Recent questions tagged calculus

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GATE Electrical 2021 | Question: 3
Let $f\left ( x \right )$ be a real-valued function such that ${f}'\left ( x_{0} \right )=0$ for some $x _{0} \in\left ( 0,1 \right ),$ and ${f}''\left ( x \right )> 0$ for all $x \in \left ( 0,1 \right )$ ... $(0,1)$ one local maximum in $(0,1)$ exactly one local minimum in $(0,1)$ two distinct local minima in $(0,1)$

asked Feb 20 in Calculus by Arjun (7.8k points)

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GATE Electrical 2021 | Question: 5
Which one of the following vector functions represents a magnetic field $\overrightarrow{B}$? $\text{($\hat{X}, \hat{Y}$ and $\hat{Z}$ are unit vectors along x-axis, y-axis, and z-axis, respectively)}$ $10x\hat{X}+20y\hat{Y}-30z\hat{Z}$ $10y\hat{X}+20x\hat{Y}-10z\hat{Z}$ $10z\hat{X}+20y\hat{Y}-30x\hat{Z}$ $10x\hat{X}-30z\hat{Y}+20y\hat{Z}$

asked Feb 20 in Calculus by Arjun (7.8k points)

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GATE Electrical 2021 | Question: 13
Suppose the circles $x^{2}+y^{2}=1$ and $\left ( x-1\right )^{2}+\left ( y-1 \right )^{2}=r^{2}$ intersect each other orthogonally at the point $\left ( u,v \right )$. Then $u+v=$ _______________.

asked Feb 20 in Calculus by Arjun (7.8k points)

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GATE Electrical 2021 | Question: 26
In the open interval $\left ( 0,1 \right )$, the polynomial $p\left ( x \right) =x^{4}-4x^{3}+2$ has two real roots one real root three real roots no real roots

asked Feb 20 in Calculus by Arjun (7.8k points)

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GATE Electrical 2021 | Question: 28
Let $\left ( -1 -j \right ), \left ( 3 -j \right ), \left ( 3 + j \right )$ and $\left ( -1+ j \right )$ be the vertices of a rectangle $C$ in the complex plane. Assuming that $C$ is traversed in counter-clockwise direction, the value of the contour integral $\oint _{C}\dfrac{dz}{z^{2}\left ( z-4 \right )}$ is $j\pi /2$ $0$ $-j\pi /8$ $j\pi /16$

asked Feb 20 in Calculus by Arjun (7.8k points)

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GATE Electrical 2020 | Question: 1
$ax^{3}+bx^{2}+cx+d$ is a polynomial on real $\text{x}$ over real coefficients $\text{a, b, c, d}$ wherein $a\neq 0.$ Which of the following statements is true? $\text{d}$ can be chosen to ensure that $\text{x = 0}$ is a root for any ... $\text{a, b, c, d}$ can be chosen to ensure that all roots are complex. $\text{c}$ alone cannot ensure that all roots are real.

asked Feb 28, 2020 in Calculus by jothee (1.8k points)

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GATE Electrical 2020 | Question: 2
Which of the following is true for all possible non-zero choices of integers $m,n;m\neq n,$ or all possible non-zero choices of real numbers $p,q;p\neq q,$ ...

asked Feb 28, 2020 in Calculus by jothee (1.8k points)

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GATE Electrical 2020 | Question: 26
For real numbers, $\text{x}$ and $\text{y}$, with $y=3x^{2}+3x+1$, the maximum and minimum value of $\text{y}$ for $\text{x}$ $\in \left [ -2,0 \right ]$ are respectively, ______. $7$ and $1/4$ $7$ and $1$ $-2$ and $-1/2$ $1$ and $1/4$

asked Feb 28, 2020 in Calculus by jothee (1.8k points)

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GATE Electrical 2020 | Question: 27
The vector function expressed by $F=a_{x}\left ( 5y-k_{1} z\right )+a_{y}\left ( 3z+k_{2}x \right )+a_{z}\left ( k_{3} y-4x\right )$ represents a conservative field, where $a_{x}, a_{y},a_{z}$ are unit vectors along $x, y$ and $z$ directions, respectively. The values of constants ... $k_{1}=3, k_{2}=8,k_{3}=5$ $k_{1}=4, k_{2}=5,k_{3}=3$ $k_{1}=0, k_{2}=0,k_{3}=0$

asked Feb 28, 2020 in Calculus by jothee (1.8k points)

