CUET Mathematics Calculus > Continuity & Differentiability PYQs

If $x = at^2, y = 2at$; then $\frac{d^2y}{dx^2}$ is equal to

Let $f(x)=\begin{cases} |x|+3 & \text{if } x\le -3 \\ -2x & \text{if } -3<x<3 \\ 6x+2 & \text{if } x\ge 3 \end{cases}$ Then, which of the following is true?

If $x^m y^n = (x + y)^{m+n}$, then $\frac{d^2y}{dx^2}$ is equal to:

If $\sin y = x \cos(a + y)$, then $\frac{dy}{dx}$ is equal to

If $y = \sqrt{ax + b}$ then $y\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 =$

If $y = \log_e(\sec e^{x^2})$ then $\frac{dy}{dx} =$

Match **List-I** with **List-II** | List-I | List-II | | :--- | :--- | | **Function** | **Points of discontinuity** | | (A) $f(x) = \frac{x^2 + 1}{x}$ | (I) $x = 4$ | | (B) $f(x) = \frac{\vert x - 1 \vert}{x - 1}$ | (II) $x = 2$ | | (C) $f(x) = \begin{cases} x - 1, & x < 2 \\ x + 1, & x \ge 2 \end{cases}$ | (III) $x = 0$ | | (D) $f(x) = \frac{1 - x}{(x - 4)}$ | (IV) $x = 1$ | Choose the **correct** answer from the options given below:

If $\frac{d}{dx}[ax^3 + ax^2 + ax + 1] = 9x^2 + 6x + 3$, then $a$ is equal to

If $e^y(x + 1) = 1$ and $\frac{d^2y}{dx^2} = k(\frac{dy}{dx})^2$, then k is equal to

If $y = \frac{1}{\sqrt[3]{1-x^3}}$ then $\frac{dy}{dx}$ is equal to

If $y = \sin^{-1}x$, then $(1-x^2)\frac{d^2y}{dx^2}$ is equal to

If $x = a\left(\cos t + \log \tan\frac{t}{2}\right), y = a\sin t$, then value of $\frac{dy}{dx}$ at $t = \frac{\pi}{4}$ is

If the function $f(x) = \begin{cases}\frac{\sin 3x}{x}, & \text{if } x \neq 0\\ \frac{3k}{2}, & \text{if } x = 0\end{cases}$ is continuous at $x = 0$, then the value of $k$ is

If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2} + 6y$ is equal to

The value of derivative of the function $\cot^{-1}\{(\cos 2x)^{1/2}\}$ at $x = \frac{\pi}{6}$ is

If $y = \left(x + \sqrt{x^2+1}\right)^m$, then $\frac{dy}{dx}$ is

If $f(x) = \begin{cases}\frac{1- \tan x}{4x-\pi}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4}\end{cases}$ is continuous at $x = \frac{\pi}{4}$, then the value of k is

If $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then $\frac{d^2y}{dx^2}$ is equal to

If $y = 5e^{2x} + 4e^{3x}$, then $\frac{d^2y}{dx^2}$ equals:

Let $y = \cos(\sin x^2)$, then the value of $\frac{dy}{dx}$ at $x = \frac{\sqrt{\pi}}{2}$ is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | Function f(x) | Points of Non-Differentiability | | --- | --- | | (A) $f(x) = \vert x\vert + 1$ | (I) Not differentiable at $x = 3$ only | | (B) $f(x) = \vert x - 3\vert $ | (II) Not differentiable at $x = -3$ only | | (C) $f(x) = \vert x + 3\vert $ | (III) Not differentiable at $x = 3, -3$ only | | (D) $f(x) = \vert x^2 - 9\vert $ | (IV) Not differentiable at $x = 0$ only | Choose the correct answer from the options given below:

For what value of $\alpha$, the function $f$ defined by $f(x) = \begin{cases} \alpha(x^2 - 2x + 1), & \text{if } x \leq 0 \\ 2x + 1, & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$?

