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Recent questions and answers in Calculus
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1
GATE Civil 2021 Set 2  Question: 1
The value of $\lim \limits_{x\rightarrow \infty } \dfrac{x \:\text{ln}\left ( x \right )}{1+x^{2}}$ is $0$ $1.0$ $0.5$ $\infty$
asked
Mar 1
in
Calculus
by
jothee
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5.2k
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gatecivil2021set2
calculus
limits
0
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0
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2
GATE Civil 2021 Set 2  Question: 3
The unit normal vector to the surface $X^{2} + Y^{2} + Z^{2} – 48 = 0$ at the point $(4, 4, 4)$ is $\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$ $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$ $\frac{2}{\sqrt{2}},\frac{2}{\sqrt{2}},\frac{2}{\sqrt{2}}$ $\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5}}$
asked
Mar 1
in
Calculus
by
jothee
(
5.2k
points)
gatecivil2021set2
calculus
vectorcalculus
vectoridentities
unitnormalvector
0
votes
0
answers
3
GATE Civil 2021 Set 2  Question: 18
The value ($\textit{round off to one decimal place}$) of $\int_{1}^{1}x\:e^{\left  x \right }dx$ is ______________
asked
Mar 1
in
Calculus
by
jothee
(
5.2k
points)
gatecivil2021set2
numericalanswers
calculus
definiteintegral
0
votes
0
answers
4
GATE Civil 2021 Set 2  Question: 36
A function is defined in Cartesian coordinate system as $f(x,y)=xe^{y}$. The value of the directional derivative of the function ($\textit{in integer}$) at the point $(2,0)$ along the direction of the straight line segment from point $(2, 0)$ to point $\left ( \dfrac{1}{2} ,2\right )$ is _________
asked
Mar 1
in
Calculus
by
jothee
(
5.2k
points)
gatecivil2021set2
numericalanswers
calculus
directionalderivatives
0
votes
0
answers
5
GATE Civil 2021 Set 1  Question: 18
Consider the limit: $\lim_{x\rightarrow 1}\left ( \frac{1}{\text{ln}\:x}  \frac{1}{x1}\right )$ The limit (correct up to one decimal place) is _____________
asked
Feb 20
in
Calculus
by
Arjun
(
8.7k
points)
gatecivil2021set1
numericalanswers
calculus
limits
0
votes
0
answers
6
GATE Civil 2021 Set 1  Question: 19
The volume determined from $\int \int \int _{V}\:8\:xyz\:dV$ for $V=\left [ 2,3 \right ]\times \left [ 1,2 \right ]\times \left [ 0,1 \right ]$ will be ($\textit{in integer}$) _____________
asked
Feb 20
in
Calculus
by
Arjun
(
8.7k
points)
gatecivil2021set1
numericalanswers
calculus
trippleintegrals
volume
0
votes
1
answer
7
GATE2020CE12
The value of $\displaystyle{} \lim_{x\to\infty}\dfrac{x^25x+4}{4x^2+2x}$ is $0 \\$ $\dfrac{1}{4} \\$ $\dfrac{1}{2} \\$ $1$
answered
Feb 15
in
Calculus
by
ijnuhb
(
140
points)
gate2020ce1
calculus
limits
0
votes
0
answers
8
GATE2020CE14
The area of an ellipse represented by an equation $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ is $\dfrac{\pi ab}{4} \\$ $\dfrac{\pi ab}{2} \\$ $\pi ab \\$ $\dfrac{4\pi ab}{3}$
asked
Feb 28, 2020
in
Calculus
by
jothee
(
5.2k
points)
gate2020ce1
calculus
curves
areaundercurve
0
votes
0
answers
9
GATE2020CE127
A continuous function $f(x)$ is defined. If the third derivative at $x_i$ is to be computed by using the fourth order central finitedivideddifference scheme (with step length $=h$ ...
