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Recent questions tagged calculus
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votes
GATE Mechanical 2021 Set 2 | Question: 11
For a two-dimensional, incompressible flow having velocity components $u$ and $v$ in the $x$ and $y$ directions, respectively, the expression $\frac{\partial \left ( u^{2} \right )}{\partial x}+\frac{\partial \left ( uv \right )}{\partial y}$ can ... $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}$
go_editor
asked
in
Calculus
Mar 1, 2021
by
go_editor
5.0k
points
gateme-2021-set2
calculus
partial-derivatives
0
answers
0
votes
GATE Mechanical 2021 Set 2 | Question: 13
A two dimensional flow has velocities in $x$ and $y$ directions given by $u = 2xyt$ and $v = -y^{2}t$, where $\text{t}$ denotes time. The equation for streamline passing through $x=1,\:y=1$ is $x^{2}y=1$ $xy^{2}=1$ $x^{2}y^{2}=1$ $x/y^{2}=1$
go_editor
asked
in
Calculus
Mar 1, 2021
by
go_editor
5.0k
points
gateme-2021-set2
calculus
derivatives
0
answers
0
votes
GATE Mechanical 2021 Set 2 | Question: 19
Value of $\left ( 1+i \right )^{8}$, where $i=\sqrt{-1}$, is equal to $4$ $16$ $4i$ $16i$
go_editor
asked
in
Calculus
Mar 1, 2021
by
go_editor
5.0k
points
gateme-2021-set2
calculus
complex-variables
0
answers
0
votes
GATE Mechanical 2021 Set 2 | Question: 26
The value of $\int_{0}^{^{\pi }/_{2}}\int_{0}^{\cos\theta }r\sin\theta \:dr\:d\theta$ is $0$ $\frac{1}{6}$ $\frac{4}{3}$ $\pi$
go_editor
asked
in
Calculus
Mar 1, 2021
by
go_editor
5.0k
points
gateme-2021-set2
calculus
definite-integrals
double-interals
0
answers
0
votes
GATE Mechanical 2021 Set 1 | Question: 2
The value of $\displaystyle{} \lim_{x \rightarrow 0} \left( \frac{1 – \cos x}{x^{2}}\right)$ is $\frac{1}{4}$ $\frac{1}{3}$ $\frac{1}{2}$ $1$
gatecse
asked
in
Calculus
Feb 22, 2021
by
gatecse
1.6k
points
gateme-2021-set1
calculus
limits
0
answers
0
votes
GATE Mechanical 2021 Set 1 | Question: 27
Let $\text{C}$ represent the unit circle centered at origin in the complex plane, and complex variable, $z=x+iy$. The value of the contour integral $\oint _{C}\dfrac{\cosh \:3z}{2z}\:dz$ (where integration is taken counter clockwise) is $0$ $2$ $\pi i$ $2 \pi i$
gatecse
asked
in
Calculus
Feb 22, 2021
by
gatecse
1.6k
points
gateme-2021-set1
calculus
complex-variables
0
answers
0
votes
GATE Mechanical 2021 Set 1 | Question: 34
Let $f\left ( x \right )=x^{2}-2x+2$ be a continuous function defined on $x \in \left [ 1,3 \right ]$. The point $x$ at which the tangent of $f\left ( x \right )$ becomes parallel to the straight line joining $f\left ( 1 \right )$ and $f\left ( 3 \right )$ is $0$ $1$ $2$ $3$
gatecse
asked
in
Calculus
Feb 22, 2021
by
gatecse
1.6k
points
gateme-2021-set1
calculus
maxima-minima
0
answers
0
votes
GATE2020-ME-2: 3
Let $I=\displaystyle \int_{x=0}^1 \int_{y=0}^{x^2} xy^2 dy \: dx$. Then, $I$ may also be expressed as $\displaystyle \int_{y=0}^1 \int_{x=0}^{\sqrt{y}} xy^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=\sqrt{y}}^1 yx^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=\sqrt{y}}^1 xy^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=0}^{\sqrt{y}} yx^2 dx \: dy$
