CBSE Class 12 Mathematics Continuity and Differentiability Assignment Set 07

This assignment has 15 short practice questions on Chapter 5 Continuity and Differentiability for Class 12 Mathematics (CBSE), on continuity, differentiability, chain rule, implicit, parametric and second-order derivatives. The answer to each question is given below it for quick self-checking.

Practice Questions

Question 1. If \( y = \sqrt{\sin x + y} \), then find \( \frac{dy}{dx} \).
Answer:
\( \frac{\cos x}{2y - 1} \)

Question 2. Find the derivative of cos⁻¹(2x² − 1) w.r.t. cos⁻¹ x.
Answer:
2

Question 3. If \( f(x) = \log_{\sqrt{x}}\left(\frac{1}{x}\right) \), then find f′(x).
Answer:
0

Question 4. If x = t² and y = t³, then find \( \frac{d^2y}{dx^2} \).
Answer:
\( \frac{3}{4t} \)

Question 5. If \( y = \log\left(\frac{1 - x^2}{1 + x^2}\right) \), then find \( \frac{dy}{dx} \).
Answer:
\( \frac{-4x}{1 - x^4} \)

Question 6. If y = (cos x²)², then find \( \frac{dy}{dx} \).
Answer:
−2x sin 2x²

Question 7. If f(x) = |cos x − sin x|, then find \( f'\left(\frac{\pi}{3}\right) \).
Answer:
\( \left(\frac{\sqrt{3} + 1}{2}\right) \)

Question 8. For the curve \( \sqrt{x} + \sqrt{y} = 1 \), find \( \frac{dy}{dx} \) at \( \left(\frac{1}{4}, \frac{1}{4}\right) \).
Answer:
−1

Question 9. Find the value of k so that the function \( f(x) = \begin{cases} 3x - 8; & \text{if } x \le 5 \\ 2k; & \text{if } x \gt 5 \end{cases} \) is continuous at the point x = 5.
Answer:
\( k = \frac{7}{2} \)

Question 10. If the function \( f(x) = \frac{1}{x + 2} \), then find the points of discontinuity of the composite function y = f{f(x)}.
Answer:
\( y = f\{f(x)\} = \frac{x + 2}{2x + 5} \) is discontinuous at x = −2 (where f(x) itself is not defined) and at \( x = -\frac{5}{2} \).

Question 11. Evaluate \( \frac{dy}{dx} \) if \( y = \log\left(x + \sqrt{x^2 + a}\right) \).
Answer:
\( \frac{dy}{dx} = \frac{1}{\left(\sqrt{x^2 + a}\right)} \)

Question 12. Evaluate \( \frac{dy}{dx} \) if \( y = \sec^{-1}\left(\frac{1}{4x^3 - 3x}\right),\ 0 \lt x \lt \frac{1}{\sqrt{2}} \).
Answer:
\( \frac{dy}{dx} = \frac{-3}{\sqrt{1 - x^2}} \) for \( \frac{1}{2} \lt x \lt \frac{1}{\sqrt{2}} \), and \( \frac{dy}{dx} = \frac{3}{\sqrt{1 - x^2}} \) for \( 0 \lt x \lt \frac{1}{2} \)

Question 13. Find \( \frac{dy}{dx} \) when x and y are connected by the equation tan⁻¹(x² + y²) = a.
Answer:
\( \frac{dy}{dx} = \frac{-x}{y} \)

Question 14. If y = tan⁻¹ x, then find \( \frac{d^2y}{dx^2} \) in terms of y alone.
Answer:
\( \frac{d^2y}{dx^2} = -\sin 2y\cdot\cos^2 y \)

Question 15. Check the differentiability of the function \( f(x) = \begin{cases} 1 + x; & \text{if } x \le 2 \\ 5 - x; & \text{if } x \gt 2 \end{cases} \) at the point x = 2.
Answer:
Not differentiable