Class 12 Continuity and Differentiability Important Questions
Class 12 continuity and differentiability important questions help students master calculus fundamentals. These problems build analytical thinking and problem-solving skills.
Continuity requires understanding limits and function behavior. Differentiability involves derivative rules and applications. Many students find these concepts challenging initially. However, regular practice makes them manageable. Moreover, focused preparation reduces exam anxiety.
Strategic Approach to Calculus Mastery
Success comes from systematic practice. Students should focus on frequently tested problem types. Understanding step-by-step derivations is equally important. This approach builds conceptual clarity and speed.
For additional free study materials, visit Udgam Welfare Foundation. You can also explore resources like Class 7 Math case study questions for foundational practice. Moreover, www.mathstudy.in offers structured guidance for mathematics learners.
Consistent revision ensures formula retention. Solving diverse problems builds adaptability. Consequently, students approach board exams with greater confidence and readiness.
Continuity And Differentiability HOTS & Important Questions Class 12 Mathematics
1. Find the values of \( a \) and \( b \) such that the function \( f \) defined by:
\[ f(x) = \begin{cases} 5 & \text{if } x \le 2 \\ ax + b & \text{if } 2 < x < 10 \\ 21 & \text{if } x \ge 10 \end{cases} \]is a continuous function.
2. Discuss the continuity of the function \( f(x) = [x] + [-x] \) where \( [x] \) is the greatest integer function. Show that it is continuous at all non-integral points.
3. If \( f(x) = \begin{cases} \frac{1 – \cos 4x}{x^2} & \text{if } x < 0 \\ a & \text{if } x = 0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4} & \text{if } x > 0 \end{cases} \) is continuous at \( x=0 \), find the value of \( a \).
4. Determine if the function \( f(x) = |x-1| + |x-2| \) is differentiable at \( x=1 \) and \( x=2 \). Justify your answer using the definition of derivatives.
5. If \( x^y + y^x = a^b \), find \( \frac{dy}{dx} \).
6. If \( y = (\sin x)^{\tan x} + (\cos x)^{\sec x} \), find \( \frac{dy}{dx} \).
7. If \( x = a(\cos t + t \sin t) \) and \( y = a(\sin t – t \cos t) \), find the value of \( \frac{d^2y}{dx^2} \) at \( t = \frac{\pi}{4} \).
8. If \( \sqrt{1-x^2} + \sqrt{1-y^2} = a(x-y) \), prove that \( \frac{dy}{dx} = \sqrt{\frac{1-y^2}{1-x^2}} \).
9. Let \( f(x) = \begin{cases} x^2 \sin(\frac{1}{x}) & \text{if } x \ne 0 \\ 0 & \text{if } x = 0 \end{cases} \). Show that \( f \) is differentiable at \( x=0 \) but \( f'(x) \) is not continuous at \( x=0 \).
10. If \( y = (x + \sqrt{x^2+1})^n \), prove that \( (x^2+1)\frac{d^2y}{dx^2} + x\frac{dy}{dx} – n^2y = 0 \).
11. Find \( \frac{dy}{dx} \) if \( y = \tan^{-1} \left( \frac{\sqrt{1+x^2} – 1}{x} \right) \).
12. If \( y = e^{a \cos^{-1} x}, -1 \le x \le 1 \), show that \( (1-x^2)\frac{d^2y}{dx^2} – x\frac{dy}{dx} – a^2y = 0 \).
13. Find the value of \( k \) so that the function \( f(x) = \begin{cases} \frac{k \cos x}{\pi – 2x} & \text{if } x \ne \frac{\pi}{2} \\ 3 & \text{if } x = \frac{\pi}{2} \end{cases} \) is continuous at \( x = \frac{\pi}{2} \).
14. Differentiate \( \tan^{-1} \left( \frac{\sqrt{1+x^2} – \sqrt{1-x^2}}{\sqrt{1+x^2} + \sqrt{1-x^2}} \right) \) with respect to \( \cos^{-1} x^2 \).
15. If \( \cos y = x \cos(a+y) \), with \( \cos a \ne \pm 1 \), prove that \( \frac{dy}{dx} = \frac{\cos^2(a+y)}{\sin a} \).