asked
Feb 28, 2020
in
Calculus
by
jothee
(
5.2k
points)
gate2020ce1
calculus
derivatives
continuousfunction
0
votes
0
answers
10
GATE2020CE139
If $C$ represents a line segment between $(0,0,0)$ and $(1,1,1)$ in Cartesian coordinate system, the value (expressed as integer) of the line integral $\int_C [(y+z)dx+(x+z)dy+(x+y)dz] $ is ______
asked
Feb 28, 2020
in
Calculus
by
jothee
(
5.2k
points)
gate2020ce1
calculus
lineintegral
0
votes
0
answers
11
GATE2020 CE22
The value of $\lim_{x\to\infty}\dfrac{\sqrt{9x^2+2020}}{x+7}\:\text{is}$ $\dfrac{7}{9}$ $1$ $3$ indeterminable
asked
Feb 13, 2020
in
Calculus
by
jothee
(
5.2k
points)
gate2020ce2
calculus
limits
0
votes
0
answers
12
GATE2020 CE224
Velocity distribution in a boundary layer is given by $\dfrac{u}{U_\infty} = \sin\large \left( \dfrac{\pi}{2}\dfrac{y}{\delta} \right)$, where $u$ is the velocity at vertical coordinate $y,\: U_\infty$ is the free stream velocity and $\delta$ is the boundary layer ... $\ s^{1}$, round off to two decimal places) at $y = 0$, is ________.
asked
Feb 13, 2020
in
Calculus
by
jothee
(
5.2k
points)
gate2020ce2
numericalanswers
calculus
gradient
+2
votes
1
answer
13
GATE2019 CE1: 1
Which one of the following is correct? $\displaystyle{} \lim_{x\rightarrow 0} \left( \dfrac{\sin4x}{\sin2x}\right)=2\;\text{and}\: \lim_{x\rightarrow 0} \left( \dfrac{\tan x}{x}\right)=1$ ...
answered
Jan 17, 2020
in
Calculus
by
KUSHAGRA गुप्ता
(
140
points)
gate2019ce1
calculus
limits
0
votes
0
answers
14
GATE2017 CE22
Let $w=f(x,y)$, where $x$ and $y$ are functions of $t$. Then, according to the chain rule, $\dfrac{dw}{dt}$ is equal to $\dfrac{dw}{dx} \dfrac{dx}{dt} + \dfrac{dw}{dy} \dfrac{dt}{dt} \\$ ... $\dfrac{d w}{dx} \dfrac{\partial x}{\partial t} + \dfrac{dw}{dy} \dfrac{\partial y}{ \partial t}$
asked
Aug 7, 2019
in
Calculus
by
gatecse
(
4k
points)
gate2017ce2
calculus
partialderivatives
0
votes
0
answers
15
GATE2017 CE219
The divergence of the vector field $V=x^2 i + 2y^3 j + z^4 k$ at $x=1, \: y=2, \: z=3$ is ________
asked
Aug 7, 2019
in
Calculus
by
gatecse
(
4k
points)
gate2017ce2
calculus
vectorcalculus
divergence
numericalanswers
0
votes
0
answers
16
GATE2017 CE226
The tangent to the curve represented by $y=x \text{ ln }x$ is required to have $45^{\circ}$ inclination with the $x$axis. The coordinates of the tangent point would be $(1,0)$ $(0, 1)$ $(1,1)$ $(\sqrt{2}, (\sqrt{2})$
asked
Aug 7, 2019
in
Calculus
by
gatecse
(
4k
points)
gate2017ce2
calculus
curves
0
votes
0
answers
17
GATE2017 CE227
Consider the following definite integral: $I= \displaystyle{} \int_0^1 \dfrac{(\sin ^{1}x)^2}{\sqrt{1x^2}} dx$. The value of the integral is $\dfrac{\pi ^3}{24} \\$ $\dfrac{\pi ^3}{12} \\$ $\dfrac{\pi ^3}{48} \\$ $\dfrac{\pi ^3}{64}$
asked
Aug 7, 2019
in
Calculus
by
gatecse
(
4k
points)
gate2017ce2
calculus
definiteintegral
+1
vote
0
answers
18
GATE2019 CE1: 4
For a small value of $h$, the Taylor series expansion for $f(x+h)$ is $f(x)+h{f}' (x) + \dfrac{h^2}{2!}{f}''(x) + \dfrac{h^3}{3!}{f}'''(x)+\dots \infty \\$ $f(x)h{f}' (x) + \dfrac{h^2}{2!}{f}''(x)  \dfrac{h^3}{3!}{f}'''(x)+ \dots \infty \\$ ... $f(x)h{f}' (x) + \dfrac{h^2}{2}{f}''(x)  \dfrac{h^3}{3}{f}'''(x)+ \dots \infty $
