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For a two-dimensional, incompressible flow having velocity components $u$ and $v$ in the $x$ and $y$ directions, respectively, the expression $\frac{\partial \left ( u^{2} \right )}{\partial x}+\frac{\partial \left ( uv \right )}{\partial y}$ ... $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}$
asked Mar 1 in Calculus jothee 4.9k points
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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 $x^{2}y=1$ $xy^{2}=1$ $x^{2}y^{2}=1$ $x/y^{2}=1$
asked Mar 1 in Calculus jothee 4.9k points
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Value of $\left ( 1+i \right )^{8}$, where $i=\sqrt{-1}$, is equal to $4$ $16$ $4i$ $16i$
asked Mar 1 in Calculus jothee 4.9k points
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The value of $\int_{0}^{^{\pi }/_{2}}\int_{0}^{\cos\theta }r\sin\theta \:dr\:d\theta$ is $0$ $\frac{1}{6}$ $\frac{4}{3}$ $\pi$
asked Mar 1 in Calculus jothee 4.9k points
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The value of $\displaystyle{} \lim_{x \rightarrow 0} \left( \frac{1 – \cos x}{x^{2}}\right)$ is $\frac{1}{4}$ $\frac{1}{3}$ $\frac{1}{2}$ $1$
asked Feb 22 in Calculus gatecse 1.6k points
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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 $0$ $2$ $\pi i$ $2 \pi i$
asked Feb 22 in Calculus gatecse 1.6k points
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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 $0$ $1$ $2$ $3$
asked Feb 22 in Calculus gatecse 1.6k points
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Let $I=\displaystyle \int_{x=0}^1 \int_{y=0}^{x^2} xy^2 dy \: dx$. Then, $I$ may also be expressed as $\displaystyle \int_{y=0}^1 \int_{x=0}^{\sqrt{y}} xy^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=\sqrt{y}}^1 yx^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=\sqrt{y}}^1 xy^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=0}^{\sqrt{y}} yx^2 dx \: dy$
asked Sep 18, 2020 in Calculus jothee 4.9k points
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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} \\ $ $7$
asked Sep 18, 2020 in Calculus jothee 4.9k points
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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
asked Sep 18, 2020 in Calculus jothee 4.9k points
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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 $\text{c} \\$ $\text{c + 1} \\$ $\dfrac{c}{c+1} \\$ $\dfrac{c+1}{c}$
asked Feb 19, 2020 in Calculus jothee 4.9k points
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Which of the following function $f(z)$, of the complex variable $z,$ is NOT analytic at all the points of the complex plane? $f\left ( z \right )=z^{2}$ $f\left ( z \right )=e^{z}$ $f\left ( z \right )=\sin z$ $f\left ( z \right )=\log z$
asked Feb 19, 2020 in Calculus jothee 4.9k points
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A vector field is defined as ... spherical shell formed by two concentric spheres with origin as the center, and internal and external radii of $1$ and $2$, respectively, is $0$ $2\pi$ $4\pi$ $8\pi$
asked Feb 19, 2020 in Calculus jothee 4.9k points
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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).
asked Feb 19, 2020 in Calculus jothee 4.9k points
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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 $\sqrt{2}$ $2$ $2 \sqrt{2}$ $4 \sqrt{2}$
asked Feb 9, 2019 in Calculus Arjun 24.6k points
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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 $v(x,y)$ ...
