Important Questions: CBSE Class 12 Mathematics Chapter 5 Continuity and Differentiability

Continuity and Differentiability is part of Unit 5 (Calculus) in the CBSE Class 12 Mathematics syllabus. The entire Calculus unit carries approximately 35 marks, and this chapter alone accounts for 15-18 marks. Expect 1-2 MCQs (1 mark each), 1-2 VSAs (2 marks each), 1-2 short-answer questions (3 marks each), and at least one long-answer or case-based question (5 marks). The 2024 and 2025 CBSE sample papers featured questions on checking continuity and differentiability at a point, applying the chain rule to composite functions, finding derivatives of inverse trigonometric functions, and differentiating parametric equations. Because this chapter is proof-heavy and requires precise algebraic manipulation, it rewards careful practice. Mastery here directly boosts your Calculus score and builds the foundation for Chapter 6 Applications of Derivatives and Chapter 7 Integrals.

One-mark questions test your recall of definitions, standard results, and quick reasoning. MCQs on continuity often present a piecewise function and ask at which point it is continuous or discontinuous. Assertion-reason questions pair a statement about differentiability with a reason about continuity. Solve these in under 90 seconds each. Always verify both left-hand and right-hand limits for continuity, and remember that differentiability implies continuity but not vice versa.

Two-mark questions require concise working in 3-4 steps. Continuity problems ask you to find an unknown constant by equating the limit to the function value. Chain rule questions involve composite functions like (3x + 5)¹⁰ or e^(sin x). Write the final derivative neatly and box your answer. Examiners award 1 mark for method and 1 mark for the correct final expression. Always state the chain rule explicitly: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x).

Three-mark questions demand clear step-by-step working. Popular topics include derivatives of inverse trigonometric functions, logarithmic differentiation, and parametric differentiation. For inverse trigonometric derivatives, substitute standard results from NCERT. For parametric forms x = f(t), y = g(t), use dy/dx = (dy/dt)/(dx/dt). Show all substitutions explicitly to secure full marks. Examiners typically allocate 1 mark for setup, 1 mark for intermediate steps, and 1 mark for the final simplified answer.

Five-mark questions test multiple concepts in one go. A classic pattern is to define a piecewise function, ask you to check continuity and differentiability at two points, then find the derivative in each interval. Another variant is a word problem (case study) where you model a real situation with a composite function and differentiate. Allocate at least 8-10 minutes per 5-mark question. Write clearly, label each part (i), (ii), and box intermediate results. Full marks require correct method, accurate algebra, and a concluding statement.

CBSE question setters follow a predictable pattern for Continuity and Differentiability. They prefer piecewise functions with absolute values, trigonometric expressions, or rational forms for continuity checks. Differentiability questions almost always involve modulus or parametric definitions. Chain rule questions use nested functions like (ax + b)^n, e^(f(x)), or log(g(x)). Inverse trigonometric derivatives appear as 3-mark standalone questions or within 5-mark proofs. Recent years have introduced case-based questions: a real-world scenario (water tank, ladder, particle motion) is described in a paragraph, followed by 4-5 sub-questions. The first sub-question may be MCQ (1 mark), the rest VSA or SA (2-3 marks each). Examiners reward students who quote NCERT theorems by name (e.g. 'By Theorem 5.2, if f is differentiable at x = a, it is continuous at x = a') and write units in application problems. Marks are often lost for skipping the continuity check before proving differentiability, so always verify continuity first.

Even bright students lose 4-6 marks in this chapter due to avoidable errors. The most frequent mistake is checking differentiability without first verifying continuity; CBSE mark schemes award zero if you skip the continuity step. Second, many students confuse left-hand and right-hand limits, especially with piecewise functions involving absolute values. Third, in chain rule problems, forgetting to multiply by the inner derivative costs 1-2 marks. Fourth, sign errors plague logarithmic and inverse trigonometric derivatives—double-check your algebra before writing the final answer. Fifth, in parametric differentiation, students write dx/dy instead of dy/dx or forget to simplify trigonometric ratios. Sixth, rounding errors or missing units in word problems lead to partial credit. Finally, many students write vague statements like 'function is continuous' without showing limit calculations; examiners need to see lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a) explicitly.

Continuity and Differentiability is one of the most technique-intensive chapters in Class 12 Mathematics, and mastering it requires repeated practice with instant feedback. CBSETUTOR.ai offers a 24×7 AI tutor that lets you upload a photo of any problem—whether from NCERT, your school worksheet, or a previous year's board paper—and receive a step-by-step solution within seconds. Unlike offline coaching or YouTube videos, CBSETUTOR.ai adapts to your individual doubts: if you are stuck on proving a function is differentiable at a point, the AI walks you through left-hand and right-hand derivative calculations, highlights where you went wrong, and suggests similar problems for practice. The platform covers all 13 chapters of CBSE Class 12 Mathematics, plus revision modules for Class 11 prerequisites. At a flat ₹999 per month for Classes 6-12, there is no per-subject or per-chapter fee, making it far more affordable than traditional tuition (typically ₹3,000-5,000/month for Maths alone). You also get a 3-day free trial—no credit card required—to test the AI tutor on your toughest Continuity and Differentiability questions. Parents and students across Delhi NCR, Mumbai, Bengaluru, and Tier-2 cities have reported 15-20% score improvements in board exams after consistent use.

First, memorise all standard derivatives from NCERT Chapters 5 and 13 (sin⁻¹x, cos⁻¹x, tan⁻¹x, cot⁻¹x, sec⁻¹x, cosec⁻¹x, and their signs). Write them on a flashcard and revise daily. Second, practice piecewise function problems until checking left-hand and right-hand limits becomes automatic. Third, for every differentiability question, write the definition in words: 'f is differentiable at x = a if lim(h→0⁻)... = lim(h→0⁺)... exists and are equal.' Fourth, use logarithmic differentiation for products or quotients of functions raised to variable powers (e.g. x^sin x). Fifth, solve at least 10 previous year questions under timed conditions—5 minutes for 3-mark, 10 minutes for 5-mark—to build speed. Sixth, after solving, cross-check your derivative using an online tool or CBSETUTOR.ai to catch algebraic slips. Finally, in the board exam, always attempt the 5-mark question from this chapter: even partial working (continuity check, or derivative in one interval) earns 2-3 marks.

Start with NCERT textbook Chapter 5 Exercises 5.1 through 5.8. Exercise 5.1 (Continuity) has 34 questions; do at least questions 1-10, 15-20, 30-34. Exercise 5.2 (Differentiability) has 10 questions; solve all. Exercise 5.3 (Chain Rule) is short (15 questions); complete every one. Exercises 5.4, 5.5 (Inverse Trig and Parametric) are crucial for 3-mark questions—do questions 1-10 from each. Miscellaneous Exercise at the end has 23 challenging problems; attempt 15-20 before your final revision. Supplement NCERT with RD Sharma Class 12 Chapter 9 (Continuity) and Chapter 10 (Differentiability)—these offer 50+ extra problems per chapter. For previous year questions, download the last five years' CBSE board papers (2020, 2021 skipped, 2022, 2023, 2024) from cbse.nic.in and solve every Chapter 5 question. Finally, use CBSETUTOR.ai's question bank to access topical tests: 25-question quizzes on Continuity only, Chain Rule only, or mixed, with instant auto-grading and explanations.