Class 12 Mathematics: Advanced Continuity and Differentiability
This worksheet challenges students with complex problems involving continuity and differentiability of functions, including piecewise functions, and applications of these concepts.
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- Q1.For what value of 'k' is the function f(x) = { (sin(k+1)x + sin x) / x , if x < 0 ; c , if x = 0 ; (sqrt(x+bx^2) - sqrt(x)) / (bx^(3/2)) , if x > 0 } continuous at x = 0?
- A.k = 1/2
- B.k = -1/2
- C.k = 0
- D.k = 1
- Q2.If f(x) = |x-1| + |x-2| + |x-3|, then f(x) is not differentiable at:
- A.x = 1, 2, 3
- B.x = 1, 2 only
- C.x = 2, 3 only
- D.x = 1, 3 only
- Q3.Let f(x) = { e^(1/x) / (1 + e^(1/x)) , if x ≠ 0 ; 0 , if x = 0 }. Which of the following statements is true regarding the continuity of f(x) at x = 0?
- A.f(x) is continuous at x = 0
- B.f(x) has a removable discontinuity at x = 0
- C.f(x) has a jump discontinuity at x = 0
- D.f(x) is continuous from the right but not from the left at x = 0
- Q4.If y = cot⁻¹( (√(1+sin x) + √(1-sin x)) / (√(1+sin x) - √(1-sin x)) ), then dy/dx is equal to:
- A.1/2
- B.-1/2
- C.1
- D.0
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What's inside this workbook
This Class 12 Mathematics Practice contains 10 questions across 2 printable pages and includes a full answer key at hard difficulty. Aligned to the CBSE and NCERT syllabus.
How this worksheet helps your child
- Builds confidence in core Mathematics concepts taught in Class 12 through repeated, varied practice.
- Aligned to CBSE and NCERT — questions mirror the pattern children see in school tests and unit assessments.
- Comes with an answer key so parents and students can self-check in minutes, without needing a teacher.
- Pitched at hard level — use it for daily homework, weekend revision or quick exam warm-ups.
About this worksheet
This practice worksheet, designed for Class 12 Mathematics students, delves deep into advanced concepts of Continuity and Differentiability. It presents challenging problems that go beyond basic definitions, encouraging students to apply their understanding to complex scenarios, including piecewise-defined functions. Mastery of continuity and differentiability is foundational for higher calculus and is a crucial component of the CBSE Class 12 curriculum. This topic not only forms a significant portion of board examinations but also builds analytical skills essential for future studies in science, engineering, and economics. This worksheet is ideal for students aiming for a thorough understanding and wishing to excel in competitive examinations, providing ample opportunity to practice intricate problems often seen in NCERT-aligned advanced texts.
What your child will learn
- Practice advanced problems related to the continuity of various types of functions.
- Solve problems involving the differentiability of functions, including at critical points.
- Apply continuity and differentiability concepts to piecewise functions.
- Analyze the properties of functions to determine their continuity and differentiability over given intervals.
- Develop problem-solving strategies for complex calculus questions.
Understanding the concept
Continuity and differentiability are two fundamental concepts in calculus that describe the behavior of functions. Imagine drawing a function's graph without lifting your pen from the paper; that's what a continuous function is like – it has no breaks, jumps, or holes. Mathematically, a function is continuous at a point if its value at that point is equal to the limit of the function as it approaches that point. Differentiability, on the other hand, is about whether a function has a well-defined and unique tangent line at every point. If you can draw a smooth, non-vertical tangent without any sharp corners or cusps, the function is differentiable there. Think of a smooth curve versus a sharp corner like the tip of a 'V'. The 'V' shape is continuous but not differentiable at its sharp point because you can't draw a single, unique tangent line there. For Class 12, this often involves checking these properties for functions defined in different ways over different parts of their domain (piecewise functions), or functions that involve absolute values, which can introduce these sharp corners. Understanding these concepts allows us to analyze rates of change and optimize various real-world situations.
Tips for parents & teachers
- Encourage your child to review the definitions and theorems of continuity and differentiability before attempting the worksheet.
- Suggest breaking down complex problems into smaller, manageable steps, especially for piecewise functions.
- Recommend drawing rough sketches of functions to visualize their behavior, even if not explicitly asked.
- Advise focusing on understanding the 'why' behind each step, not just memorizing formulas.
- Provide a quiet environment conducive to concentration, as these problems require deep analytical thought.
Topics covered
How to use this worksheet at home
Open the print view and use your browser's Print dialog to print on regular A4 paper or save as a PDF. Hand the printed sheet to your child (Class 12) and set a relaxed time limit — 15 to 25 minutes works for most Mathematics worksheets. When they're done, open the "With answer key" view and let them self-mark question by question. Re-print as many times as you need; the file never expires and there's no download limit.
Frequently asked questions
- Is this Mathematics worksheet really free?
- Yes — every workbook on WorkbookWala is 100% free to download and print. No login, no email, no paywall. We're free for students, parents and teachers.
- Is it aligned to the CBSE / NCERT syllabus?
- Yes. All Class 12 worksheets are built around the CBSE and NCERT pattern so they map cleanly to what your child is studying in school.
- Can I print this worksheet at home?
- Yes. Click "Open print view" and use your browser's Print dialog to print on standard A4 paper, or save it as a PDF for later. The layout is designed to fit cleanly without scaling.
- Does it include an answer key?
- Yes — click "With answer key" to print a version with answers included, so parents and students can self-mark in minutes.
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