GATE Overflow for GATE IN - Questions without answers in Calculus

GATE IN 2026 | Question: 2

Mon, 23 Feb 2026 13:37:56 +0000

Which one among the following statements is true for the function $f(x)=x\mid x \mid $?

$f(x)$ is discontinuous but not differentiable at $x=0$.
$f(x)$ is discontinuous and differentiable at $x=0$.
$f(x)$ is continuous but not differentiable at $x=0$.
$f(x)$ is continuous and differentiable at $x=0$.

GATE IN 2025 | GA | Question: 3

Thu, 06 Mar 2025 17:23:09 +0000

If $P e^{x}=Q e^{-x}$ for all real values of $x$, which one of the following statements is true?

$P=Q=0$
$P=Q=1$
$P=1 ; Q=-1$
$\frac{P}{Q}=0$

GATE IN 2025 | Question: 26

Thu, 06 Mar 2025 17:18:03 +0000

The value of the surface integral $\iint_{S}(2 x+z) \mathrm{d} y \mathrm{~d} z+(2 x+z) \mathrm{d} x \mathrm{~d} z+(2 z+y) \mathrm{d} x \mathrm{~d} y$ over the sphere $S: x^{2}+y^{2}+z^{2}=9$ is

$72 \pi$
$144 \pi$
$36 \pi$
$432 \pi$

GATE IN 2025 | Question: 28

Thu, 06 Mar 2025 17:17:59 +0000

The value of the integral $\int_{-\pi}^{\pi}\left(\cos ^{6} x+\cos ^{4} x\right) \mathrm{d} x$ is

$\frac{\pi}{2}$
$\frac{5 \pi}{8}$
$\frac{11 \pi}{8}$
$\frac{9 \pi}{8}$

GATE IN 2023 | Question: 13

Mon, 22 May 2023 00:14:48 +0000

What is $\lim _{x \rightarrow 0} f(x)$, where $f(x)=x \sin \frac{1}{x} ?$

$0$
$1$
$\infty$
Limit does not exist

GATE IN 2022 | Question: 18

Sun, 20 Mar 2022 17:33:58 +0000

The global minimum of $x^{3} e^{-|x|}$ for $x \in (-\infty, \infty)$ occurs at $x = $ _________ (round off to one decimal place)

GATE IN 2021 | Question: 2

Fri, 19 Feb 2021 15:48:41 +0000

Consider the sequence $x_{n}=0.5x_{n-1}+1,n=1,2, \dots \:\dots$ with $x_0=0$ . Then $\displaystyle \lim_{n\rightarrow \infty} x_n$ is

$0$
$1$
$2$
$\infty$

GATE2020 IN: 28

Tue, 03 Nov 2020 14:26:44 +0000

A straight line drawn on an x-y plane intercepts the x-axis at -0.5 and the y-axis at 1.

The equation that describes this line is __________.

$\text{y=-0.5x+1}$
$\text{y=x-0.5}$
$\text{y=0.5x-1}$
$\text{y=2x+1}$

GATE2020 IN: 31

Tue, 03 Nov 2020 14:26:43 +0000

Consider the function $f(x,y)=x^2+y^2.$ The minimum value the function attains on the line $x+y=1$ (rounded off to one decimal place) is __________.

GATE2020: 1

Tue, 03 Nov 2020 04:24:37 +0000

The unit vectors along the mutually perpendicular x,y and z axes are $\hat{i},\;\hat{j}\; and \;\hat{k}$ respectively. Consider the plane $z=0$ and two vectors $\overrightarrow{a} and\;\overrightarrow{b}$ on that plane such that $\overrightarrow{a}\neq \alpha \overrightarrow{b}$ for any scalar $\alpha$. A vector perpendicular to both $\overrightarrow{a} and\;\overrightarrow{b}$ is ___________

$\hat{k}$
$\hat{i}-\hat{j}$
$-\hat{j}$
$\hat{i}$

GATE IN 2017 | Question: 27

Mon, 02 Nov 2020 00:00:21 +0000

The angle between two vectors $X_1=\begin{bmatrix}2 & 6 & 14\end{bmatrix}^T$ and $X_2=\begin{bmatrix}-12 & 8 & 16\end{bmatrix}^T$ in radian is __________.

GATE 2016 - 1

Mon, 26 Mar 2018 18:43:26 +0000

A straight line of the form $y=mx+c$ passes through the origin and the point $(x,y)$ =$(2,6)$. The value of $m$ is _____.