If $xy + \frac{x^2}{y} = x^3y + y$, then $\frac{dy}{dx}$ is equal to

If $xy = e^{(x-y)}$, then $\frac{dy}{dx}$ is equal to:

$$\frac{d^2}{dx^2} \left\{ \det \begin{bmatrix} x^3 & x \\ 2 & e^x \end{bmatrix} \right\}$$ equals

If $x = a\sin 2t(1 + \cos 2t)$ and $y = b\cos 2t(1 - \cos 2t)$, then $(\frac{dy}{dx})_{\text{at } x=\frac{\pi}{4}}$ is equal to

If $f(x) = |x| + |x - 5|$, then which of the following statements are TRUE? (A) f is a continuous function every where (B) f is a continuous function except $x = 5$ and $x = 0$ (C) f is a continuous function except $x = 0$ but not differentiable at $x = 5$ (D) f is a continuous function everywhere but not differentiable at $x = 0$ and $x = 5$ Choose the correct answer from the options given below:

Differentiation of $\frac{x^3}{1 - x^3}$ with respect to $x^3$ is equal to:

If $x = \frac{a}{1 + t}$ and $y = \frac{a}{(1 + t)^2}$ where $a > 0$ , then $\frac{d^2y}{dx^2}$ at $t = 1$ is

If $y = (x+1)(x^2+1)(x^4+1)(x^8+1)$ then $\frac{dy}{dx}$ at $x = -1$ is

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{d^2y}{dx^2}$ equals.

If $y = \sqrt{\sin x + \sqrt{\sin x + \sqrt{\sin x + \cdots + \infty}}}$ then $\frac{dy}{dx}$ equals to

If $f(x) = \begin{cases} ax - 1 & {if } x \ > 1\\ \ 2x + 1 & {if } x < 1 \end{cases}$ is continuous at $x = 1$, then $a$ equals

$\frac{d}{dx}\left(e^{2\log_e x^3}\right)$ equals

Derivative of $x^x$ with respect to $x\log x$ is

If $x = a\sec^3 \theta$, $y = a \tan^3 \theta$, then $\frac{dy}{dx}$ at $ \theta = \frac{\pi}{3}$ is

The function $f(x) = \begin{cases} \frac{(\sin 2x)}{x} + \cos x & , if \ x \neq 0 \\ K & , if \ x = 0 \end{cases}$ is continuous at $x = 0$, then the value of K is:

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

If $x = e^t$ and $y = e^{2t}$ then $\frac{d^2y}{dx^2} =$

If the function $f(x) = \begin{cases} ax + 2, & x \leq 1 \\ x^2 + 3x + b, & x > 1 \end{cases}$ is differentiable at $x = 1$, then the value of $(2a + b)$ is

If $y = \log_e\left(\frac{e^2}{x^2}\right)$ for $x \neq 0$, then $\frac{d^2y}{dx^2}$ equals

Match List-I with List-II Where $\mathbb{R}$ is set of real numbers | List-I | List-II | |---|---| | (A) $\sin x$ is continuous on: | (I) $\mathbb{R} - \{0\}$ | | (B) $ \tan x$ is continuous on: | (II) $\mathbb{R}$ | | (C) $\cot x$ is continuous on: | (III) $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$ | | (D) $x^{-n}, n \in \mathbb{N}$ is continuous on: | (IV) $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$ | Choose the correct answer from the options given below:

Let $x = t^2, y = t^3$. Then $\frac{d^2y}{dx^2}$ is equal to

Which of the following functions $f(x)$ are differentiable at $x = 0$? (A) $|x|$ (B) $|x - 1|$ (C) $[x]$, where $[t]$ denotes the greatest integer $\leq t$ (D) $|x + 1|$ (E) $x^2$ Choose the correct answer from the options given below:

If $y = \sin^{-1} x + \sin^{-1} \sqrt{1-x^2}, x \in (-1, 0)$, then $\frac{dy}{dx}$ is equal to

