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GATE ECE 2020 | Question: 3
The partial derivative of the function $f(x, y, z) = e^{1-x\cos y} + xze^{-1/(1+y^{2})}$ with respect to $x$ at the point $(1,0,e)$ is $-1$ $0$ $1 \\$ $\dfrac{1}{e}$
The partial derivative of the function$$f(x, y, z) = e^{1-x\cos y} + xze^{-1/(1+y^{2})}$$with respect to $x$ at the point $(1,0,e)$ is$-1$$0$$1 \\$$\dfrac{1}{e}$
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GATE ECE 2020 | Question: 51
For the solid $S$ shown below, the value of $\underset{S}{\iiint} xdxdydz$ (rounded off to two decimal places) is _______________.
For the solid $S$ shown below, the value of $\underset{S}{\iiint} xdxdydz$ (rounded off to two decimal places) is _______________.
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GATE ECE 2018 | Question: 24
Taylor series expansion of $f\left ( x \right )=\int ^{x}_{0}e^{-\left ( \frac{t^{2}}{2} \right )}dt$ around $x=0$ has the form $f\left ( x \right )={a}_{0}+a_{1}x+a_{2}x^{2}+...$ The coefficient $a_{2}$ (correct to two decimal places) is equal to ________.
Taylor series expansion of $f\left ( x \right )=\int ^{x}_{0}e^{-\left ( \frac{t^{2}}{2} \right )}dt$ around $x=0$ has the form $$f\left ( x \right )={a}_{0}+a_{1}x+a_{2}...
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GATE ECE 2019 | Question: 19
The value of the integral $ \displaystyle{}\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.
The value of the integral $ \displaystyle{}\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.
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GATE ECE 2017 Set 2 | Question: 30
The minimum value of the function $f(x)=\frac{1}{3} x(x^2-3)$ in the interval $-100≤x≤100$ occurs at $x =$ ________.
The minimum value of the function $f(x)=\frac{1}{3} x(x^2-3)$ in the interval $-100≤x≤100$ occurs at $x =$ ________.
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GATE ECE 2015 Set 2 | Question: 27
The value of the integral $\int_{-\infty}^{\infty} 12\cos(2\pi t) \dfrac{\sin(4\pi t)}{4 \pi t}dt$ is _________.
The value of the integral $\int_{-\infty}^{\infty} 12\cos(2\pi t) \dfrac{\sin(4\pi t)}{4 \pi t}dt$ is _________.
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GATE ECE 2016 Set 2 | Question: 4
How many distinct value of $x$ satisfy the equation $\sin(x)=x/2$, where $x$ is in radians? $1$ $2$ $3$ $4$ or more
How many distinct value of $x$ satisfy the equation $\sin(x)=x/2$, where $x$ is in radians?$1$$2$$3$$4$ or more
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GATE ECE 2016 Set 3 | Question: 4
The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1-x)}}$ is equal to _______
The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1-x)}}$ is equal to _______
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GATE ECE 2019 | Question: 16
The value of the contour integral $\frac{1}{2\pi j} \oint\left(z+\frac{1}{z}\right)^{2}dz$ evaluated over the unit circle $\mid z \mid=1$ is_______.
The value of the contour integral$$\frac{1}{2\pi j} \oint\left(z+\frac{1}{z}\right)^{2}dz$$evaluated over the unit circle $\mid z \mid=1$ is_______.
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GATE ECE 2013 | Question: 20
A polynomial $f(x) = a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x - a_{0}$ with all coefficients positive has no real roots no negative real root odd number of real roots at least one positive and one negative real root
A polynomial $f(x) = a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x - a_{0}$ with all coefficients positive hasno real rootsno negative real rootodd number of real roots a...
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GATE ECE 2015 Set 3 | Question: 2
The contour on the $x-y$ plane, where the partial derivative of $x^{2} + y^{2}$ with respect to $y$ is equal to the partial derivative of $6y+4x$ with respect to $x$, is $y=2$ $x=2$ $x+y=4$ $x-y=0$
The contour on the $x-y$ plane, where the partial derivative of $x^{2} + y^{2}$ with respect to $y$ is equal to the partial derivative of $6y+4x$ with respect to $x$, is$...
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GATE ECE 2014 Set 4 | Question: 1
The series $\Sigma_{n=0}^{\infty} \frac{1}{n!}$ converges to $2 \text{ ln } 2$ $\sqrt{2}$ $2$ $e$
The series $\Sigma_{n=0}^{\infty} \frac{1}{n!}$ converges to$2 \text{ ln } 2$$\sqrt{2}$$2$$e$
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GATE ECE 2015 Set 1 | Question: 2
A function $f(x)=1-x^2+x^3$ is defined in the closed interval $[-1,1]$. The value of $x$, in the open interval $(-1,1)$ for which the mean value theorem is satisfied, is $-1/2$ $-1/3$ $1/3$ $1/2$
A function $f(x)=1-x^2+x^3$ is defined in the closed interval $[-1,1]$. The value of $x$, in the open interval $(-1,1)$ for which the mean value theorem is satisfied, is$...
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TIFR ECE 2012 | Question: 1
The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is $\ln \left(1+e^{-1 / 4}\right)$ $\ln (5 / 3)$ $0$ $\ln \left(1+e^{2}\right)$ None of the above
The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is$\ln \left(1+e^{-1 / 4}\...