go_editor
asked
in
Calculus
Sep 18, 2020
by
go_editor
5.0k
points
gateme-2020-set2
calculus
definite-integrals
double-interals
0
answers
0
votes
GATE2020-ME-2: 26
The directional derivative of $f(x,y,z) = xyz$ at point $(-1,1,3)$ in the direction of vector $\hat{i} – 2 \hat{j} +2 \hat{k}$ is $3\hat{i} – 3 \hat{j} - \hat{k} \\$ $- \dfrac{7}{3} \\$ $\dfrac{7}{3} \\ $ $7$
go_editor
asked
in
Calculus
Sep 18, 2020
by
go_editor
5.0k
points
gateme-2020-set2
calculus
vector-identities
directional-derivatives
0
answers
0
votes
GATE2020-ME-2: 27
The function $f(z)$ of complex variable $z=x+iy$, where $i=\sqrt{-1}$, is given as $f(z)=(x^3-3xy^2)+i \: v(x,y)$. For this function to be analytic, $v(x,y)$ should be $(3xy^2-y^3) +$ constant $(3x^2y^2-y^3) +$ constant $(x^3-3x^2 y) +$ constant $(3x^2y-y^3) +$ constant
go_editor
asked
in
Calculus
Sep 18, 2020
by
go_editor
5.0k
points
gateme-2020-set2
calculus
complex-variables
analytic-functions
0
answers
0
votes
GATE2020-ME-1: 2
The value of $\displaystyle{}\lim_{x \to \infty}\left ( \dfrac{1 -e^{-c\left ( 1-x \right )}}{1-x\:e^{-c\left ( 1-x \right )}} \right )$ is $\text{c} \\$ $\text{c + 1} \\$ $\dfrac{c}{c+1} \\$ $\dfrac{c+1}{c}$
go_editor
asked
in
Calculus
Feb 19, 2020
by
go_editor
5.0k
points
gateme-2020-set1
calculus
limits
0
answers
0
votes
GATE2020-ME-1: 4
Which of the following function $f(z)$, of the complex variable $z,$ is NOT analytic at all the points of the complex plane? $f\left ( z \right )=z^{2}$ $f\left ( z \right )=e^{z}$ $f\left ( z \right )=\sin z$ $f\left ( z \right )=\log z$
go_editor
asked
in
Calculus
Feb 19, 2020
by
go_editor
5.0k
points
gateme-2020-set1
calculus
complex-variables
0
answers
0
votes
GATE2020-ME-1: 27
A vector field is defined as ... shell formed by two concentric spheres with origin as the center, and internal and external radii of $1$ and $2$, respectively, is $0$ $2\pi$ $4\pi$ $8\pi$
go_editor
asked
in
Calculus
Feb 19, 2020
by
go_editor
5.0k
points
gateme-2020-set1
calculus
vector-identities
0
answers
0
votes
GATE2020-ME-1: 36
An analytic function of a complex variable $z=x + iy \left ( i=\sqrt{-1} \right )$ is defined as $f\left ( z \right )=x^{2}-y^{2}+i\psi \left ( x,y \right ),$ where $\psi \left ( x,y \right )$ is a real function. The value of the imaginary part of $f(z)$ at $z=\left ( 1+i \right )$ is __________ (round off to $2$ decimal places).
go_editor
asked
in
Calculus
Feb 19, 2020
by
go_editor
5.0k
points
gateme-2020-set1
numerical-answers
calculus
complex-variables
analytic-functions
0
answers
0
votes
GATE2019 ME-2: 2
The directional derivative of the function $f(x,y)=x^2+y^2$ along a line directed from $(0,0)$ to $(1,1)$, evaluated at the point $x=1, y=1$ is $\sqrt{2}$ $2$ $2 \sqrt{2}$ $4 \sqrt{2}$
Arjun
asked
in
Calculus
Feb 9, 2019
by
Arjun
27.4k
points
gateme-2019-set2
calculus
directional-derivatives
1
answer
0
votes
GATE2019 ME-2: 4
An analytic function $f(z)$ of complex variable $z=x+iy$ may be written as $f(z)=u(x,y)+iv(x,y)$. Then $u(x,y)$ and $v(x,y)$ ...
