Continuity and Differentiability: 1 mark Questions (CBSE Class 12)

If $e^{-x}+e^{-y}=2$, then $\frac{dy}{dx}$ is

1. (A)$e^{x-y}$
2. (B)$e^{y-x}$
3. (C)$-e^{x-y}$
4. (D)$-e^{y-x}$

If $e^{x+y}=3x$, then $\frac{dy}{dx}$ is

1. (A)$\frac{3}{e^{x+y}}$
2. (B)$\frac{1}{e^{x+y}}$
3. (C)$\frac{1-e^{x+y}}{e^{x+y}}$
4. (D)$\frac{3-e^{x+y}}{e^{x+y}}$

If $x+y=xy$, then $\frac{dy}{dx}$ is

1. (A)$\frac{y}{x-1}$
2. (B)$\frac{1}{x-1}$
3. (C)$\frac{y-1}{x-1}$
4. (D)$\frac{1-y}{x-1}$

The value of k for which the function $f(x)=\{\begin{array}{ll}{x}^2\sin \frac{1}{x},& x\ne 0\\ k(x+1),& x=0\end{array}$ is a continuous function, is :

1. (A)$\frac{1}{4}$
2. (B)$2$
3. (C)$\frac{1}{2}$
4. (D)$0$

If ${\sin }^{-1}x=y$, then $\frac{dy}{dx}$ is :

1. (A)${\cos }^{-1}x$
2. (B)$\cos y$
3. (C)$\frac{1}{1-x^2}$
4. (D)$\sec y$

If ${\tan }^{-1}x=y$, then $\frac{dy}{dx}$ is equal to :

1. (A)$({\sec }^{-1}x{)}^2$
2. (B)${\sec }^{2}y$
3. (C)$\frac{1}{\sqrt{1+x^2}}$
4. (D)${\cos }^{2}y$

If $2{\cos }^{-1}x=y$, then $\frac{dy}{dx}$ is :

1. (A)$-2{\sin }^{-1}x$
2. (B)$-\frac{1}{2}\sin \frac{y}{2}$
3. (C)$\frac{-1}{\sqrt{1-2x^2}}$
4. (D)$-2\text{cosec}\frac{y}{2}$

Differential of $e^{e^x}$ with respect to x is :

1. (A)$\log x$
2. (B)$e^{e^x}$
3. (C)$e^x{e}^{e^x}$
4. (D)$(e^x{)}^2$

If $f(x)=\{\begin{array}{ll}\frac{\sin x}{x}+\cos x,& x\ne 0\\ k,& x=0\end{array}$ is continuous at x = 0, then the value of k is :

1. (A)0
2. (B)$-2$
3. (C)$-1$
4. (D)2

The greatest integer function, $f(x)=[x]$, $0<x<3$ is not differentiable at how many points ?

1. (A)At only one point
2. (B)At only two points
3. (C)At no point
4. (D)At three points

If $f(x)=\{\begin{array}{ll}\frac{x^2-4x-5}{x+1},& x\ne -1\\ k,& x=-1\end{array}$
is continuous at $x=-1$, then the value of k is :

1. (A)Any real value
2. (B)6
3. (C)$-1$
4. (D)$-6$

Derivative of ${\cos }^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right)$, $-\frac{\pi }{4}<x<\frac{\pi }{4}$ with respect to x is :

1. (A)$-1$
2. (B)1
3. (C)$\frac{\pi }{4}$
4. (D)$-\frac{\pi }{4}$