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Recent questions and answers in Calculus
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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.
answered
Mar 17
in
Calculus
by
Adarsh Joshi
(
150
points)
gate2020ee
calculus
polynomials
0
votes
0
answers
2
GATE Electrical 2021  Question: 3
Let $f\left ( x \right )$ be a realvalued 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)
gateee2021
calculus
maximaminima
0
votes
0
answers
3
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 xaxis, yaxis, and zaxis, 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)
gateee2021
calculus
fieldvectors
0
votes
0
answers
4
GATE Electrical 2021  Question: 13
Suppose the circles $x^{2}+y^{2}=1$ and $\left ( x1\right )^{2}+\left ( y1 \right )^{2}=r^{2}$ intersect each other orthogonally at the point $\left ( u,v \right )$. Then $u+v=$ _______________.
asked
Feb 20
in
Calculus
by
Arjun
(
7.8k
points)
gateee2021
numericalanswers
calculus
curves
0
votes
0
answers
5
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)
gateee2021
calculus
polynomials
0
votes
0
answers
6
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 counterclockwise direction, the value of the contour integral $\oint _{C}\dfrac{dz}{z^{2}\left ( z4 \right )}$ is $j\pi /2$ $0$ $j\pi /8$ $j\pi /16$
asked
Feb 20
in
Calculus
by
Arjun
(
7.8k
points)
gateee2021
calculus
contourplots
0
votes
0
answers
7
GATE Electrical 2020  Question: 2
Which of the following is true for all possible nonzero choices of integers $m,n;m\neq n,$ or all possible nonzero choices of real numbers $p,q;p\neq q,$ ...
asked
Feb 28, 2020
in
Calculus
by
jothee
(
1.8k
points)
gate2020ee
calculus
definiteintegral
0
votes
0
answers
8
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)
gate2020ee
calculus
maximaminima
0
votes
0
answers
9
GATE Electrical 2020  Question: 27
The vector function expressed by $F=a_{x}\left ( 5yk_{1} z\right )+a_{y}\left ( 3z+k_{2}x \right )+a_{z}\left ( k_{3} y4x\right )$ 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)
gate2020ee
calculus
fieldvectors
0
votes
0
answers
10
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)
gate2019ee
numericalanswers
calculus
lineintegral
0
votes
0
answers
11
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)
gate2019ee
calculus
fourierseries
0
votes
0
answers
12
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)
gate2019ee
numericalanswers
calculus
divergence
0
votes
0
answers
13
GATE Electrical 2012  Question: 38
The direction of vector $\textbf{A}$ is radically outward from the origin, with $\mid \textbf{A} \mid k r ^n$ where $r^2=x^2+y^2+z^2$ and $k$ is a constant. The value of $n$ for which $\nabla \cdot \textbf{A} = 0$ is $2$ $2$ $1$ $0$
asked
Mar 24, 2018
in
Calculus
by
Andrijana3306
(
1.4k
points)
gate2012ee
differentialequations
0
votes
0
answers
14
GATE Electrical 2012  Question: 27
The maximum value of $f(x) = x^39x^2+24x+5$ in the interval $[1,6]$ is $21$ $25$ $41$ $46$
asked
Mar 24, 2018
in
Calculus
by
Andrijana3306
(
1.4k
points)
gate2012ee
calculus
maximaminima
0
votes
0
answers
15
GATE Electrical 2018  Question: 43
Let $f(x) = 3x^37x^2+5x+6$. 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)
gate2018ee
numericalanswers
calculus
maximaminima
0
votes
0
answers
16
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)
gate2018ee
numericalanswers
calculus
definiteintegral
0
votes
0
answers
17
GATE Electrical 2018  Question: 18
Let $f$ be a realvalued 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)
gate2018ee
numericalanswers
calculus
definiteintegral
0
votes
0
answers
18
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)
gate2018ee
calculus
directionalderivatives
0
votes
0
answers
19
GATE Electrical 2018  Question: 11
Let $f$ be a realvalued 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)
gate2018ee
calculus
continuityanddifferentiability
0
votes
0
answers
20
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} 1x & \ 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)
gate2017ee2
calculus
continuity
0
votes
0
answers
21
GATE Electrical 2017 Set 2  Question: 27
The value of the contour integral in the complex plane $\oint \frac{z^{3}2z+3}{z2} dz$ 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)
gate2017ee2
calculus
contourintegral
0
votes
