CBSE Class 12 Continuity and Differentiability PYQs Analysis (2021-2025)

Based on detailed analysis of last 5 years' papers. Perfect for 2026 Boards prep!

Key Trends

Weightage Breakdown: 1-2 MCQs/assertion-reason (1 mark), 2-3 short answers/numericals on derivatives (2-3 marks), 1 long answer/proof (4-5 marks) on Rolle's/LMVT or higher-order.
Repeated Topics: Checking continuity/differentiability at a point, chain rule, logarithmic differentiation, parametric functions, second-order derivatives, Rolle's theorem & LMVT applications, derivatives of inverse trig functions.
Difficulty: Medium; algebraic manipulation & careful checking of limits key. More proofs and case-based in recent years.
Changes: Increased focus on application-based questions and verifying theorems (2023–2025).

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
Discuss the continuity of f(x) = \|x-1\| + \|x-2\| at x=1 and x=2. Or check continuity of f(x) = {x sin(1/x) if x≠0, 0 if x=0} at x=0.	Short Answer (2-3 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; Use left/right-hand limits; continuous everywhere but not differentiable at points.
Differentiate using logarithmic differentiation: y = (x^x)(x^{sin x}) or y = (sin x)^x or similar composite functions.	Short Answer (3 marks)	2021 Term 2, 2022, 2023, 2024	Repeated 4x; Take ln y → (1/y)y' = ... → y' = y(...).
If y = sin^{-1}x + cos^{-1}x, prove dy/dx = 0. Or differentiate inverse trig functions like tan^{-1}( (x-a)/(1+ax) ).	Short Answer/Proof (2 marks)	2022, 2023, 2024, 2025	Repeated 4x; Direct derivative or identity sin^{-1}x + cos^{-1}x = π/2.
Assertion: If f is differentiable at x=a, then f is continuous at x=a. Reason: Limit of [f(x)-f(a)]/(x-a) exists.	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason explains.
Find dy/dx if x = a cos θ, y = b sin θ (parametric) or x = sin t, y = cos 2t.	Short Answer (2-3 marks)	2021 Term 1, 2022, 2023, 2025	Repeated 4x; dy/dx = (dy/dθ)/(dx/dθ).
Verify Rolle's theorem for f(x) = x^2 - 4x + 3 in [1,3]. Find c in (1,3) such that f'(c)=0.	Short Answer (2 marks)	2022, 2024, 2025	Repeated 3x; f(1)=f(3)=0, continuous & differentiable → c=2 (f'=2x-4=0).
Find second-order derivative: If y = e^x sin x or y = log(sin x) or parametric second derivative.	Short Answer (3 marks)	2023, 2024	Repeated 2x; Use product rule repeatedly or formula for parametric.
MCQ: The function f(x) = \|x\| is differentiable at x=0? (a) Yes (b) No	MCQ (1 mark)	2021 Term 1, 2023, 2024	Repeated 3x; Answer (b) No (left/right derivatives differ).
Case-based: Given function with piecewise definition or parametric, check continuity/differentiability and find derivative.	Case-Based (4 marks)	2023, 2025	Repeated 2x; Focus on limit matching and chain rule.
Using LMVT, prove inequalities or find c such that f'(c) = [f(b)-f(a)]/(b-a) for given interval.	Short Answer (2-3 marks)	2021 Term 2, 2024	Repeated 2x; Application of mean value theorem.

Best Format for Preparation and Answering

Study Format: Tables for standard derivatives (trig, inverse trig, exponential). Practice chain rule + log diff on 10 PYQs daily. Memorize Rolle's/LMVT statements & conditions.
Answer Format: Continuity: Show lim x→a- = lim x→a+ = f(a). Differentiability: Show limit of [f(x)-f(a)]/(x-a) exists. Derivatives: Step-by-step application of rules.
Tips: Focus on checking continuity/differentiability at points & logarithmic differentiation (high repeat & marks). Time: 10–12 mins for long questions. Very scoring if steps are clear.

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