Recent questions tagged calculus

0 votes

0 answers

1

GATE Civil 2021 Set 2 | Question: 1
The value of $\lim \limits_{x\rightarrow \infty } \dfrac{x \:\text{ln}\left ( x \right )}{1+x^{2}}$ is $0$ $1.0$ $0.5$ $\infty$

go_editor asked in Calculus Mar 1, 2021

by go_editor

5.3k points

0 votes

0 answers

2

GATE Civil 2021 Set 2 | Question: 3
The unit normal vector to the surface $X^{2} + Y^{2} + Z^{2} – 48 = 0$ at the point $(4, 4, 4)$ is $\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$ $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$ $\frac{2}{\sqrt{2}},\frac{2}{\sqrt{2}},\frac{2}{\sqrt{2}}$ $\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5}},\frac{1}{\sqrt{5}}$

go_editor asked in Calculus Mar 1, 2021

by go_editor

5.3k points

0 votes

0 answers

3

GATE Civil 2021 Set 2 | Question: 18
The value ($\textit{round off to one decimal place}$) of $\int_{-1}^{1}x\:e^{\left | x \right |}dx$ is ______________

go_editor asked in Calculus Mar 1, 2021

by go_editor

5.3k points

0 votes

0 answers

4

GATE Civil 2021 Set 2 | Question: 36
A function is defined in Cartesian coordinate system as $f(x,y)=xe^{y}$. The value of the directional derivative of the function ($\textit{in integer}$) at the point $(2,0)$ along the direction of the straight line segment from point $(2, 0)$ to point $\left ( \dfrac{1}{2} ,2\right )$ is _________

go_editor asked in Calculus Mar 1, 2021

by go_editor

5.3k points

0 votes

0 answers

5

GATE Civil 2021 Set 1 | Question: 18
Consider the limit: $\lim_{x\rightarrow 1}\left ( \frac{1}{\text{ln}\:x} - \frac{1}{x-1}\right )$ The limit (correct up to one decimal place) is _____________

Arjun asked in Calculus Feb 20, 2021

by Arjun

11.6k points

0 votes

0 answers

6

GATE Civil 2021 Set 1 | Question: 19
The volume determined from $\int \int \int _{V}\:8\:xyz\:dV$ for $V=\left [ 2,3 \right ]\times \left [ 1,2 \right ]\times \left [ 0,1 \right ]$ will be ($\textit{in integer}$) _____________

Arjun asked in Calculus Feb 20, 2021

by Arjun

11.6k points

0 votes

1 answer

7

GATE2020-CE-1-2
The value of $\displaystyle{} \lim_{x\to\infty}\dfrac{x^2-5x+4}{4x^2+2x}$ is $0 \\$ $\dfrac{1}{4} \\$ $\dfrac{1}{2} \\$ $1$

go_editor asked in Calculus Feb 28, 2020

by go_editor

5.3k points

0 votes

0 answers

8

GATE2020-CE-1-4
The area of an ellipse represented by an equation $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ is $\dfrac{\pi ab}{4} \\$ $\dfrac{\pi ab}{2} \\$ $\pi ab \\$ $\dfrac{4\pi ab}{3}$

go_editor asked in Calculus Feb 28, 2020

by go_editor

5.3k points

0 votes

0 answers

9

GATE2020-CE-1-27
A continuous function $f(x)$ is defined. If the third derivative at $x_i$ is to be computed by using the fourth order central finite-divided-difference scheme (with step length $=h$ ...