0
answers
22
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)
gate2017ee2
numericalanswers
calculus
curves
0
votes
0
answers
23
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)
gate2017ee2
numericalanswers
calculus
curves
0
votes
0
answers
24
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)
gate2017ee2
numericalanswers
calculus
derivatives
partialderivatives
0
votes
0
answers
25
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)
gate2017ee2
calculus
fieldvectors
0
votes
0
answers
26
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)
gate2017ee1
calculus
lineintegral
0
votes
0
answers
27
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)
gate2017ee1
calculus
continuityanddifferentiability
0
votes
0
answers
28
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)
gate2017ee1
numericalanswers
calculus
doubleintegral
0
votes
0
answers
29
GATE Electrical 2017 Set 1  Question: 2
For a complex number $z,\displaystyle{} \lim_{z \rightarrow i} \frac{z^{2}+1}{z^{3}+2zi (z^{2}+2)}$ is $2i$ $i$ $i$ $2i$
asked
Feb 27, 2017
in
Calculus
by
Arjun
(
7.8k
points)
gate2017ee1
calculus
limits
complexnumber
0
votes
0
answers
30
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)
gate2013ee
calculus
derivatives
0
votes
0
answers
31
GATE Electrical 2013  Question: 24
Given a vector field $\textbf{F}=y^2x \textbf{a}_xyz \textbf{a}_yx^2 \textbf{a}_z$ 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)
gate2013ee
calculus
fieldvector
integral
0
votes
0
answers
32
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 $x^2y+y^2z+z^2x$ $2xy+2yz+2zx$ $x+y+z$ $0$
asked
Feb 12, 2017
in
Calculus
by
makhdoom ghaya
(
9.3k
points)
gate2014ee3
calculus
fieldvectors
0
votes
0
answers
33
GATE Electrical 2014 Set 2  Question: 26
To evaluate the double integral $\displaystyle \int_{0}^{8} \bigg (\int_{(y/2)}^{y/2+1} \bigg (\dfrac{2xy}{2} \bigg)dx \bigg)dy$ , we make the substitution $u=\bigg (\dfrac{2xy}{2} \bigg)$ and $v=\dfrac{y}{2}$ ... $\displaystyle \int_{0}^{4} \bigg (\int_{0}^{2}u \: du \bigg ) dv$
asked
Feb 12, 2017
in
Calculus
by
makhdoom ghaya
(
9.3k
points)
gate2014ee2
calculus
definiteintegral
doubleintegral
0
votes
0
answers
34
GATE Electrical 2014 Set 2  Question: 28
The minimum value of the function $f(x)=x^33x^224x+100$ in the interval $[3,3]$ is $20$ $28$ $16$ $32$
asked
Feb 12, 2017
in
Calculus
by
makhdoom ghaya
(
9.3k
points)
gate2014ee2
calculus
maximaminima
0
votes
0
answers
35
GATE Electrical 2014 Set 2  Question: 3
Minimum of the real valued function $f(x)=(x1)^{2/3}$ occurs at $x$ equal to $\infty$ $0$ $1$ $\infty$
asked
Feb 12, 2017
in
Calculus
by
makhdoom ghaya
(
9.3k
points)
gate2014ee2
calculus
maximaminima
0
votes
0
answers
36
GATE Electrical 2014 Set 1  Question: 28
The line integral of function $F = yzi$, in the counterclockwise direction, along the circle $x^2+y^2 = 1$ at $z = 1$ is $2\pi$ $\pi$ $\pi$ $2\pi$
asked
Feb 12, 2017
in
Calculus
by
makhdoom ghaya
(
9.3k
points)
gate2014ee1
calculus
lineintegral
circleequation
0
votes
0
answers
37
GATE Electrical 2014 Set 1  Question: 17
In the formation of RouthHurwitz array for a polynomial, all the elements of a row have zero values. This premature termination of the array indicates the presence of only one root at the origin Imaginary roots only positive real roots only negative real roots
asked
Feb 12, 2017
in
Calculus
by
makhdoom ghaya
(
9.3k
points)
gate2014ee1
calculus
polynomial
routhhurwitzarray
0
votes
0
answers
38
GATE Electrical 2014 Set 1  Question: 2
Let $f(x)=xe^{x}$ . The maximum value of the function in the interval $(0,\infty)$ is $e^{1}$ $e$ $1e^{1}$ $1+e^{1}$
asked
Feb 12, 2017
in
Calculus
by
makhdoom ghaya
(
9.3k
points)
gate2014ee1
calculus
maximaminima
0
votes
0
answers
39
GATE Electrical 2015 Set 2  Question: 26
The volume enclosed by the surface $f(x, y) = e^{x}$ over the triangle bounded by the lines $x = y; x = 0; y = 1$ in the $xy$ plane is ________.
asked
Feb 12, 2017
in
Calculus
by
makhdoom ghaya
(
9.3k
points)
gate2015ee2
calculus
volumeintegral
numericalanswers
0
votes
0
answers
40
GATE Electrical 2015 Set 2  Question: 3
Match the following. ... $P4; Q1; R3; S2$ $P4; Q3; R1; S2$ $P3; Q4; R2; S1$
asked
Feb 12, 2017
in
Calculus
by
makhdoom ghaya
(
9.3k
points)
gate2015ee2
calculus
divergence
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