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GATE Electrical 2019 | Question: 18
If $f=2x^{3}+3y^{2}+4z$, the value of line integral $\int_{c} \text{grad}f \cdot d \textbf{r}$ evaluated over contour $C$ formed by the segments $(-3,-3,2)\rightarrow(2,-3,2)\rightarrow(2,6,2) \rightarrow(2,6,-1) $ is___________.

asked Feb 12, 2019 in Calculus by Arjun (7.8k points)

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GATE Electrical 2019 | Question: 28
A periodic function $f(t)$, with a period of $2 \pi$, is represented as its Fourier series, $f(t) = a_0 + \sum_{n=1}^{\infty }a_n \cos nt + \sum_{n=1}^{\infty} b_n \sin nt.$ ... $a_1 = \frac{A}{2}; \: b_1 = 0$ $a_1 = 0; \: b_1 = \frac{A}{\pi}$ $a_1 = 0;b_1 = \frac{A}{2}$

asked Feb 12, 2019 in Calculus by Arjun (7.8k points)

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GATE Electrical 2019 | Question: 39
If $\textbf{A}= 2x \textbf{i} + 3y \textbf{j} +4z \textbf{k}$ and $u=x^2+y^2+z^2$, then $\text{div} \big(u \textbf{A} \big)$ at $(1,1,1)$ is _______

asked Feb 12, 2019 in Calculus by Arjun (7.8k points)

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GATE Electrical 2012 | Question: 27
The maximum value of $f(x) = x^3-9x^2+24x+5$ in the interval $[1,6]$ is $21$ $25$ $41$ $46$

asked Mar 24, 2018 in Calculus by Andrijana3306 (1.4k points)

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GATE Electrical 2018 | Question: 43
Let $f(x) = 3x^3-7x^2+5x+6$. The maximum value of $f(x)$ over the interval $[0,2]$ is ________ (up to one decimal place).

asked Feb 19, 2018 in Calculus by Arjun (7.8k points)

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GATE Electrical 2018 | Question: 42
As shown in the figure, $C$ is the arc from the point $(3,0)$ to the point $(0,3)$ on the circle $x^2+y^2=9$. The value of the integral $\int_C (y^2+2yx) dx +(2xy+x^2)dy$ is ________ (up to $2$ decimal places).

asked Feb 19, 2018 in Calculus by Arjun (7.8k points)

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GATE Electrical 2018 | Question: 18
Let $f$ be a real-valued function of a real variable defined as $f(x)=x – [x]$, where $[x]$ denotes the largest integer less than or equal to $x$. The value of $\int_{0.25}^{1.25} f(x) dx$ is _______ (up to $2$ decimal places).

asked Feb 19, 2018 in Calculus by Arjun (7.8k points)

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GATE Electrical 2018 | Question: 11
Let $f$ be a real-valued function of a real variable defined as $f(x)=x^2$ for $x \geq 0$, and $f(x)=-x^2$ for $x<0$. Which one of the following statements is true? $f(x)$ is discontinuous at $x=0$ $f(x)$ ... is differentiable but its first derivative is not continuous at $x=0$ $f(x)$ is differentiable but its first derivative is not differentiable at $x=0$

asked Feb 19, 2018 in Calculus by Arjun (7.8k points)

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GATE Electrical 2018 | Question: 12
The value of the directional derivative of the function $\Phi (x,y,z) = xy^2 +yz^2+zx^2$ at the point $(2,-1,1)$ in the direction of the vector $\textbf{p}= \textbf{i} +2 \textbf{j} + 2 \textbf{k}$ is $1$ $0.95$ $0.93$ $0.9$

asked Feb 19, 2018 in Calculus by Arjun (7.8k points)

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GATE Electrical 2017 Set 2 | Question: 26
Let $ g(x)= \begin{cases} -x & \ x \leq 1 \\ x+1 & \ x \geq 1 \end{cases}$ and $ f(x)= \begin{cases} 1-x & \ x \leq 0 \\ x^{2} & \ x > 0 \end{cases}$. Consider the composition of $f$ and $g$ ... $(f {\circ} g) (x)$ present in the interval $(-\infty, 0)$ is: $0$ $1$ $2$ $4$

asked Feb 27, 2017 in Calculus by Arjun (7.8k points)

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GATE Electrical 2017 Set 2 | Question: 27
The value of the contour integral in the complex plane $\oint \frac{z^{3}-2z+3}{z-2} dz$ along the contour $\mid z \mid =3$, taken counter- clockwise is $-18 \pi i$ $0$ $14 \pi i$ $48 \pi i$

asked Feb 27, 2017 in Calculus by Arjun (7.8k points)