asked Feb 9, 2019 in Calculus Arjun 24.6k points
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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 $- \dfrac{\pi}{2} \\$ $\dfrac{\pi}{3} \\$ $\dfrac{\pi}{2} \\$ $\pi$
asked Feb 9, 2019 in Calculus Arjun 24.6k points
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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 $f(x)$ at $x=\pi /6$ ... $<0.1 \%$ $> 0.1 \% \text{ and } <1 \%$ $> 1 \% \text{ and } <5 \%$ $>5 \%$
asked Feb 9, 2019 in Calculus Arjun 24.6k points
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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 $\dfrac{\pi}{4} \\$ $\dfrac{\pi}{2} \\$ ${\pi} \\$ $2 \pi$
asked Feb 9, 2019 in Calculus Arjun 24.6k points
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The value of the following definite integral is __________ (round off to three decimal places) $\int_1^e (x \: \ln \: x) dx$
asked Feb 9, 2019 in Calculus Arjun 24.6k points
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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. ... $a-b-e-g-h$ $a-c-g-h$ $a-d-f-h$ $a-b-c-f-h$
asked Mar 20, 2018 in Calculus Andrijana3306 1.5k points
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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. ... duration to complete the project remain the same critical path changes but the total duration to complete the project changes to $17$ days
asked Mar 20, 2018 in Calculus Andrijana3306 1.5k points
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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 $\dfrac{1}{\sqrt{2}} \hat{i} +\dfrac{1}{\sqrt{2}} \hat{j} \\$ $\dfrac{1}{\sqrt{2}} \hat{i} -\dfrac{1}{\sqrt{2}} \hat{j} \\ $ $\hat{k} \\$ $\dfrac{1}{\sqrt{3}} \hat{i} +\dfrac{1}{\sqrt{3}} \hat{j} +\dfrac{1}{\sqrt{3}} \hat{k}$
asked Mar 20, 2018 in Calculus Andrijana3306 1.5k points
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At $x=0$, the function $f(x)=x^3+1$ has a maximum value a minimum value a singularity a point of inflection
asked Mar 20, 2018 in Calculus Andrijana3306 1.5k points
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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
asked Mar 20, 2018 in Calculus Andrijana3306 1.5k points
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$\underset{x \rightarrow 0}{\lim} \bigg( \dfrac{1- \cos x}{x^2} \bigg)$ is $1/4$ $1/2$ $1$ $2$
asked Mar 20, 2018 in Calculus Andrijana3306 1.5k points
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The area enclosed between the straight line $y=x$ and the parabola $y=x^2$ in the $x-y$ plane is $1/6$ $1/4$ $1/3$ $1/2$
asked Mar 20, 2018 in Calculus Andrijana3306 1.5k points
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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 $\phi =\text{ln} \mid \overrightarrow{r} \mid$ ... $\dfrac{\overrightarrow{r}}{\overrightarrow{r} \cdot \overrightarrow{r} } \\ $ $\dfrac{\overrightarrow{r}}{\mid \overrightarrow{r} \mid^3} $
asked Feb 17, 2018 in Calculus Arjun 24.6k points
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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 $2/5$ $1/2$ $2$ $4/5$
asked Feb 17, 2018 in Calculus Arjun 24.6k points
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The divergence of the vector field $\overrightarrow{u}=e^x(\cos \: y\hat{i}+\sin \: y \hat{j})$ is $0$ $e^x \cos y + e^x \sin y$ $2e^x \cos y$ $2e^x \sin y$
asked Feb 17, 2018 in Calculus Arjun 24.6k points
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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 $-0.5$ $0.0$ $0.5$ $1.0$
asked Feb 17, 2018 in Calculus Arjun 24.6k points
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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 $V$ $2V$ $3V$ $4V$
asked Feb 17, 2018 in Calculus Arjun 24.6k points
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$F(z)$ 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? $0$ $1$ $-1$ $y$
asked Feb 17, 2018 in Calculus Arjun 24.6k points
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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 = $ $f(\xi)(b-a)$ $f(b)(\xi-a)$ $f(a)(b-\xi)$ $0$
asked Feb 17, 2018 in Calculus Arjun 24.6k points
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If $f(z)=(x^{2}+ay^{2})+i bxy$ is a complex analytic function of $z=x+iy$, where $i=\sqrt{-1}$, then $a=-1, b=-1$ $a=-1, b=2$ $a=1, b= 2$ $a=2, b=2$
asked Feb 27, 2017 in Calculus Arjun 24.6k points
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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 ________.
asked Feb 27, 2017 in Calculus Arjun 24.6k points
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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 $\dfrac{\Pi }{2} \\$ $\pi \\$ $2 \pi \\$ $4 \pi$
asked Feb 27, 2017 in Calculus Arjun 24.6k points
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For the vector $\vec{V}=2yz\hat{i}+3xz \hat{j}+4xy \hat{k}$, the value of $\bigtriangledown$. $(\bigtriangledown \times \vec{V})$ is ________.
asked Feb 27, 2017 in Calculus Arjun 24.6k points
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The value of $\displaystyle{}\lim_{x \rightarrow 0}\dfrac{x^{3}-\sin(x)}{x}$ is $0$ $3$ $1$ $-1$
asked Feb 27, 2017 in Calculus Arjun 24.6k points
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