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Recent questions and answers in Calculus

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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 $x^{2}y=1$ $xy^{2}=1$ $x^{2}y^{2}=1$ $x/y^{2}=1$
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 ...
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GATE Mechanical 2021 Set 2 | Question: 19
Value of $\left ( 1+i \right )^{8}$, where $i=\sqrt{-1}$, is equal to $4$ $16$ $4i$ $16i$
Value of $\left ( 1+i \right )^{8}$, where $i=\sqrt{-1}$, is equal to$4$$16$$4i$$16i$
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GATE Mechanical 2021 Set 2 | Question: 26
The value of $\int_{0}^{^{\pi }/_{2}}\int_{0}^{\cos\theta }r\sin\theta \:dr\:d\theta$ is $0$ $\frac{1}{6}$ $\frac{4}{3}$ $\pi$
The value of $\int_{0}^{^{\pi }/_{2}}\int_{0}^{\cos\theta }r\sin\theta \:dr\:d\theta$ is $0$$\frac{1}{6}$$\frac{4}{3}$$\pi$
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GATE Mechanical 2021 Set 1 | Question: 2
The value of $\displaystyle{} \lim_{x \rightarrow 0} \left( \frac{1 – \cos x}{x^{2}}\right)$ is $\frac{1}{4}$ $\frac{1}{3}$ $\frac{1}{2}$ $1$
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}$$1$
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GATE2020-ME-2: 26
The directional derivative of $f(x,y,z) = xyz$ at point $(-1,1,3)$ in the direction of vector $\hat{i} – 2 \hat{j} +2 \hat{k}$ is $3\hat{i} – 3 \hat{j} - \hat{k} \\$ $- \dfrac{7}{3} \\$ $\dfrac{7}{3} \\ $ $7$
The directional derivative of $f(x,y,z) = xyz$ at point $(-1,1,3)$ in the direction of vector $\hat{i} – 2 \hat{j} +2 \hat{k}$ is$3\hat{i} – 3 \hat{j} - \hat{k} \\$$-...
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#MADE EASY MOCK TEST GATE2021
The area common to both circles r=a$\sqrt{2}$ and r=2a cos$\theta$.
The area common to both circles r=a$\sqrt{2}$ and r=2a cos$\theta$.
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GATE MOCK TEST(MADE EASY)
The area enclosed between the circles r=2a cos$\theta$ and r=a$\sqrt{2}$.
The area enclosed between the circles r=2a cos$\theta$ and r=a$\sqrt{2}$.
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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 $\text{c} \\$ $\text{c + 1} \\$ $\dfrac{c}{c+1} \\$ $\dfrac{c+1}{c}$
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} \\$$...
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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? $f\left ( z \right )=z^{2}$ $f\left ( z \right )=e^{z}$ $f\left ( z \right )=\sin z$ $f\left ( z \right )=\log z$
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}$$f\left ( z \right ...
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GATE2020-ME-1: 27
A vector field is defined as ... shell formed by two concentric spheres with origin as the center, and internal and external radii of $1$ and $2$, respectively, is $0$ $2\pi$ $4\pi$ $8\pi$
A vector field is defined as $$\overrightarrow{f}\left ( x,y,z \right )=\dfrac{x}{\left [ x^{2}+y^{2}+z^{2} \right ]^{\frac{3}{2}}}\widehat{i}\:+\:\dfrac{y}{\left [ x^{2}...
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GATE2019 ME-1: 51
The value of the following definite integral is __________ (round off to three decimal places) $\int_1^e (x \: \ln \: x) dx$
The value of the following definite integral is __________ (round off to three decimal places)$$\int_1^e (x \: \ln \: x) dx$$
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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 $v(x,y)$ ...
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 $v(x,y)$ must satisfy$\dfrac{\partial u}{ \partial x} ...
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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 $a=-1, b=-1$ $a=-1, b=2$ $a=1, b= 2$ $a=2, b=2$
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$$a=2, b=2$
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GATE2019 ME-2: 2
The directional derivative of the function $f(x,y)=x^2+y^2$ along a line directed from $(0,0)$ to $(1,1)$, evaluated at the point $x=1, y=1$ is $\sqrt{2}$ $2$ $2 \sqrt{2}$ $4 \sqrt{2}$
The directional derivative of the function $f(x,y)=x^2+y^2$ along a line directed from $(0,0)$ to $(1,1)$, evaluated at the point $x=1, y=1$ is$\sqrt{2}$$2$$2 \sqrt{2}$$4...
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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 $\dfrac{\pi}{4} \\$ $\dfrac{\pi}{2} \\$ ${\pi} \\$ $2 \pi$
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 ...
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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 $a-b-e-g-h$ $a-c-g-h$ $a-d-f-h$ $a-b-c-f-h$
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.$\begi...
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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
At $x=0$, the function $f(x)=x^3+1$ hasa maximum valuea minimum valuea singularitya point of inflection
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GATE ME 2012 | Question: 12
$\underset{x \rightarrow 0}{\lim} \bigg( \dfrac{1- \cos x}{x^2} \bigg)$ is $1/4$ $1/2$ $1$ $2$
$\underset{x \rightarrow 0}{\lim} \bigg( \dfrac{1- \cos x}{x^2} \bigg)$ is$1/4$$1/2$$1$$2$
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GATE ME 2012 | Question: 11
The area enclosed between the straight line $y=x$ and the parabola $y=x^2$ in the $x-y$ plane is $1/6$ $1/4$ $1/3$ $1/2$
The area enclosed between the straight line $y=x$ and the parabola $y=x^2$ in the $x-y$ plane is$1/6$$1/4$$1/3$$1/2$
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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 $2/5$ $1/2$ $2$ $4/5$
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 $A$ i...
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GATE2018-2-2
The divergence of the vector field $\overrightarrow{u}=e^x(\cos \: y\hat{i}+\sin \: y \hat{j})$ is $0$ $e^x \cos y + e^x \sin y$ $2e^x \cos y$ $2e^x \sin y$
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$$2e^x \sin y$
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GATE2018-2-1
The Fourier cosine series for an even function $f(x)$ is given by $ f(x)=a_0 + \Sigma_{n=1}^\infty a_n \cos (nx).$ The value of the coefficient $a_2$ for the function $f(x)=\cos ^2 (x)$ in $[0, \pi]$ is $-0.5$ $0.0$ $0.5$ $1.0$
The Fourier cosine series for an even function $f(x)$ is given by $$ f(x)=a_0 + \Sigma_{n=1}^\infty a_n \cos (nx).$$ The value of the coefficient $a_2$ for the function $...
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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 $V$ $2V$ $3V$ $4V$
The value of the integralover 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 $\over...
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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? $0$ $1$ $-1$ $y$
$F(z)$ 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 equa...
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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 = $ $f(\xi)(b-a)$ $f(b)(\xi-a)$ $f(a)(b-\xi)$ $0$
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 = $...
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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 ________.
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 surfa...
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GATE2017 ME-2: 2
The divergence of the vector $-yi+xj$ is ________.
The divergence of the vector $-yi+xj$ is ________.
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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 ________.
For the vector $\vec{V}=2yz\hat{i}+3xz \hat{j}+4xy \hat{k}$, the value of $\bigtriangledown$. $(\bigtriangledown \times \vec{V})$ is ________.
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