Let $[x]$ denote the greatest integer $\leq t$ and $a \mathbb{Z} = [ax: x \in \mathbb{Z}, a \in \mathbb{R}]$ (where $\mathbb{Z}$ is set of integer and $\mathbb{R}$ is set of real number). The set of points of discontinuity of the function $f(x) = [2x]$ is given by

The derivative of $(\log x)^x$ with respect to $\log x$ is

If $x = t^{1/2}$, $y = t^{3/2}$, then $\frac{dy}{dx}$ =

If $f(x) = \begin{cases} mx + 1,\ x \geq \pi/2 \\sin x + n, x \leq \pi/2, & \end{cases}$ is continuous at $x = \pi/2$, where $m \in \mathbb{Z}$ (set of integers), then $\sin 2n =$

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

Differentiation of $\log[\log(\log x^5)]$ with respect to $x$ is

If $x = t^3$, $y = t^2$ then $\frac{d^2y}{dx^2}$ is equal to:

If $y = t - \frac{1}{t}$ and $x = t + \frac{1}{t}$, then $\frac{dy}{dx}$ is equal to

If $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$, where $|x| < 1, |y| < 1$ and $x ≠ y$, then

The greatest integer function $f(x) = [x]$ is differentiable for all values of

If $f(x) = \begin{cases} \frac{\tan(\frac{\pi}{4} - x)}{\cot 2x} & , x ≠ \frac{\pi}{4} \\ 2K + 1 & , x = \frac{\pi}{4} \end{cases}$ is continuous at $x = \frac{\pi}{4}$, then the value of K is equal to

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

If $y^{1/m} + y^{-1/m} = 2x$, then the value of $(x^2 - 1)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ is:

The value of k for which the function $f(x) = \begin{cases} \frac{1-\cos 8x}{16x^2}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases}$ is continuous at $x = 0$ is:

If $x = e^{\cos 2t}$, $y = e^{\sin 2t}$, then $\frac{dy}{dx}$ equals to

If $y = \sqrt{x + \sqrt{x + \sqrt{x + ...\ ...\ ...}}}$, then

If $y = (log x)^{(log x)}$, $x > 1$ then $\frac{dy}{dx}$ is equal to

If $g(x) = \begin{cases} \frac{αx}{|x|}, & \text{if } x < 0 \\ 5, & \text{if } x ≥ 0 \end{cases}$ is continuous at x = 0, then the value of α is

If $y = e^{acos^{-1}x}, -1 < x < 1$, then $(1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx}$ is equal to

Match List-I with List-II | List-I | List-II | | --- | --- | | (A) $f(x) = x \sin x$ | (I) is not continuous at $x = -3$ | | (B) $f(x) = \frac{\vert x\vert }{x}, x \neq 0$ and $f(x) = 1 \text{ at } x = 0$ | (II) is continuous everywhere | | (C) $f(x) = x - [x]$, $[x]$ denotes greatest integer function | (III) is not differentiable at $x = 1$ | | (D) $f(x) = e^{\vert x - 1\vert }$ | (IV) is not continuous at $x = 0$ | Choose the correct answer from the options given below:

If $y = \left(\log\left(x + \sqrt{x^2+a^2}\right)\right)^2$ and $x \neq \frac{1-a^2}{2}$, then $(x^2+a^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ is equal to:

Match List-I with List-II | List-I | List-II | |---|---| | (Parametric equations) | $\left(\frac{dy}{dx}\right)$ | | (A) $x = \frac{2}{t}, y = 2t$ | (I) $4t^2$ | | (B) $x = t^3, y = 3t + 2$ | (II) $2(t+1)$ | | (C) $x = \log t, y = 2t^2$ | (III) $-t^2$ | | (D) $x = e^t, y = 2te^t$ | (IV) $t^{-2}$ | Choose the correct answer from the options given below:

For $x > y > 0$, if $x^5 y^6 = (x + y)^{11}$, then $\frac{d^2y}{dx^2}$ is

If $x = a\cos\alpha + b\sin\alpha$ and $y = a\sin\alpha - b\cos\alpha$, then $\left(x\frac{dy}{dx} - y^2\frac{d^2y}{dx^2}\right)$ is equal to:

The value of k for which the function, defined by, $f(x) = \begin{cases} \frac{3x + 4 \tan x}{x} & : x \neq 0 \\ k & : x = 0 \end{cases}$ is continuous at $x = 0$, is

Which of the following functions $f(x)$ are differentiable at $x = 0$? (A) $|x|$ (B) $|x - 1|$ (C) $|\sin x|$ (D) $|\cos x|$ (E) $x^2$ Choose the correct answer from the options given below:

If $e^y(x + 1) = 1$, then

If $x = 4t$ and $y = \frac{4}{t}$, then $\frac{d^2y}{dx^2}$ is

If the function defined by $f(x) = \begin{cases} \ kx^2 + 1, & \text{if } x \le 1 \\ 2 , & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$, then k is equal to

The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \begin{cases} x^2, & x \ge 1 \\ x, & x < 1 \end{cases}$ is

If $y = \frac{1}{1+x^{b-a}+x^{c-a}} + \frac{1}{1+x^{c-b}+x^{a-b}} + \frac{1}{1+x^{a-c}+x^{b-c}}$ then $\frac{d^2y}{dx^2}$ is

Match List-I with List-II | List-I | List-II | | --- | --- | | Function | Derivative | | --- | --- | | (A) $y = \sin^{-1} x + \sin^{-1} \sqrt{1 - x^2}; \vert x\vert < 1$ | (I) $\frac{dy}{dx} = \frac{1}{2y-1}$ | | (B) $y = \sqrt{x + y}, x+y > 0 \text{ and } y \neq \frac{1}{2}$ | (II) $\frac{dy}{dx} = 10^x \log_e 10$ | | (C) $y = \log_{10} x, x > 0$ | (III) $\frac{dy}{dx} = 0$ | | (D) $y = 10^x$ | (IV) $\frac{dy}{dx} = \frac{1}{x \log_e 10}$ | Choose the correct answer from the options given below:

Match **List-I** with **List-II**. Here [x] denotes the greatest integer function $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) } f(x) = [x] & \text{(I) is continuous everywhere but not differentiable at } x=-1 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) } f(x) = |x-1| & \text{(II) is continuous everywhere except at all integral values} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) } f(x) = e^{|x|} & \text{(III) is continuous everywhere but not differentiable at } x=1 \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) } f(x) = |x+1| & \text{(IV) is continuous everywhere but not differentiable at } x=0 \\[1.2ex] \hline \end{array}$ Choose the **correct** answer from the options given below:

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

If $x^2 - y^2 = t - \frac{1}{t}$, and $x^4 + y^4 = t^2 + \frac{1}{t^2}$, then which of the following is correct?

If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2} + 6y$ is equal to

Match List-I with List-II $\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) } f(x) = |x| & \text{(I) Not differentiable at } x=-2 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) } f(x) = |x+2| & \text{(II) Not differentiable at } x=0 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) } f(x) = |x^2-4| & \text{(III) Not differentiable at } x=2 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) } f(x) = |x-2| & \text{(IV) Not differentiable at } x=2,-2 \text{ only} \\[1.2ex] \hline \end{array}$ Choose the correct answer from the options given below:

Let $y = \sin(\cos x^2)$, then the value of $\frac{dy}{dx}$ at $x = \frac{\sqrt{\pi}}{2}$ is equal to

If the function $f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x} & : x \neq \frac{\pi}{2} \\ 3 & : x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, then $k$ is equal to

If $e^y = \log x$, then which of the following is true?