Answers and Hints
1. Answer: \(a = 2, b = 1\). Hint: Ensure continuity at \(x = 2\) and \(x = 10\).
2. Hint: For non-integral \(x\), \([x] + [-x] = 0\). At integral points, check left and right limits.
3. Answer: \(a = 8\). Hint: Evaluate \(\lim_{x \to 0^-} f(x)\) and \(\lim_{x \to 0^+} f(x)\), then set equal to \(a\).
4. Hint: Check left and right derivatives at \(x=1\) and \(x=2\). The function is not differentiable at these points due to sharp corners.
5. Answer: \(\frac{dy}{dx} = -\frac{y \cdot x^{y-1} + y^x \log y}{x^y \log x + x \cdot y^{x-1}}\). Hint: Use implicit differentiation and logarithmic differentiation.
6. Hint: Use logarithmic differentiation for both terms.
7. Answer: \(\frac{8\sqrt{2}}{a\pi}\). Hint: Compute \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\), then find \(\frac{dy}{dx}\) and differentiate again.
8. Hint: Differentiate implicitly and isolate \(\frac{dy}{dx}\).
9. Hint: For differentiability at \(x=0\), show \(\lim_{h \to 0} \frac{f(h) – f(0)}{h} = 0\). For discontinuity of \(f'(x)\), consider \(\lim_{x \to 0} f'(x)\).
10. Hint: Differentiate \(y\) twice and substitute into the given differential equation.
11. Answer: \(\frac{1}{2(1+x^2)}\). Hint: Simplify the expression inside \(\tan^{-1}\) and differentiate.
12. Hint: Differentiate \(y\) once and twice, then substitute into the given equation.
13. Answer: \(k = 6\). Hint: Evaluate \(\lim_{x \to \frac{\pi}{2}} \frac{k \cos x}{\pi – 2x}\) and set equal to 3.
14. Answer: \(\frac{1}{2}\). Hint: Simplify the \(\tan^{-1}\) expression and use chain rule.
15. Hint: Differentiate implicitly and solve for \(\frac{dy}{dx}\).
Frequently Asked Questions (FAQs)
Where can I download Class 12 continuity and differentiability important questions pdf?
You can download the Class 12 continuity and differentiability important questions pdf from www.mathstudy.in. This resource includes diverse problem types. Consequently, students gain comprehensive practice for board examinations.
How to get Class 12 continuity and differentiability important questions pdf download for offline use?
Class 12 continuity and differentiability important questions pdf download is available on Udgam Welfare Foundation. Simply visit the platform and access the free resource. Thus, offline practice becomes convenient anytime.
Are Class 12 continuity and differentiability important questions with solutions available free?
Yes, Class 12 continuity and differentiability important questions with solutions are freely accessible. These solutions explain each derivation step clearly. Therefore, students learn proper methodology and avoid common errors.
Where can I find Previous Year questions of continuity and Differentiability Class 12 PDF download?
Previous Year questions of continuity and Differentiability Class 12 PDF download is offered on www.mathstudy.in. These papers reveal exam trends. Hence, practicing them builds familiarity with question patterns.
Where can I get Class 12 continuity and differentiability important questions and answers?
Class 12 continuity and differentiability important questions and answers are compiled on Udgam Welfare Foundation. These resources follow NCERT guidelines. Moreover, answers are explained for easy understanding.
What topics are covered in Class 12 differentiability important questions?
Class 12 differentiability important questions cover derivative rules, chain rule, implicit differentiation, and higher-order derivatives. These topics carry significant weightage. Therefore, focused practice yields better scores.
Where can I download Differentiation Class 12 Important questions PDF?
Differentiation Class 12 Important questions PDF is available on www.mathstudy.in. This compilation includes solved examples. Consequently, students master various differentiation techniques effectively.
How do PYQs of Continuity and Differentiability Class 12 help in preparation?
PYQs of Continuity and Differentiability Class 12 provide insight into exam patterns. They highlight frequently repeated concepts. Thus, students can prioritize topics accordingly for efficient revision.
Can I get chapter-wise questions with answer keys?
Yes, chapter-wise questions with answer keys are available on Udgam Welfare Foundation. These help in targeted practice. Additionally, answer keys enable self-assessment and improvement tracking.
How many questions should I practice daily?
Practicing 8 to 10 quality questions daily is ideal. This includes both continuity and differentiability problems. Consistent practice builds speed and accuracy over time.