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GATE ECE 2018 | Question: 52
Let $r=x^{2}+y-z$ and $z^{3}-xy+yz+y^{3}=1.$ Assume that $x$ and $y$ are independent variables. At $\left( x,y,z \right)=\left ( 2,-1,1 \right ),$ the value (correct to two decimal places) of $\dfrac{\partial r}{\partial x}$ is _________ .
Let $r=x^{2}+y-z$ and $z^{3}-xy+yz+y^{3}=1.$ Assume that $x$ and $y$ are independent variables. At $\left( x,y,z \right)=\left ( 2,-1,1 \right ),$ the value (correct to t...
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GATE ECE 2014 Set 2 | Question: 45
The value of the integral $\int_{-\infty }^{\infty } \text{sinc}^{2}(5t) \: dt$ is _______.
The value of the integral $\int_{-\infty }^{\infty } \text{sinc}^{2}(5t) \: dt$ is _______.
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GATE ECE 2018 | Question: 6
Consider $p(s)=s^{3}+ a_{2}s^{2}+a_{1}s+a_{0}$ with all real coefficients. It is known that its derivatives ${p}'(s)$ has no real roots. The number of real roots of $p(s)$ is $0$ $1$ $2$ $3$
Consider $p(s)=s^{3}+ a_{2}s^{2}+a_{1}s+a_{0}$ with all real coefficients. It is known that its derivatives ${p}'(s)$ has no real roots. The number of real roots of $p(s)...
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GATE ECE 2018 | Question: 55
Let $X\left[ k \right ] = k + 1,0\leq k\leq 7$ be $8$-point $\:\text{DFT}\:$ of a sequence $x[n]$. where $X\left [ k \right ]=\sum_{n=0}^{N-1}x \left [ n \right ]e^{-j2\pi nk/N}$. The value (correct to two decimal places) of $\sum_{n=0}^{3}x \left [ 2n \right ]$ is ________.
Let $X\left[ k \right ] = k + 1,0\leq k\leq 7$ be $8$-point $\:\text{DFT}\:$ of a sequence $x[n]$.where $X\left [ k \right ]=\sum_{n=0}^{N-1}x \left [ n \right ]e^{-j2\pi...
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GATE ECE 2015 Set 1 | Question: 28
Which one of the following graphs describes the function $f(x)=e^{-x}(x^2+x+1)$?
Which one of the following graphs describes the function $f(x)=e^{-x}(x^2+x+1)$?
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TIFR ECE 2011 | Question: 13
If $a_k$ is an increasing function of $k$, i.e. $a_1<a_2<\ldots<a_k \ldots$. Then which of the following is $\text{TRUE.}$ $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k}}=\infty$ ... . Either $(a)$ or $(b)$. $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k}}=0$. None of the above.
If $a_k$ is an increasing function of $k$, i.e. $a_1<a_2<\ldots<a_k \ldots$. Then which of the following is $\text{TRUE.}$$\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \fr...
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TIFR ECE 2010 | Question: 20
The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is $2 t$ $t^{2}$ $2 t+t e^{-t^{2} / 2}$ Does not have a simple closed form expression None of the above
The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is$2 t$$t^{2}$$2 t+t e^{-t^{2} / 2}$Does not have a sim...
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TIFR ECE 2022 | Question: 13
Calculate the minimum value attained by the function \[\sin (\pi x)-\sqrt{2} \pi x^{2}\] for values of $x$ which lie in the interval $[0,1]$. $\frac{1}{\sqrt{2}}\left(1-\frac{\pi}{8}\right)$ $0$ $1-\frac{\pi}{2 \sqrt{2}}$ $-\frac{1}{\sqrt{2}}\left(1+\frac{9 \pi}{2}\right)$ $-\sqrt{2} \pi$
Calculate the minimum value attained by the function\[\sin (\pi x)-\sqrt{2} \pi x^{2}\]for values of $x$ which lie in the interval $[0,1]$.$\frac{1}{\sqrt{2}}\left(1-\fra...
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TIFR ECE 2011 | Question: 10
Let $f(x)=|x|$, for $x \in(-\infty, \infty)$. Then $f(x)$ is not continuous but differentiable. $f(x)$ is continuous and differentiable. $f(x)$ is continuous but not differentiable. $f(x)$ is neither continuous nor differentiable. None of the above.
Let $f(x)=|x|$, for $x \in(-\infty, \infty)$. Then$f(x)$ is not continuous but differentiable.$f(x)$ is continuous and differentiable.$f(x)$ is continuous but not differe...
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GATE ECE 2015 Set 1 | Question: 29
The maximum area (in square units) of a rectangle whose vertices lie on the eclipse $x^2+4y^2=1$ is __________.
The maximum area (in square units) of a rectangle whose vertices lie on the eclipse $x^2+4y^2=1$ is __________.
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GATE ECE 2016 Set 1 | Question: 26
The integral $\frac{1}{2\pi} \iint_D(x+y+10) \,dx\,dy$, where $D$ denotes the disc: $x^2+y^2\leq 4$,evaluates to _________
The integral $\frac{1}{2\pi} \iint_D(x+y+10) \,dx\,dy$, where $D$ denotes the disc: $x^2+y^2\leq 4$,evaluates to _________
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GATE ECE 2015 Set 3 | Question: 5
The value of $\displaystyle{}\sum_{n=0}^{\infty} n \left(\dfrac{1}{2}\right)^{n}$ is ________.
The value of $\displaystyle{}\sum_{n=0}^{\infty} n \left(\dfrac{1}{2}\right)^{n}$ is ________.
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