go_editor asked in Calculus Feb 28, 2020

by go_editor

5.3k points

0 votes

0 answers

10

GATE2020-CE-1-39
If $C$ represents a line segment between $(0,0,0)$ and $(1,1,1)$ in Cartesian coordinate system, the value (expressed as integer) of the line integral $\int_C [(y+z)dx+(x+z)dy+(x+y)dz] $ is ______

go_editor asked in Calculus Feb 28, 2020

by go_editor

5.3k points

0 votes

0 answers

11

GATE2020 CE-2-2
The value of $\lim_{x\to\infty}\dfrac{\sqrt{9x^2+2020}}{x+7}\:\text{is}$ $\dfrac{7}{9}$ $1$ $3$ indeterminable

go_editor asked in Calculus Feb 13, 2020

by go_editor

5.3k points

0 votes

0 answers

12

GATE2020 CE-2-24
Velocity distribution in a boundary layer is given by $\dfrac{u}{U_\infty} = \sin\large \left( \dfrac{\pi}{2}\dfrac{y}{\delta} \right)$, where $u$ is the velocity at vertical coordinate $y,\: U_\infty$ is the free stream velocity and $\delta$ is the boundary layer ... $\ s^{-1}$, round off to two decimal places) at $y = 0$, is ________.

go_editor asked in Calculus Feb 13, 2020

by go_editor

5.3k points

0 votes

0 answers

13

GATE2017 CE-2-2
Let $w=f(x,y)$, where $x$ and $y$ are functions of $t$. Then, according to the chain rule, $\dfrac{dw}{dt}$ is equal to $\dfrac{dw}{dx} \dfrac{dx}{dt} + \dfrac{dw}{dy} \dfrac{dt}{dt} \\$ ... $\dfrac{d w}{dx} \dfrac{\partial x}{\partial t} + \dfrac{dw}{dy} \dfrac{\partial y}{ \partial t}$

gatecse asked in Calculus Aug 7, 2019

by gatecse

4.0k points

0 votes

0 answers

14

GATE2017 CE-2-19
The divergence of the vector field $V=x^2 i + 2y^3 j + z^4 k$ at $x=1, \: y=2, \: z=3$ is ________

gatecse asked in Calculus Aug 7, 2019

by gatecse

4.0k points

0 votes

0 answers

15

GATE2017 CE-2-26
The tangent to the curve represented by $y=x \text{ ln }x$ is required to have $45^{\circ}$ inclination with the $x$-axis. The coordinates of the tangent point would be $(1,0)$ $(0, 1)$ $(1,1)$ $(\sqrt{2}, (\sqrt{2})$

gatecse asked in Calculus Aug 7, 2019

by gatecse

4.0k points

0 votes

0 answers

16

GATE2017 CE-2-27
Consider the following definite integral: $I= \displaystyle{} \int_0^1 \dfrac{(\sin ^{-1}x)^2}{\sqrt{1-x^2}} dx$. The value of the integral is $\dfrac{\pi ^3}{24} \\$ $\dfrac{\pi ^3}{12} \\$ $\dfrac{\pi ^3}{48} \\$ $\dfrac{\pi ^3}{64}$

gatecse asked in Calculus Aug 7, 2019

by gatecse

4.0k points

2 votes

1 answer

17

GATE2019 CE-1: 1
Which one of the following is correct? $\displaystyle{} \lim_{x\rightarrow 0} \left( \dfrac{\sin4x}{\sin2x}\right)=2\;\text{and}\: \lim_{x\rightarrow 0} \left( \dfrac{\tan x}{x}\right)=1$ ...

Arjun asked in Calculus Feb 14, 2019

by Arjun

11.6k points

1 vote

0 answers

18

GATE2019 CE-1: 4
For a small value of $h$, the Taylor series expansion for $f(x+h)$ is $f(x)+h{f}' (x) + \dfrac{h^2}{2!}{f}''(x) + \dfrac{h^3}{3!}{f}'''(x)+\dots \infty \\$ $f(x)-h{f}' (x) + \dfrac{h^2}{2!}{f}''(x) - \dfrac{h^3}{3!}{f}'''(x)+ \dots \infty \\$ ... $f(x)-h{f}' (x) + \dfrac{h^2}{2}{f}''(x) - \dfrac{h^3}{3}{f}'''(x)+ \dots \infty $