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GATE Electrical 2017 Set 2 | Question: 18
Consider a function $f(x, y, z)$ given by $f(x, y, z)=(x^{2}+y^{2}-2z^{2})(y^{2}+z^{2})$ The partial derivative of this function with respect to $x$ at the point, $x=2, y=1$ and $z=3$ is _______.

asked Feb 27, 2017 in Calculus by Arjun (7.8k points)

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GATE Electrical 2017 Set 2 | Question: 19
Let $x$ and $y$ be integers satisfying the following equations $2x^{2}+y^{2}=34$ $x+2y=11$ The value of $(x+y)$ is _______.

asked Feb 27, 2017 in Calculus by Arjun (7.8k points)

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GATE Electrical 2017 Set 2 | Question: 20
Let $y^{2}-2y+1=x$ and $\sqrt{x}+y=5$. The value of $x+\sqrt{y}$ equals _________. (Give the answer up to three decimal places).

asked Feb 27, 2017 in Calculus by Arjun (7.8k points)

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GATE Electrical 2017 Set 2 | Question: 3
The figures show diagramatic representations of vector fields $\vec{X}, \vec{Y}, \text{and} \vec{Z}$ ... $\bigtriangledown . \vec{X}=0,\bigtriangledown \times \vec{Y} = 0, \bigtriangledown \times \vec{Z}=0$

asked Feb 27, 2017 in Calculus by Arjun (7.8k points)

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GATE Electrical 2017 Set 1 | Question: 26
A function $f(x)$ is defined as $f(x)= \begin{cases} e^{x}, & x < 1 \\ \text{In } x+ax^{2}+bx, & x\geq 1 \end{cases}$, where $x \in \mathbb{R}$ Which one of the following statement is TRUE? $f(x)$ is NOT differentiable at $x=1$ ... for all values of $a$ and $b$ such that $a+b=e$. $f(x)$ is differentiable at $x=1$ for all values of $a$ and $b$.

asked Feb 27, 2017 in Calculus by Arjun (7.8k points)

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GATE Electrical 2017 Set 1 | Question: 28
Consider the line integral $I=\int_{c} (x^{2}+iy^{2})dz$, where $z=x+iy$. The line $c$ is shown in the figure below. The value of $I$ is $\frac{1}{2}i \\ $ $\frac{2}{3}i \\ $ $\frac{3}{4}i \\ $ $\frac{4}{5}i$

asked Feb 27, 2017 in Calculus by Arjun (7.8k points)

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GATE Electrical 2017 Set 1 | Question: 17
Let $I= c\int \int _{R} xy^{2} dxdy$, where $R$ is the region shown in the figure and $c= 6 \times 10^{-4}$. The value of $I$ equals _________. (Give the answer up to two decimal places.)

asked Feb 27, 2017 in Calculus by Arjun (7.8k points)

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GATE Electrical 2017 Set 1 | Question: 2
For a complex number $z,\displaystyle{} \lim_{z \rightarrow i} \frac{z^{2}+1}{z^{3}+2z-i (z^{2}+2)}$ is $-2i$ $-i$ $i$ $2i$

asked Feb 27, 2017 in Calculus by Arjun (7.8k points)

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GATE Electrical 2013 | Question: 46
A function $y=5x^2+10x$ is defined over an open interval $x$ = $(1, 2)$ . At least at one point in this interval, $\dfrac{\mathrm{dy} }{\mathrm{d} x}$ is exactly $20$ $25$ $30$ $35$

asked Feb 12, 2017 in Calculus by piyag476 (1.5k points)

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GATE Electrical 2013 | Question: 24
Given a vector field $\textbf{F}=y^2x \textbf{a}_x-yz \textbf{a}_y-x^2 \textbf{a}_z$ the line integral $\int \textbf{F} \cdot d \textbf{l}$ evaluated along a segment on the $x$-axis from $x=1$ to $x=2$ is $-2.33$ $0$ $2.33$ $7$

asked Feb 12, 2017 in Calculus by piyag476 (1.5k points)

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GATE Electrical 2014 Set 3 | Question: 3
Let $\nabla .(fv)=x^2y+y^2z+z^2x$ , where $f$ and $v$ are scalar and vector fields respectively. If $v=yi+zj+xk$ then $v.\Delta f$ is $x^2y+y^2z+z^2x$ $2xy+2yz+2zx$ $x+y+z$ $0$