Continuity and Differentiability — Class 12 Mathematics PYQs

If $f(x)=|x|+|x-1|$, then which of the following is correct ?

(A) $f(x)$ is both continuous and differentiable, at $x=0$ and $x=1$.
(B) $f(x)$ is differentiable but not continuous, at $x=0$ and $x=1$.
(C) $f(x)$ is continuous but not differentiable, at $x=0$ and $x=1$.
(D) $f(x)$ is neither continuous nor differentiable, at $x=0$ and $x=1$.

Assertion (A): $f(x)=\{\begin{array}{ll}3x-8,& x\le 5\\ 2k,& x>5\end{array}$ is continuous at $x=5$ for $k=\frac{5}{2}$.

Reason (R): For a function $f$ to be continuous at $x=a$, ${\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a^{+}}f(x)=f(a)$.

(A) Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(C) Assertion (A) is true, but Reason (R) is false.
(D) Assertion (A) is false, but Reason (R) is true.

(a) Differentiate $2^{{\cos }^{2}x}$ w.r.t. ${\cos }^{2}x$.

OR

(b) If ${\tan }^{-1}(x^2+y^2)=a^2$, then find $\frac{dy}{dx}$.

(a) If $x=e^{x/y}$, then prove that $\frac{dy}{dx}=\frac{x-y}{x\log x}$.

OR

(b) If $f(x)=\{\begin{array}{ll}2x-3,& -3\le x\le -2\\ x+1,& -2<x\le 0\end{array}$

Check the differentiability of $f(x)$ at $x=-2$.

(a) If $\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y)$, then prove that $\frac{dy}{dx}=\frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}$.

OR

(b) If $x=a\left(\cos \theta +\log \tan \frac{\theta }{2}\right)$ and $y=\sin \theta$, then find $\frac{d^{2}y}{d{x}^2}$ at $\theta =\frac{\pi }{4}$.

(a) Differentiate ${\tan }^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)$ w.r.t. ${\cos }^{-1}\left(2x\sqrt{1-x^2}\right)$, $x\in \left(\frac{1}{\sqrt{2}},1\right)$

OR

(b) Find $\frac{dy}{dx}$, if $y=x^{\tan x}+\frac{\sqrt{x^2+1}}{2}$.

If $x=at$, $y=\frac{a}{t}$, then $\frac{dy}{dx}$ is :

Which of the following statements is true for the function

$$f(x)=\{\begin{array}{ll}{x}^2+3,& x\ne 0\\ 1,& x=0\end{array}$$

The number of points of discontinuity of

$$f(x)=\{\begin{array}{ll}|x|+3,& \text{if }x\le -3\\ -2x,& \text{if }-3<x<3\\ 6x+2,& \text{if }x\ge 3\end{array}$$

is :

The derivative of $\sin (x^2)$ w.r.t. $x$, at $x=\sqrt{\pi }$ is :