Limits, Continuity & Differentiability
Continuity and Differentiability
nta_pyq_2025_jan
Grade 12
Question:
If the function 2 ⎧ {sin(k1 + 1)x + sin(k2 - 1)x} , x < 0 ⎪ ⎪ x ⎪ 4, x = 0 f (x) = ⎨ ⎪ ⎪ 2 2+k1 x ⎩ ⎪ log ( ), x > 0 x e 2+k2 x is continuous at x = 0, then k + k is equal to 2 1 2 2
Step-by-Step Solution
Key Concept: Apply the core result for continuity and differentiability at a point and simplify using the given constraints.
lim 2 {sin(k1 + 1)x + sin(k2 - 1)x} = 4 - x (4) x\to0 \Rightarrow 2 (k1 + 1) + 2 (k2 - 1) = 4 \Rightarrow k1 + k2 = 2 2 2 + k1 x \Rightarrow lim ln( ) = 4 x\to0 + x 2 + k2 x 1 (k1 - k2 ) x \Rightarrow lim ln(1 + ) = 2 x\to0 + x 2 + k2 x k1 - k2 \Rightarrow = 2 2 \Rightarrow k1 - k2 = 4 \therefore k1 = 3, k2 = -1 2 2 k1 + k2 = 9 + 1 = 10
Correct Answer: 4