Master Continuity and Differentiability: Download Class 12 Maths Chapter 5 PDF for Exam Success (2025-26)
FAQs on Free PDF: Class 12 Maths Continuity and Differentiability NCERT Book (2025-26)
1. What are the most important topics in Chapter 5, Continuity and Differentiability, for the CBSE Class 12 Maths board exam 2025-26?
Based on previous years' trends, the most important topics from which questions are frequently asked are:
- Determining the value of a constant (k, a, b) when a function is given as continuous.
- Proving the differentiability of a function at a specific point.
- Derivatives of functions in parametric forms.
- Logarithmic differentiation for complex functions like xsin(x).
- Finding the second-order derivative (d2y/dx2).
- Application of Rolle's and Lagrange's Mean Value Theorems.
Focusing on these areas will cover the majority of high-marks questions.
2. What is the typical marks distribution for important questions from this chapter in the board exams?
In the CBSE Class 12 Maths exam, questions from Continuity and Differentiability are a significant part of the Calculus unit. You can expect:
- 1-mark questions (MCQs): These often test basic continuity rules or direct differentiation formulas.
- 2-mark questions: Simple problems on finding derivatives using the chain rule or implicit differentiation.
- 3-mark questions: Problems involving logarithmic differentiation or derivatives of parametric functions.
- 5-mark questions (Long Answer): These are typically complex problems, such as finding the values of constants in a piecewise function to ensure continuity, or proving a result using second-order derivatives.
3. Why is checking a function's continuity considered an important step before testing for differentiability in board exam questions?
This is a crucial conceptual point. A fundamental theorem in calculus states that if a function is differentiable at a point, it must be continuous at that point. The reverse is not always true. Therefore, if you find that a function is not continuous at a given point, you can directly conclude that it is not differentiable there, saving valuable time in an exam. Forgetting this can lead to attempting unnecessary and complex differentiation steps.
4. What kind of Higher Order Thinking Skills (HOTS) questions are expected from logarithmic and parametric differentiation?
For HOTS questions, expect problems that require multiple steps or concepts. For example:
- Logarithmic Differentiation: Problems where you need to differentiate a function which is a sum of multiple functions with variables in the exponent, like y = xx + (sin x)cos x.
- Parametric Differentiation: You might be asked to find the second-order derivative (d2y/dx2) for parametric equations, which is a common point of error. Another HOTS type is differentiating one function with respect to another function, both given in parametric form.
5. How can I avoid the most common mistakes when solving questions on Rolle's or Lagrange's Mean Value Theorem (MVT)?
To avoid common errors in MVT questions, follow this checklist:
- Verify the Conditions: Always explicitly state and verify the two main conditions before applying the theorem: (a) The function must be continuous on the closed interval [a, b] and (b) differentiable on the open interval (a, b).
- Check for Rolle's Theorem: For Rolle's Theorem specifically, you must also verify the third condition: f(a) = f(b).
- Differentiate Correctly: Be careful while finding the derivative, f'(x), as any mistake here will lead to an incorrect value of 'c'.
- Final Answer: Ensure that the value of 'c' you calculate lies strictly within the open interval (a, b). If it doesn't, you may have made a calculation error.
6. What is the conceptual difference between a function being continuous and differentiable at a point, and why is this important for solving exam problems?
Understanding the difference is key to tackling tricky questions.
- Continuity at a point 'c' means the graph of the function has no breaks or jumps at that point. Mathematically, the Left-Hand Limit (LHL), Right-Hand Limit (RHL), and the function's value all must be equal (LHL = RHL = f(c)).
- Differentiability at a point 'c' is a stricter condition. It means the graph has a unique, non-vertical tangent line at that point, implying it is smooth and has no sharp corners or kinks. Mathematically, the Left-Hand Derivative (LHD) must equal the Right-Hand Derivative (RHD).
This is important because a function can be continuous but not differentiable (e.g., y = |x| at x=0), which is a frequent basis for board questions.
7. Which types of problems are most expected for 5-mark important questions from Continuity and Differentiability?
For 5-mark questions, focus on problems that require detailed, step-by-step solutions. The most frequently asked types are:
- Piecewise Functions: Finding the values of two unknown constants (e.g., 'a' and 'b') when a function is given to be continuous or differentiable over its domain.
- Parametric Second-Order Derivatives: Questions that ask you to prove a relationship involving y, dy/dx, and d2y/dx2 when x and y are given in parametric form.
- Logarithmic Differentiation Proofs: Complex proofs involving functions like y = (x)cos(x) or similar forms where you need to show that it satisfies a given differential equation.