asked
Feb 14, 2019
in
Calculus
by
Arjun
(
8.7k
points)
gate2019ce1
calculus
taylorseries
0
votes
0
answers
19
GATE2019 CE1: 26
Which one of the following is NOT a correct statement? The function $\sqrt[x]{x}, \: (x>0)$, has the global maxima at $x=e$ The function $\sqrt[x]{x}, \: (x>0)$, has the global minima at $x=e$ The function $x^3$ has neither global minima nor global maxima The function $\mid x \mid$ has the global minima at $x=0$
asked
Feb 14, 2019
in
Calculus
by
Arjun
(
8.7k
points)
gate2019ce1
calculus
maximaminima
0
votes
0
answers
20
GATE2019 CE1: 30
Consider two functions: $x=\psi \text{ ln } \phi$ and $y= \phi \text{ ln } \psi$. Which one of the following is the correct expression for $\frac{\partial \psi}{\partial x}$? $\dfrac{x \: \text{ln } \psi}{\text{ln } \phi \text{ ln } \psi 1} \\$ ... $\dfrac{\: \text{ln } \psi}{\text{ln } \phi \text{ ln } \psi 1}$
asked
Feb 14, 2019
in
Calculus
by
Arjun
(
8.7k
points)
gate2019ce1
calculus
partialderivatives
0
votes
0
answers
21
GATE2019 CE2: 3
The following inequality is true for all $x$ close to $0$. $2\dfrac{x^2}{3} < \dfrac{x \sin x}{1 \cos x} <2$ What is the value of $\underset{x \to 0}{\lim} \dfrac{x \sin x}{1 – \cos x}$? $0$ $1/2$ $1$ $2$
asked
Feb 12, 2019
in
Calculus
by
Arjun
(
8.7k
points)
gate2019ce2
calculus
limits
0
votes
0
answers
22
GATE2019 CE2: 4
What is curl of the vector field $2x^2y \textbf{i} + 5z^2 \textbf{j} – 4yz \textbf{k}$? $6z \textbf{i} + 4x \textbf{j} – 2x^2 \textbf{k}$ $6z \textbf{i}  8xy \textbf{j} + 2x^2 y\textbf{k}$ $ 14 z \textbf{i} + 6y \textbf{j} + 2x^2 \textbf{k}$ $14z \textbf{i} – 2x^2 \textbf{k}$
asked
Feb 12, 2019
in
Calculus
by
Arjun
(
8.7k
points)
gate2019ce2
calculus
vectorcalculus
vectoridentities
fieldvectors
curl
0
votes
0
answers
23
GATE201622
The optimum value of the function $f(x)=x^24x+2$ is $2$ (maximum) $2$ (minimum) $ – 2$ (maximum) $ – 2$ (minimum)
asked
Mar 28, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2016ce2
calculus
maximaminima
0
votes
0
answers
24
GATE201625
What is the value of $\underset{x \rightarrow 0 \\ y \rightarrow 0}{\lim} \dfrac{xy}{x^2+y^2}$? $1$ $1$ $0$ Limit does not exist
asked
Mar 28, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2016ce2
calculus
limits
0
votes
0
answers
25
GATE2016225
A circular curve of radius R connects two straights with a deflection angle of $60^{\circ}$. The tangent length is $0.577 \: R$ $1.155 \: R$ $1.732 \:R$ $3.464 \: R$
asked
Mar 28, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2016ce2
calculus
circularcurve
0
votes
0
answers
26
GATE2016228
The angle of intersection of the curves $x^{2}=4y$ and $y^{2} = 4x$ at point $(0,0)$ is $0^{\circ}$ $30^{\circ}$ $45^{\circ}$ $90^{\circ}$
asked
Mar 28, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2016ce2
calculus
curves
0
votes
0
answers
27
GATE2016229
The area between the parabola $x^2=8y$ and the straight line $y=8$ is _______.