Arjun asked in Calculus Feb 14, 2019

by Arjun

11.6k points

0 votes

0 answers

19

GATE2019 CE-1: 26
Which one of the following is NOT a correct statement? The function $\sqrt[x]{x}, \: (x>0)$, has the global maxima at $x=e$ The function $\sqrt[x]{x}, \: (x>0)$, has the global minima at $x=e$ The function $x^3$ has neither global minima nor global maxima The function $\mid x \mid$ has the global minima at $x=0$

Arjun asked in Calculus Feb 14, 2019

by Arjun

11.6k points

0 votes

0 answers

20

GATE2019 CE-1: 30
Consider two functions: $x=\psi \text{ ln } \phi$ and $y= \phi \text{ ln } \psi$. Which one of the following is the correct expression for $\frac{\partial \psi}{\partial x}$? $\dfrac{x \: \text{ln } \psi}{\text{ln } \phi \text{ ln } \psi -1} \\$ ... $\dfrac{\: \text{ln } \psi}{\text{ln } \phi \text{ ln } \psi -1}$

Arjun asked in Calculus Feb 14, 2019

by Arjun

11.6k points

0 votes

0 answers

21

GATE2019 CE-2: 3
The following inequality is true for all $x$ close to $0$. $2-\dfrac{x^2}{3} < \dfrac{x \sin x}{1- \cos x} <2$ What is the value of $\underset{x \to 0}{\lim} \dfrac{x \sin x}{1 – \cos x}$? $0$ $1/2$ $1$ $2$

Arjun asked in Calculus Feb 12, 2019

by Arjun

11.6k points

0 votes

0 answers

22

GATE2019 CE-2: 4
What is curl of the vector field $2x^2y \textbf{i} + 5z^2 \textbf{j} – 4yz \textbf{k}$? $6z \textbf{i} + 4x \textbf{j} – 2x^2 \textbf{k}$ $6z \textbf{i} - 8xy \textbf{j} + 2x^2 y\textbf{k}$ $- 14 z \textbf{i} + 6y \textbf{j} + 2x^2 \textbf{k}$ $-14z \textbf{i} – 2x^2 \textbf{k}$

Arjun asked in Calculus Feb 12, 2019

by Arjun

11.6k points

0 votes

0 answers

23

GATE2016-2-2
The optimum value of the function $f(x)=x^2-4x+2$ is $2$ (maximum) $2$ (minimum) $ – 2$ (maximum) $ – 2$ (minimum)

Milicevic3306 asked in Calculus Mar 28, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

24

GATE2016-2-5
What is the value of $\underset{x \rightarrow 0 \\ y \rightarrow 0}{\lim} \dfrac{xy}{x^2+y^2}$? $1$ $-1$ $0$ Limit does not exist

Milicevic3306 asked in Calculus Mar 28, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

25

GATE2016-2-25
A circular curve of radius R connects two straights with a deflection angle of $60^{\circ}$. The tangent length is $0.577 \: R$ $1.155 \: R$ $1.732 \:R$ $3.464 \: R$

Milicevic3306 asked in Calculus Mar 28, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

26

GATE2016-2-28
The angle of intersection of the curves $x^{2}=4y$ and $y^{2} = 4x$ at point $(0,0)$ is $0^{\circ}$ $30^{\circ}$ $45^{\circ}$ $90^{\circ}$

Milicevic3306 asked in Calculus Mar 28, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

27

GATE2016-2-29
The area between the parabola $x^2=8y$ and the straight line $y=8$ is _______.

Milicevic3306 asked in Calculus Mar 28, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

28

GATE2016-1-27
The value of $\int_0^{\infty} \dfrac{1}{1+x^2} dx + \int _0^{\infty} \dfrac{\sin x}{x} dx$ is $\dfrac{\pi}{2} \\$ $\pi \\$ $\dfrac{3 \pi}{2} \\$ $1$

Milicevic3306 asked in Calculus Mar 28, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

29

GATE2016-1-28
The area of the region bounded by the parabola $y=x^2+1$ and the straight line $x+y=3$ is $\dfrac{59}{6} \\$ $\dfrac{9}{2} \\$ $\dfrac{10}{3} \\$ $\dfrac{7}{6}$

