Continuity and Differentiability: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. Differentiate $\sin (3x+2)$.
2. Differentiate $e^{\tan x}$.
3. Find $k$: $kx+1$ ($x\le 2$), $3x-1$ ($x>2$) continuous.
4. Differentiate $\log (\log x)$.
5. $\frac{dy}{dx}$ if $x^2+y^2=25$.
6. Differentiate $\sqrt{\sin x}$.
7. Differentiate $2^{x^2}$.
8. $\frac{dy}{dx}$ if $x=\cos t$, $y=\sin t$.
9. Second derivative of $\sin 2x$.
10. Differentiate ${\cos }^{-1}(2x)$.

D. Short answer I (2 marks each)

1. Check the continuity of $f=\frac{x^2-9}{x-3}$ ($x\ne 3$), $f(3)=6$ at 3.
2. Show that $f(x)=|x-2|$ is not differentiable at 2.
3. Differentiate ${\tan }^{-1}\frac{2x}{1-x^2}$, $|x|<1$.
4. Find $\frac{dy}{dx}$ if $xy=\sin (x+y)$.
5. Differentiate $x^{\sin x}$.
6. If $y=\log (1+x^2)$, find $y_2$.
7. Find $\frac{dy}{dx}$ for $x=a\sec \theta$, $y=a\tan \theta$.
8. Differentiate ${\sin }^{2}x$ w.r.t. ${\cos }^{2}x$.

Rough work

E. Short answer II (3 marks each)

1. Find $a,b$ if $f=ax+1$ ($x<1$), 3 ($x=1$), $b{x}^2+1$ ($x>1$) is continuous at 1.
2. Differentiate $(\cos x{)}^x+(\sin x{)}^{1/x}$.
3. If $y=\sqrt{x+\sqrt{x+\sqrt{x+\dots }}}$, show $\frac{dy}{dx}=\frac{1}{2y-1}$.
4. If $x=a(\theta +\sin \theta )$, $y=a(1-\cos \theta )$, find $\frac{d^{2}y}{d{x}^2}$ at $\theta =\frac{\pi }{2}$.
5. If $y={\tan }^{-1}\sqrt{\frac{1-\cos x}{1+\cos x}}$, $0<x<\pi$, find $y^{\prime }$.
6. If $y=({\sin }^{-1}x{)}^2$, show $(1-x^2)y_2-x{y}_1=2$.

F. Long answer (5 marks each)

1. (a) If $x^m{y}^n=(x+y{)}^{m+n}$, prove $\frac{dy}{dx}=\frac{y}{x}$. (b) Find $\frac{dy}{dx}$ if $y=x^{\tan x}+\sqrt{\frac{x^2+1}{2}}$.
2. (a) If $y=\sin (m{\sin }^{-1}x)$, prove $(1-x^2)y_2-x{y}_1+m^{2}y=0$. (b) Discuss the continuity and differentiability of $f(x)=|x|+|x-1|$ at $x=1$.

G. HOTS (higher-order thinking)

1. Give a function continuous everywhere but not differentiable at $x=3$.
2. Why is $\frac{d}{dx}|x|=\frac{x}{|x|}$ only for $x\ne 0$?
3. If $f$ and $g$ are both discontinuous at $a$, can $f+g$ be continuous?
4. Find $\frac{d}{dx}\left(x^{x^x}\right)$ in outline.