asked
Mar 28, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2016ce2
calculus
definiteintegral
areaundercurve
numericalanswers
0
votes
0
answers
28
GATE2016127
The value of $\int_0^{\infty} \dfrac{1}{1+x^2} dx + \int _0^{\infty} \dfrac{\sin x}{x} dx$ is $\dfrac{\pi}{2} \\$ $\pi \\$ $\dfrac{3 \pi}{2} \\$ $1$
asked
Mar 28, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2016ce1
calculus
definiteintegral
0
votes
0
answers
29
GATE2016128
The area of the region bounded by the parabola $y=x^2+1$ and the straight line $x+y=3$ is $\dfrac{59}{6} \\$ $\dfrac{9}{2} \\$ $\dfrac{10}{3} \\$ $\dfrac{7}{6}$
asked
Mar 28, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2016ce1
calculus
definiteintegral
areaundercurve
0
votes
0
answers
30
GATE2016129
The magnitudes of vectors $\textbf{P, Q}$ and $\textbf{R}$ are $100$ kN, $250$ kN and $150$ kN, respectively as shown in the figure. The respective values of the magnitude (in kN) and the direction (with respect to the xaxis) of the resultant vector are $290.9$ and $96.0^{\circ}$ $368.1$ and $94.7^{\circ}$ $330.4$ and $118.9^{\circ}$ $400.1$ and $113.5^{\circ}$
asked
Mar 28, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2016ce1
calculus
vectoridentities
0
votes
0
answers
31
GATE2017 CE1: 3
Let $x$ be a continuous variable defined over the interval $( \infty, \infty)$, and $f(x) = e^{xe^{x}}$. The integral $g(x)= \int f(x) \: dx$ is equal to $e^{e^{x}}$ $e^{e^{x}}$ $e^{e^{x}}$ $e^{x}$
asked
Mar 26, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2017ce1
calculus
integrals
continuousvariable
0
votes
0
answers
32
GATE2017 CE1: 21
$\underset{x \to 0}{\lim} \bigg( \dfrac{\tan x}{x^2x} \bigg)$ is equal to _________
asked
Mar 26, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2017ce1
numericalanswers
calculus
limits
0
votes
0
answers
33
GATE201521
While minimizing the function $f(x)$, necessary and sufficient conditions for a point, $x_0$ to be a minima are: $f’(x_0) > 0$ and $f’’(x_0)=0$ $f’(x_0) < 0$ and $f’’(x_0)=0$ $f’(x_0) = 0$ and $f’’(x_0)<0$ $f’(x_0) = 0$ and $f’’(x_0)>0$
asked
Mar 26, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2015ce2
calculus
maximaminima
0
votes
0
answers
34
GATE201523
Given, $i=\sqrt{1}$, the value of the definite integral, $I= \int_0^{\pi/2} \dfrac{\cos x+ i \sin x}{\cos x – i \sin x} dx$ is: $1$ $1$ $i$ $i$
asked
Mar 26, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2015ce2
calculus
definiteintegral
0
votes
0
answers
35
GATE201524
$\underset{x \to \infty}{\lim} \bigg( 1+ \dfrac{1}{x} \bigg)^{2x} $ is equal to $e^{2}$ $e$ $1$ $e^2$
asked
Mar 26, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2015ce2
calculus
limits
0
votes
0
answers
36
GATE2015129
Consider the following complex function: $f(z) = \frac{9}{(z1)(z+2)^2}$ Which of the following is one of the residues of the above function? $1$ $9/16$ $2$ $9$
asked
Mar 26, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2015ce1
calculus
complexfunction
0
votes
0
answers
37
GATE2015130
The directional derivative of the field $u(x,y,z) = x^23yz$ in the direction of the vector $(\hat{i}+\hat{j}2 \hat{k})$ at point $(2, 1, 4)$ is _________.
asked
Mar 26, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2015ce1
numericalanswers
calculus
vectoridentities
directionalderivatives
0
votes
0
answers
38
GATE2014226
The expression $\displaystyle{} \lim_{a \to 0} \dfrac{x^a1}{a}$ is equal to $\log x$ $0$ $x \log x$ $\infty$
asked
Mar 26, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2014ce2
calculus
limits
0
votes
0
answers
39
GATE201423
$z=\dfrac{23i}{5+i}$ can be expressed as $0.50.5i$ $0.5+0.5i$ $0.50.5i$ $0.5+0.5i$
asked
Mar 26, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2014ce2
calculus
complexnumber
0
votes
0
answers
40
GATE2014128
A particle moves along a curve whose parametric equations are: $x=t^3+2t$, $y=3e^{2t}$ and $z=2 \sin (5t)$, where $x$, $y$ and $z$ show variations of the distance covered by the particle (in cm) with time $t$ (in s). The magnitude of the acceleration of the particle (in $cm/s^2$) at $t=0$ is ________
asked
Mar 26, 2018
in
Calculus
by
Milicevic3306
(
11.9k
points)
gate2014ce1
numericalanswers
engineeringmathematics
calculus
parametricequations
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