Let $e^y(x+1) = 1$. Then which of the following are TRUE? (A) $\frac{d^2y}{dx^2} = -\frac{1}{(x+1)^2}$ (B) $\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$ (C) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = -1$ (D) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = 1$ (E) $\left.\frac{d^2y}{dx^2}\right|_{x=1} = \frac{1}{4}$ Choose the correct answer from the options given below:

Let $f(x)=\begin{cases}\dfrac{|x|}{x},&x\ne0\\1,&x=0\end{cases}$ and $g(x)=\begin{cases}x\sin\left(\dfrac{1}{x}\right),&x\ne0\\0,&x=0\end{cases}$ Then at the origin, which one of the following is true?

If $x = \frac{1-t}{1+t}$ and $y = \frac{3t}{1+t}$, then $\frac{d^2y}{dx^2}$ is equal to

If $y = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$, then $\frac{dy}{dx}$ is equal to

If $f(x) = x^3 e^{-x}$, then the value of $f''(1)$ is equal to

If $x^2 - y^2 = 1$, then which of the following is correct? (A) $(x^2 - 1)\left(\frac{dy}{dx}\right)^2 = x^2$ (B) $(x^2 - 1)\left(\frac{d^2y}{dx^2}\right)^2 = x^2$ (C) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = x^2$ (D) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = 1$ Choose the correct answer from the options given below:

If $f(x) = x^3\log_e x$, Then $f''(e^2)$ is equal to

Match **List-I** with **List-II** | List-I | List-II | | :--- | :--- | | **Function** | **Property** | | (A) $f(x) = \begin{cases} \frac{x}{\vert x \vert} & : x \neq 0 \\ 0 & : x = 0 \end{cases}$ | (I) continuous but not differentiable at $x= 0$ | | (B) $f(x) = \vert x \vert$ | (II) continuous but not differentiable at $x=1$ | | (C) $f(x) = \vert x^2 - 1 \vert$ | (III) discontinuous at $x = 0$ | | (D) $f(x) = \vert x - 1 \vert$ | (IV) continuous but not differentiable at $x = 1, -1$ | Choose the **correct** answer from the options given below:

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

The function $f(x) = [x]$, where $[x]$ denotes the greatest integer function, is continuous at $x =$ (A) 2.9 (B) 5 (C) -3 (D) 6.5 Choose the correct answer from the options given below:

If $y = \sqrt{2024x + 2025}$, then which of the following is correct?

Match List-I with List-II | List-I | List-II | |---|---| | (Function) | (Derivative with respect to 'x') | | (A) $f(x) = x^x$ | (I) $ax^{a-1}$ | | (B) $f(x) = a^x$ | (II) 0 | | (C) $f(x) = a^a$ | (III) $a^x log_e a$ | | (D) $f(x) = x^a$ | (IV) $x^x(1 + log_e x)$ | Choose the correct answer from the options given below:

If $t=e^{2 x}$ and $y=\log _{e} t^{2}$, then $\frac{d^{2} y}{d x^{2}}$ is :

The second order derivative of which of the following functions is $5^{\mathrm{x}}$ ?

Let $[x]$ denote the greatest integer function. Then match List-I with List-II:</p> | List-I | List-II | | --- | --- | | (A) $ \vert x - 1\vert + \vert x - 2\vert $ | (I) is differentiable everywhere except at $ x = 0 $ | | (B) $ x - \vert x\vert $ | (II) is continuous everywhere | | (C) $ x - [x] $ | (III) is not differentiable at $ x = 1 $ | | (D) $ x \, \vert x\vert $ | (IV) is differentiable at $ x = 1 $ |

If $f(x)$, defined by $f(x)=\left\{\begin{array}{lll}k x+1 & \text { if } & x \leq \pi \\ \cos x & \text { if } & x>\pi\end{array}\right.$ is continuous at $x=\pi$, then the value of $k$ is :

If $\mathrm{e}^{\mathrm{y}}=\mathrm{x}^{\mathrm{x}}$, then which of the following is true ?