Arjun
asked
in
Calculus
Feb 9, 2019
by
Arjun
27.4k
points
gateme-2019-set2
calculus
partial-derivatives
complex-variables
analytic-functions
0
answers
0
votes
GATE2019 ME-2: 26
Given a vector $\overrightarrow{u} = \dfrac{1}{3} \big(-y^3 \hat{i} + x^3 \hat{j} + z^3 \hat{k} \big)$ and $\hat{n}$ as the unit normal vector to the surface of the hemipshere $(x^2+y^2+z^2=1; \: z \geq 0)$ ... $S$ is $- \dfrac{\pi}{2} \\$ $\dfrac{\pi}{3} \\$ $\dfrac{\pi}{2} \\$ $\pi$
Arjun
asked
in
Calculus
Feb 9, 2019
by
Arjun
27.4k
points
gateme-2019-set2
calculus
vector-identities
0
answers
0
votes
GATE2019 ME-2: 28
The derivative of $f(x)= \cos x$ can be estimated using the approximation $f'(x)=\dfrac{f(x+h)-f(x-h)}{2h}$. The percentage error is calculated as $\bigg( \dfrac{\text{Exact value - Approximate value}}{\text{Exact value}} \bigg) \times 100$. The percentage error in the derivative of $f(x)$ ... $> 0.1 \% \text{ and } <1 \%$ $> 1 \% \text{ and } <5 \%$ $>5 \%$
Arjun
asked
in
Calculus
Feb 9, 2019
by
Arjun
27.4k
points
gateme-2019-set2
calculus
derivatives
0
answers
0
votes
GATE2019 ME-1: 2
A parabola $x=y^2$ with $0 \leq x \leq 1$ is shown in the figure. The volume of the solid of rotation obtained by rotating the shaded area by $360^{\circ}$ around x-axis is $\dfrac{\pi}{4} \\$ $\dfrac{\pi}{2} \\$ ${\pi} \\$ $2 \pi$
Arjun
asked
in
Calculus
Feb 9, 2019
by
Arjun
27.4k
points
gateme-2019-set1
calculus
area-under-curve
1
answer
0
votes
GATE2019 ME-1: 51
The value of the following definite integral is __________ (round off to three decimal places) $\int_1^e (x \: \ln \: x) dx$
Arjun
asked
in
Calculus
Feb 9, 2019
by
Arjun
27.4k
points
gateme-2019-set1
numerical-answers
calculus
definite-integrals
0
answers
1
vote
GATE ME 2012 | Question: 54
For a particular project, eight activities are to be carried out. Their relationships with other activities and expected durations are mentioned in the table below. ... critical path for the project is $a-b-e-g-h$ $a-c-g-h$ $a-d-f-h$ $a-b-c-f-h$
Andrijana3306
asked
in
Calculus
Mar 20, 2018
by
Andrijana3306
1.5k
points
gateme-2012
calculus
0
answers
0
votes
GATE ME 2012 | Question: 55
For a particular project, eight activities are to be carried out. Their relationships with other activities and expected durations are mentioned in the table below. ... the project remain the same critical path changes but the total duration to complete the project changes to $17$ days
Andrijana3306
asked
in
Calculus
Mar 20, 2018
by
Andrijana3306
1.5k
points
gateme-2012
calculus
0
answers
0
votes
GATE ME 2012 | Question: 25
For the spherical surface $x^2+y^2+z^2=1$, the unit outward normal vector at th point $\left( \dfrac{1}{\sqrt{2}}, \dfrac{1}{\sqrt{2}}, 0\right)$ is given by $\dfrac{1}{\sqrt{2}} \hat{i} +\dfrac{1}{\sqrt{2}} \hat{j} \\$ ... $\hat{k} \\$ $\dfrac{1}{\sqrt{3}} \hat{i} +\dfrac{1}{\sqrt{3}} \hat{j} +\dfrac{1}{\sqrt{3}} \hat{k}$
Andrijana3306
asked
in
Calculus