Milicevic3306 asked in Calculus Mar 28, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

30

GATE2016-1-29
The magnitudes of vectors $\textbf{P, Q}$ and $\textbf{R}$ are $100$ kN, $250$ kN and $150$ kN, respectively as shown in the figure. The respective values of the magnitude (in kN) and the direction (with respect to the x-axis) of the resultant vector are $290.9$ and $96.0^{\circ}$ $368.1$ and $94.7^{\circ}$ $330.4$ and $118.9^{\circ}$ $400.1$ and $113.5^{\circ}$

Milicevic3306 asked in Calculus Mar 28, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

31

GATE2017 CE-1: 3
Let $x$ be a continuous variable defined over the interval $(- \infty, \infty)$, and $f(x) = e^{-x-e^{-x}}$. The integral $g(x)= \int f(x) \: dx$ is equal to $e^{e^{-x}}$ $e^{-e^{-x}}$ $e^{-e^{x}}$ $e^{-x}$

Milicevic3306 asked in Calculus Mar 26, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

32

GATE2017 CE-1: 21
$\underset{x \to 0}{\lim} \bigg( \dfrac{\tan x}{x^2-x} \bigg)$ is equal to _________

Milicevic3306 asked in Calculus Mar 26, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

33

GATE2015-2-1
While minimizing the function $f(x)$, necessary and sufficient conditions for a point, $x_0$ to be a minima are: $f’(x_0) > 0$ and $f’’(x_0)=0$ $f’(x_0) < 0$ and $f’’(x_0)=0$ $f’(x_0) = 0$ and $f’’(x_0)<0$ $f’(x_0) = 0$ and $f’’(x_0)>0$

Milicevic3306 asked in Calculus Mar 26, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

34

GATE2015-2-3
Given, $i=\sqrt{-1}$, the value of the definite integral, $I= \int_0^{\pi/2} \dfrac{\cos x+ i \sin x}{\cos x – i \sin x} dx$ is: $1$ $-1$ $i$ $-i$

Milicevic3306 asked in Calculus Mar 26, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

35

GATE2015-2-4
$\underset{x \to \infty}{\lim} \bigg( 1+ \dfrac{1}{x} \bigg)^{2x} $ is equal to $e^{-2}$ $e$ $1$ $e^2$

Milicevic3306 asked in Calculus Mar 26, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

36

GATE2015-1-29
Consider the following complex function: $f(z) = \frac{9}{(z-1)(z+2)^2}$ Which of the following is one of the residues of the above function? $-1$ $9/16$ $2$ $9$

Milicevic3306 asked in Calculus Mar 26, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

37

GATE2015-1-30
The directional derivative of the field $u(x,y,z) = x^2-3yz$ in the direction of the vector $(\hat{i}+\hat{j}-2 \hat{k})$ at point $(2, -1, 4)$ is _________.

Milicevic3306 asked in Calculus Mar 26, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

38

GATE2014-2-26
The expression $\displaystyle{} \lim_{a \to 0} \dfrac{x^a-1}{a}$ is equal to $\log x$ $0$ $x \log x$ $\infty$

Milicevic3306 asked in Calculus Mar 26, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

39

GATE2014-2-3
$z=\dfrac{2-3i}{-5+i}$ can be expressed as $-0.5-0.5i$ $-0.5+0.5i$ $0.5-0.5i$ $0.5+0.5i$

Milicevic3306 asked in Calculus Mar 26, 2018

by Milicevic3306

11.9k points

0 votes

0 answers

40

GATE2014-1-28
A particle moves along a curve whose parametric equations are: $x=t^3+2t$, $y=-3e^{-2t}$ and $z=2 \sin (5t)$, where $x$, $y$ and $z$ show variations of the distance covered by the particle (in cm) with time $t$ (in s). The magnitude of the acceleration of the particle (in $cm/s^2$) at $t=0$ is ________

Milicevic3306 asked in Calculus Mar 26, 2018

by Milicevic3306

11.9k points