = axf (x) + af (x) ′ \Rightarrow (1 - a)f (x) = axf (x) ′ f (x) (1 - a) 1 \Rightarrow = ⋅ f (x) a x Integrate both side w.r.t. ( x ) ′ f (x) (1 - a) 1 \Rightarrow \int dx = \int dx f (x) a x 1 - a \Rightarrow ln f (x) = ( ) ln x + c a Now at x = 1f (1) = 1 \Rightarrow c = 0 Also given f (16) = 1 8 1 1-a \therefore = (16) a 8 4-4a -3 \Rightarrow 2 = 2 a 4 - 4a \Rightarrow -3 = a \Rightarrow -3a = 4 - 4a \Rightarrow a = 4 -3/4 \therefore f (x) = x -3 - 7 f (x) = x 4 4 1 Put x = 16 -7/4 -7 1 -3 1 -3 -4x( ) ′ 4 f ( ) = ( ) = ⋅ 2 = -96 16 4 16 4 1 ′ \therefore 16 - f ( ) \Rightarrow 16 - (-96) = 112 16
Correct Answer: 112