Mar 20, 2018
by
Andrijana3306
1.5k
points
gateme-2012
calculus
vector-identities
0
answers
0
votes
GATE ME 2012 | Question: 24
At $x=0$, the function $f(x)=x^3+1$ has a maximum value a minimum value a singularity a point of inflection
Andrijana3306
asked
in
Calculus
Mar 20, 2018
by
Andrijana3306
1.5k
points
gateme-2012
calculus
functions-of-single-variable
maxima-minima
0
answers
0
votes
GATE ME 2012 | Question: 12
Consider the function $f(x) = \mid x \mid $ in the interval $-1 \leq x \leq 1$. At the point $x=0, \: f(x)$ is continuous and differentiable non-continuous and differentiable continuous and non-differentiable neither continuous nor differentiable
Andrijana3306
asked
in
Calculus
Mar 20, 2018
by
Andrijana3306
1.5k
points
gateme-2012
calculus
continuity-and-differentiability
0
answers
0
votes
GATE ME 2012 | Question: 12
$\underset{x \rightarrow 0}{\lim} \bigg( \dfrac{1- \cos x}{x^2} \bigg)$ is $1/4$ $1/2$ $1$ $2$
Andrijana3306
asked
in
Calculus
Mar 20, 2018
by
Andrijana3306
1.5k
points
gateme-2012
calculus
limits
0
answers
0
votes
GATE ME 2012 | Question: 11
The area enclosed between the straight line $y=x$ and the parabola $y=x^2$ in the $x-y$ plane is $1/6$ $1/4$ $1/3$ $1/2$
Andrijana3306
asked
in
Calculus
Mar 20, 2018
by
Andrijana3306
1.5k
points
gateme-2012
calculus
definite-integrals
area-under-curve
0
answers
0
votes
GATE2018-2-28
For a position vector $\overrightarrow{r} = x \hat{i}+y \hat{j} + z\hat{k}$ the norm of the vector can be defined as $\mid \overrightarrow{r} \mid = \sqrt{x^2+y^2+z^2}$. Given a function $\phi =\text{ln} \mid \overrightarrow{r} \mid$, its ... $\dfrac{\overrightarrow{r}}{\overrightarrow{r} \cdot \overrightarrow{r} } \\ $ $\dfrac{\overrightarrow{r}}{\mid \overrightarrow{r} \mid^3} $
Arjun
asked
in
Calculus
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set2
calculus
vector-identities
0
answers
0
votes
GATE2018-2-26
Let $z$ be a complex variable. For a counter-clockwise integration around a unit circle $C$, centered at origin, $\oint_C \frac{1}{5z-4} dz=A \pi i$, the value of $A$ is $2/5$ $1/2$ $2$ $4/5$
Arjun
asked
in
Calculus
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set2
calculus
complex-variables
0
answers
0
votes
GATE2018-2-2
The divergence of the vector field $\overrightarrow{u}=e^x(\cos \: y\hat{i}+\sin \: y \hat{j})$ is $0$ $e^x \cos y + e^x \sin y$ $2e^x \cos y$ $2e^x \sin y$
Arjun
asked
in
Calculus
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set2
calculus
divergence-and-curl
0
answers
0
votes
GATE2018-2-1
The Fourier cosine series for an even function $f(x)$ is given by $ f(x)=a_0 + \Sigma_{n=1}^\infty a_n \cos (nx).$ The value of the coefficient $a_2$ for the function $f(x)=\cos ^2 (x)$ in $[0, \pi]$ is $-0.5$ $0.0$ $0.5$ $1.0$
Arjun
asked
in
Calculus
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set2
calculus
fourier-series
0
answers
0
votes
GATE2018-1-27
