Calculus

Continuity & Differentiability

Easy

common

The derivative of $|x|$ with respect to $x$ , where $x \in R \sim \{0\}$ is

$\frac{x}{|x|}$

-1

$\frac{1}{x}$

✅ Correct Option: 2

For $x>0$ , $|x|=x$ and the derivative is 1; for $x<0$ , $|x|=-x$ and the derivative is $-1$ . Both cases are expressed compactly as $\frac{x}{|x|}$ .

2022: 25 May Shift 1

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

2026: 31 May Shift 2

Match the LIST-I with LIST-II

	LIST-I Function $f(x)$		LIST-II Second order derivative of $f(x)$ w.r.t $'x'$
A.	$x^2 \log_e x$	I.	$e^x(2 + 4x + x^2)$
B.	$e^x \log_e x$	II.	$e^x\left(\log_e x + \dfrac{2}{x} - \dfrac{1}{x^2}\right)$
C.	$x^2 e^x$	III.	$2\log_e x + 3$
D.	$x^{-2} e^x$	IV.	$\dfrac{e^x}{x^2}\left(\dfrac{6}{x^2} - \dfrac{4}{x} + 1\right)$

Choose the correct answer from the options given below:

2022: 6 Aug Shift 2

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

2025: 15 May Shift 1

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

2022: 4 Aug Shift 1

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

2025: 22 May Shift 2

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:

The derivative of ∣x∣|x|∣x∣ with respect to xxx, where x∈R∼{0}x \in R \sim \{0\}x∈R∼{0} is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The derivative of $|x|$ with respect to $x$ , where $x \in R \sim \{0\}$ is