GATE2016-2

Mon, 26 Mar 2018 18:43:25 +0000

$\underset{n\rightarrow \infty }{\lim}\:\left(\sqrt{n^2+n}-\sqrt{n^2+1}\right)$ is __________.

GATE2016-4

Mon, 26 Mar 2018 18:43:25 +0000

The vector that is $NOT$ perpendicular to the vectors $(i+j+k)$ and $(i+2j+3k)$ is ______.

$(i-2j+k)$
$(-i+2j-k)$
$(0i+0j+0k)$
$(4i+3j+5k)$

GATE2016-26

Mon, 26 Mar 2018 18:43:22 +0000

Let $f: [-1,1]\rightarrow \mathbb{R}$, where $f(x)=2x^3-x^4-10$. The minimum value of $f(x)$ is $\_\_\_\_\_\_\_.$

GATE2015-13

Mon, 26 Mar 2018 18:32:03 +0000

The double integral $\int_0^a \int_0^y f(x,y) dx\;dy$ is equivalent to

$\int_0^x\int_0^y f(x,y) dx\;dy$
$\int_0^a \int_x^y f(x,y) dx\;dy$
$\int_0^a \int_x^a f(x,y) dy\;dx$
$\int_0^a \int_0^a f(x,y) dx\;dy$

GATE2015-14

Mon, 26 Mar 2018 18:32:03 +0000

The magnitude of the directional derivative of the function $f(x,y)=x^2+3y^2$ in a direction normal to the circle $x^2+y^2=2,$ at the point $(1,1),$ is

$4\sqrt{2}$
$5\sqrt{2}$
$7\sqrt{2}$
$9\sqrt{2}$

GATE2015-57

Mon, 26 Mar 2018 18:31:56 +0000

The fundamental period of the signal $x(t)=2\cos(\frac{2\pi t}{3})+\cos(\pi t)$, in seconds, is ________________ s.

GATE2014-4

Sun, 25 Mar 2018 23:10:51 +0000

A vector is defined as

$$f=y\hat{i}+x\hat{j}+z\hat{k}$$

where $\hat{i},\hat{j},\hat{k}$ are unit vectors in Cartesian $(x,y,z)$ coordinate system.

The surface integral $f.ds$ over the closed surface S of a cube with vertices having the following coordinates:

$(0,0,0), (1,0,0),(1,1,0),(0,1,0),(0,0,1),(1,0,1),(1,1,1),(0,1,1)$ is __________.

GATE IN 2013 | Question: 5

Sun, 25 Mar 2018 19:30:29 +0000

For a vector $E$, which one of the following statements is $\text{NOT TRUE}$?

If $\Delta.E=0,\;E$ is called solenoidal
If $\Delta \times E=0,\;E$ is called conservative
If $\Delta\times E=0,\;E$ is called irrotational
If $\Delta.E=0,\;E$ is called irrotational

GATE IN 2012 | Question: 29

Sun, 25 Mar 2018 09:53:04 +0000

The maximum value of $f(x)=x^3-9x^2+24x+5$ in the interval $[1, 6]$ is

$21$
$25$
$41$
$46$

GATE IN 2012 | Question: 28

Sun, 25 Mar 2018 09:53:04 +0000

The direction of vector $\text{A}$ is radially outward from the origin, with $|A|=kr^n$ where $r^2=x^2+y^2+z^2$ and $k$ is a constant. The value of $n$ for which $\nabla.$ $\text{A=0}$ is

$-2$
$2$
$1$
$0$

GATE IN 2012 | Question: 1

Sun, 25 Mar 2018 09:52:59 +0000

If $x=\sqrt{-1},$ then the value of $x^x$ is

$e^{-\pi/2}$
$e^{\pi/2}$
$x$
$1$

GATE2018IN: 26

Tue, 20 Feb 2018 12:03:44 +0000

Given $\overrightarrow{F}$ = (x$^2$ – 2y) $\overrightarrow{i}$ – 4xz$\overrightarrow{j}$ + 4xz$^2$$\overrightarrow{k}$, the value of the linear integral $\int_c$$\overrightarrow{F}$ . d$\overrightarrow{l}$ along the straight line c from (0,0,0) to (1,1,1) is

3/16
0
-5/12
-1

GATE2018IN: 4

Tue, 20 Feb 2018 12:03:42 +0000

Consider two functions f(x) = (x – 2)$^2$ and g(x) = 2x – 1, where x is real. The smallest value of x for which f(x) = g(x) is $\_\_\_\_\_\_$