The equation of the tangent to the curve $\mathrm{x}^{\frac{5}{2}}+\mathrm{y}^{\frac{5}{2}}=33$ at the point $(1,4)$ is :

If $y = x^x$, $\frac{dy}{dx}$ will be:

If $y = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ then $\frac{dy}{dx} =$

Points of discontinuity of the greatest integer function $f(x) = [x]$, where $[x]$ denotes integer less than or equal to $x$, are

If $y = x^{(x \sin x)}$ then $\frac{dy}{dx} = ?$

If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then which of the following can be a discontinuous function ?

If $f(x) = \begin{cases} ax^2 + b, & x < -1 \\ bx^2 + ax + 4, & x \geq -1 \end{cases}$ is everywhere differentiable, then :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $x = 2at^2, y = at^4$ | (I) | Inverse trignometric function | | (B) | $f(x) = (2x + 3)^3$ | (II) | Implicit function | | (C) | $xy + y^2 = \tan(x + y)$ | (III) | Parametric function | | (D) | $y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right), -\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}$ | (IV) | Composite function | Choose the **correct** answer from the options given below :

Let $y = \log_e \left(\frac{a + b \sin x}{a - b \sin x}\right)$, then value of $\frac{dy}{dx}$ is :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | $y = \log(\sin x)$ | (I) | $\frac{d^2y}{dx^2} = -\frac{1}{x^2}$ | | (B) | $y = e^{(1 + \log x)}$ | (II) | $\frac{d^2y}{dx^2} = 2$ | | (C) | $y = \log\lvert x \rvert$ | (III) | $\frac{d^2y}{dx^2} = 0$ | | (D) | $y = x^2 + 4x - 1$ | (IV) | $\frac{d^2y}{dx^2} = -\csc^2 x$ | Choose the **correct** answer from the options given below :

The given function $f(x) = [x]$ is discontinuous at :

If $x = a\left(t - \frac{1}{t}\right)$, $y = b\left(t + \frac{1}{t}\right)$, then $\frac{dy}{dx} =$

Which of the following statements are correct ? (A) If $f : R \to R$ then $f(x) = |x|$ is continuous everywhere. (B) If $f : R \to R$ then $f(x) = |x|$ is continuous everywhere but not differentiable at $x = 0$. (C) Let $f : R - \{0\} \to R$ then $f(x) = \frac{1}{x}$ is continuous everywhere. (D) Let $f : R \to R$ then $f(x) = |x - 1| + |x - 2|$ is continuous everywhere but not differentiable at exactly 2 points. (E) If $f : R \to R$ then $f(x) = \cot x$ is continuous everywhere. Choose the correct answer from the options given below :

If $f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$, then k is :

The derivative of $\sec(\tan \sqrt{x})$ with respect to x is :

If $y = \log\left[\frac{x^2}{e^2}\right]$ then value of $\frac{d^2y}{dx^2}$ is :

The condition on a and b, such that for $y = \frac{a}{x} - \frac{b}{x^2}$, $\frac{dy}{dx} = 0$ at $x=1$ is :

The points of discontinuity of the function f defined by $f(x) = \begin{cases} x+2 & x \leq 1 \\ x-2 & 1 < x < 2 \\ 0 & x \geq 2 \end{cases}$ are :

If $\cos y = x\cos(a+y)$, then $\frac{dy}{dx} = $

If $f(x) = \begin{cases} \frac{x^2 - 9}{x - 3}, & x \neq 3 \\ 5, & x = 3 \end{cases}$ then $f(x)$ :

If $y = x^x$, then value of $\frac{dy}{dx}$ at $x = 2$ is :

A function $f(x)$ is defined by : $f(x) = \begin{cases} x + 2, & \text{if } x < 0 \\ -x + 2, & \text{if } x > 0 \end{cases}$ Which of the following is true ?