20 MCQs

1. $\frac{d}{dx}{\sin }^{-1}x$
   (a) $\frac{1}{1+x^2}$ (b) $\frac{-1}{\sqrt{1-x^2}}$ (c) $\frac{1}{\sqrt{1-x^2}}$ (d) $\frac{1}{x\sqrt{x^2-1}}$
2. $f=\frac{\sin x}{x}$, $f(0)=k$ continuous at 0 if $k=$
   (a) 1 (b) 0 (c) $-1$ (d) 2
3. Not differentiable at 0
   (a) $x^2$ (b) $\sin x$ (c) $x|x|$ (d) $|x|$
4. $\frac{d}{dx}{e}^{\sin x}$
   (a) $e^{\sin x}$ (b) $\cos x{e}^{\sin x}$ (c) $\sin x{e}^{\cos x}$ (d) $e^{\cos x}$
5. $\frac{d}{dx}{x}^x$
   (a) $x^x$ (b) $x^x(1+\log x)$ (c) $x\cdot x^{x-1}$ (d) $x^x\log x$
6. If $x^2+y^2=1$, $y^{\prime }=$
   (a) $\frac{x}{y}$ (b) $-\frac{y}{x}$ (c) $\frac{y}{x}$ (d) $-\frac{x}{y}$
7. $\frac{d}{dx}\log (\cos x)$
   (a) $-\tan x$ (b) $\cot x$ (c) $\tan x$ (d) $-\cot x$
8. $[x]$ is discontinuous at
   (a) $x=0.5$ (b) $x=1.5$ (c) $x=2$ (d) $x=\pi$
9. $x=t^2$, $y=t^3$: $\frac{dy}{dx}=$
   (a) $\frac{2}{3}t$ (b) $t$ (c) $6t^5$ (d) $\frac{3}{2}t$
10. $\frac{d^2}{d{x}^2}\sin x$
    (a) $\cos x$ (b) $-\sin x$ (c) $\sin x$ (d) $-\cos x$
11. $\frac{d}{dx}{\tan }^{-1}\frac{2x}{1-x^2}$ for $|x|<1$
    (a) $\frac{1}{1+x^2}$ (b) $\frac{-2}{1+x^2}$ (c) $\frac{2}{1+x^2}$ (d) $\frac{2x}{1+x^2}$
12. $\frac{d}{dx}{2}^x$
    (a) $2^x\log 2$ (b) $x{2}^{x-1}$ (c) $2^x$ (d) $\frac{2^x}{\log 2}$
13. $k{x}^2$ ($x\le 1$), 4 ($x>1$) continuous: $k=$
    (a) 2 (b) 4 (c) 1 (d) 16
14. If $y=\log x$, $y_2=$
    (a) $\frac{1}{x}$ (b) $\frac{1}{x^2}$ (c) $-\frac{1}{x^2}$ (d) $-\frac{1}{x}$
15. $\frac{d}{dx}\sqrt{x}$
    (a) $2\sqrt{x}$ (b) $\sqrt{x}$ (c) $\frac{1}{\sqrt{x}}$ (d) $\frac{1}{2\sqrt{x}}$
16. Derivative of $x^2$ w.r.t. $x^3$
    (a) $\frac{2}{3x}$ (b) $\frac{3x}{2}$ (c) $6x^5$ (d) $\frac{2x}{3}$
17. $\frac{d}{dx}{\cos }^{-1}x+\frac{d}{dx}{\sin }^{-1}x$
    (a) 0 (b) $\frac{2}{\sqrt{1-x^2}}$ (c) 1 (d) $\frac{\pi }{2}$
18. $\frac{d}{dx}{\log }_{10}x$
    (a) $\frac{1}{x}$ (b) $10^x$ (c) $\frac{\log 10}{x}$ (d) $\frac{1}{x\log 10}$
19. If $y=A{e}^{2x}+B{e}^{-2x}$, $y_2=$
    (a) $2y$ (b) $4y$ (c) $-4y$ (d) $y$
20. $\frac{d}{dx}{\sec }^{2}x$
    (a) $2\sec x$ (b) $\sec x\tan x$ (c) $2{\sec }^{2}x\tan x$ (d) $2\tan x$

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: $|x|$ is continuous but not differentiable at 0. R: LHD $=-1\ne$ RHD $=1$ at 0.
2. A: $\frac{d}{dx}({\sin }^{-1}x+{\cos }^{-1}x)=0$. R: ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$.
3. A: $f(x)=[x]$ is differentiable at $x=1$. R: $[x]$ is discontinuous at every integer.
4. A: Every differentiable function is continuous. R: $|x|$ is continuous everywhere.
5. A: $\frac{d}{dx}(x^2)$ w.r.t. $x$ at $x=1$ is 2. R: $\frac{d}{dx}{x}^n=n{x}^{n-1}$.
6. A: $f(x)=\frac{1}{x}$ is continuous on R. R: it is continuous at every point of its domain.

Key: 1 (a) · 2 (a) · 3 (d) · 4 (b) · 5 (a) · 6 (d)

Answers to the practice set