The value of the integral over the closed surface $S$ bounding a volume $V$, where $\overrightarrow{r} = x \hat{i} + y \hat{j}+z \hat{k}$ is the position vector and $\overrightarrow{n}$ is the normal to the surface $S$, is $V$ $2V$ $3V$ $4V$
Arjun
asked
in
Calculus
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set1
calculus
surface-integral
vector-identities
0
answers
0
votes
GATE2018-1-4
$F(z)$ is a function of the complex variable $z=x+iy$ given by $F(z)+ i \: z + k \: Re(z) + i \: Im(z)$. For what value of $k$ will $F(z)$ satisfy the Cauchy-Riemann equations? $0$ $1$ $-1$ $y$
Arjun
asked
in
Calculus
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set1
calculus
complex-variables
euler-cauchy-equations
0
answers
0
votes
GATE2018-1-3
According to the Mean Value Theorem, for a continuous function $f(x)$ in the interval $[a,b]$, there exists a value $\xi$ in this interval such that $\int_a^b f(x) dx = $ $f(\xi)(b-a)$ $f(b)(\xi-a)$ $f(a)(b-\xi)$ $0$
Arjun
asked
in
Calculus
Feb 17, 2018
by
Arjun
27.4k
points
gateme-2018-set1
calculus
mean-value-theorems
definite-integrals
1
answer
0
votes
GATE2017 ME-2: 29
If $f(z)=(x^{2}+ay^{2})+i bxy$ is a complex analytic function of $z=x+iy$, where $i=\sqrt{-1}$, then $a=-1, b=-1$ $a=-1, b=2$ $a=1, b= 2$ $a=2, b=2$
Arjun
asked
in
Calculus
Feb 27, 2017
by
Arjun
27.4k
points
gateme-2017-set2
calculus
complex-variables
0
answers
0
votes
GATE2017 ME-2: 26
The surface integral $\int \int _{s} F.n $ dS over the surface $S$ of the sphere $x^{2}+y^{2}+z^{2}=9$, where $F=(x+y) i+(x+z) j+(y+z)k$ and $n$ is the unit outward surface normal, yields ________.
Arjun
asked
in
Calculus
Feb 27, 2017
by
Arjun
27.4k
points
gateme-2017-set2
numerical-answers
calculus
surface-integral
0
answers
0
votes
GATE2017 ME-2: 2
The divergence of the vector $-yi+xj$ is ________.
Arjun
asked
in
Calculus
Feb 27, 2017
by
Arjun
27.4k
points
gateme-2017-set2
numerical-answers
calculus
vector-identities
divergence-and-curl
0
answers
0
votes
GATE2017 ME-1: 28
A parametric curve defined by $x= \cos \left ( \dfrac{\Pi u}{2} \right ), y= \sin \left ( \dfrac{\Pi u}{2} \right )$ in the range $0 \leq u \leq 1$ is rotated about the $X$-axis by $360$ degrees. Area of the surface generated is $\dfrac{\Pi }{2} \\$ $\pi \\$ $2 \pi \\$ $4 \pi$
Arjun
asked
in
Calculus
Feb 27, 2017
by
Arjun
27.4k
points
gateme-2017-set1
calculus
area-under-curve
0
answers
0
votes
GATE2017 ME-1: 27
For the vector $\vec{V}=2yz\hat{i}+3xz \hat{j}+4xy \hat{k}$, the value of $\bigtriangledown$. $(\bigtriangledown \times \vec{V})$ is ________.
Arjun
asked
in
Calculus
Feb 27, 2017
by
Arjun
27.4k
points
gateme-2017-set1
numerical-answers
calculus
vector-identities
0
answers
0
votes
GATE2017 ME-1: 2
The value of $\displaystyle{}\lim_{x \rightarrow 0}\dfrac{x^{3}-\sin(x)}{x}$ is $0$ $3$ $1$ $-1$
Arjun
asked
in
Calculus
Feb 27, 2017
by
Arjun
27.4k
points
gateme-2017-set1
calculus
limits
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