If $y = 500 e^{7x} + 600 e^{-7x}$ and $\frac{d^2 y}{dx^2} = ky$, then the value of k is :

If $y = x^{\sin x}$, then value of $x^{-\sin x} \frac{dy}{dx} - \cos x \log x$ is :

$f(x) = [x]$, where $[\,]$ represents greatest integer function. (A) For $2 \leq x < 3$, $[x] = 3$ (B) For $2 \leq x < 3$, $[x] = 2$ (C) Right hand derivative of f at $x = 2$ is not defined (D) Left hand derivative of f at $x = 2$ is zero (E) $f(x)$ is not differentiable at $x = 2$ Choose the correct answer from the options given below :

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

Let $y = \log(x + \sqrt{x^2+1})$, and $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} = c$. Then identify the correct statements about the values of $a$, $b$ and $c$ : (A) $a = 1 + x^2$ (B) $b = 0$ (C) $c = 0$ (D) $b = x$ (E) $c = 2$ Choose the correct answer from the options given below :

If $\sqrt{y+x} + \sqrt{y-x} = a$, $a > 1$, $\frac{d^2y}{dx^2}$ is equal to :

If $x = \int_0^y \frac{dt}{\sqrt{1+9t^2}}$ and $\frac{d^2y}{dx^2} = \lambda y$, then, $\lambda$ is equal to

The value of k, for which the function $f(x) = \begin{cases} \frac{\sin kx}{x} + 3\cos x, & x \neq 0 \\ 7, & x = 0 \end{cases}$ is continuous at $x = 0$, is

If $f$ is a function defined by $f(x) = \begin{cases} 5x^2 - x + 3, & x < 1 \\ 3x + 4, & x \geq 1 \end{cases}$, then, at $x = 1$, $f$ is

If $y = \frac{\log_e x}{x}$, then $\frac{d^2y}{dx^2} =$

If $y = \frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{x+1} - \sqrt{x-1}}$, then $(x^2-1)^{3/2} \frac{d^2y}{dx^2} =$

If $f(x) = \begin{cases} \frac{\tan(\pi/4 - x)}{\cot 2x}, & x \neq \pi/4 \\ k, & x = \pi/4 \end{cases}$ is continuous at $x = \pi/4$, then the value of k is

The derivative of $\sin^{-1}\left(\frac{2x}{1+x^2}\right)$ w.r.t. $\tan^{-1}\left(\frac{2x}{1-x^2}\right)$ is

Derivative of $x^3 + 1$ with respect to $x^2 + 1$ is

If $x = 2\sin\theta$ and $y = 2\cos\theta$, then the value of $\frac{d^2y}{dx^2}$ at $\theta = 0$ is

If $x = e^{y + e^{y + e^{y + \ldots \infty}}}$, $x > 0$, then $\frac{dy}{dx}$ is equal to

The function $f(x) = e^{|x|}$ is (a) continuous everywhere on $R$ (b) not continuous at $x = 0$ (c) Differentiable everywhere on $R$ (d) not differentiable at $x = 0$ (e) continuous and differentiable on $R$ Choose the most appropriate answer from the options given below :

Let $y = m\sin rx + n\cos rx$. What is the value of $\frac{d^2y}{dx^2}$?

Match List I with List II | List I | List II | |---|---| | A. If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then $\frac{g(x)}{f(x)}$ is | I. discontinuous at exactly three points. | | B. The function $f(x) = \frac{4-x^2}{4x-x^3}$ is | II. continuous everywhere | | C. The function $f(x) = \lvert x \rvert + \lvert x-1 \rvert$ is | III. discontinuous at $x = 0$. | | D. The function $f(x) = \lvert \sin x \rvert$ is | IV. continuous at $x = 0$ and $x = 1$ | Choose the correct answer from the options given below:

The function $f(x) = |x - 1|$ is

If $x = at^2$ and $y = 2at$, then the value of $\frac{d^2y}{dx^2}$ is ( where t is a parameter )

If $y = \frac{1}{x+1}$, then $\frac{d^2y}{dx^2}$ at $x = 2$ is:

The derivative of $\sin(\tan^{-1} e^{2x})$ with respect to $x$ is:

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

The function $f(x) = \frac{x-1}{x(x^2-1)}, x \neq 1, f(1) = 1$, is discontinuous at