Limits, Continuity & Differentiability
Continuity and Differentiability
nta_pyq_2025_jan
Grade 12
Question:
Let the function, 2 -3ax - 2, x < 1 f (x) = { 2 a + bx, x \ge 1 be differentiable for all x \in R, where a > 1, b \in R. If the area of the region enclosed by y = f (x) and the line y = -20 is \alpha + \beta\sqrt3, \alpha, \beta \in Z , then the value of \alpha + \beta is ________
Step-by-Step Solution
Key Concept: Apply the core result for continuity and differentiability at a point and simplify using the given constraints.
f (x) is continuous and differentiable (34) at x = 1, LHL = RHL, LHD = RHD 2 - 3a - 2 = a + b, -6a = b a = 2; b = -12 2 -6x - 2, x < 1 f (x) = { 4 - 12x, x \ge 1 1 2 2 Area = \int (-6x - 2 + 20) dx + \int (4 - 12x + 20)dx] -\sqrt3 1 = 16 + 12\sqrt3 + 6 = 22 + 12\sqrt3 \therefore \alpha + \beta = 34
Correct Answer: 34