Integration

Q. A person pushes a box on a rough horizontal platform surface. He applies a force of $200\text{N}$ over a distance of $15\text{m}$. Thereafter, he gets progressively tired, and his applied force reduces linearly with distance to $100\text{N}$. The total distance through which the box has been moved, is $30\text{m}$. What is the work done by the person, during the total movement of the box ?

1. $3280\text{J}$
2. $2780\text{J}$
3. $5690\text{J}$
4. $5250\text{J}$

Q. At the moment t = 0 particle leaves origin and moves in the positive direction of x-axis. Its velocity as a function of time is given as v = (5-t) m/s. x-coordinate of particle at t = 2 sec is

1. 6 m
2. 8 m
3. 10 m
4. 12 m

Q. If v = (5 - t) m/s and particle starts from origin then its position time graph is

1. 
2. 
3. 
4. 

Q. The flux passing through the surface $S_5$ will be

1. $-0.135N{m}^2{C}^{-1}$
2. $-0.054N{m}^2{C}^{-1}$
3. $-0.081N{m}^2{C}^{-1}$
4. $-0.054N{m}^2{C}^{-1}$

Q. A point moves in a straight line under the retardation $a{v}^2$. If the initial velocity is $u$, the distance covered in '$t$' seconds is

1. $aut$
2. $\frac{1}{a}\ln \left(aut\right)$
3. $\frac{1}{a}\ln (1+aut)$
4. $a\ln \left(aut\right)$

Q. The displacement of a particle along the x-axis is given by $x=as{in}^2\omega t$. The motion of the particle comes ponds to

1. SHM of frequency $\frac{2\omega }{2\pi }$
2. SHM of frequency $\frac{3\omega }{2\pi }$
3. non SHM
4. SHM of frequency $\frac{\omega }{2\pi }$

Q. At the moment $t=0$ particle leaves origin and moves in the positive direction of $x$-axis. Its velocity as a function of time is given as $v=(5-t)\text{m/s}$. $x$-coordinate of particle at $t=2\text{s}$ is

1. 8 m
2. 6 m
3. 10 m
4. 12 m

Q. The equation of projectile is $y=16x-\frac{x^2}{4}$ the horizontal range is:

1. $8$ m
2. $16$ m
3. $64$ m
4. $12.8$ m

Q. If $v=(5-t)\text{m/s}$ and particle starts from origin then its position time graph is

1. 
2. 
3. 
4. 

Q. The mathematical tool that we use to find the change in velocity from a function of acceleration with respect to time is called integration.

1. True
2. False

Q. Find the integration of $\int \frac{e^{{\tan }^{-1}x}}{1+x^2}.dx$

1. $e^{{\tan }^{-1}x}+C$
2. $e^{-{\tan }^{-1}x}+C$
3. $\frac{e^{{\tan }^{-1}x}}{2}+C$
4. $e^{\tan x}+C$

Q. The mathematical tool that we use to find the change in velocity from a function of acceleration w.r.t time is

Q. If $v=(5-t)\text{m/s}$ and particle starts from origin then its position time graph is

1. 
2. 
3. 
4. 

Q. A particle moves along a straight line and its velocity depends on time as $v=4t-t^2.$ Then for first $5\text{s}$

1. Average velocity is $\frac{25}{3}\text{m}$
2. Average speed is $10{\text{ms}}^{-1}$
3. Average velocity is $\frac{5}{3}{\text{ms}}^{-1}$
4. Acceleration is $4{\text{ms}}^{-2}\text{at}t=0$

Q. If v = (5 - t) m/s and particle starts from origin then its position time graph is

1. 
2. 
3. 
4. 

Q. A body of mass $2\text{kg}$ is driven by an engine delivering a constant power of $1\text{J/s}$. The body starts from rest and moves in a straight line. After $9$ seconds, the body has moved a distance (in $\text{m}$)

Q. The value of ${\int }_1^e(\frac{{\tan }^{-1}x}{x}+\frac{\log x}{1+x^2})dx$ is

1. $\tan e$
2. ${\tan }^{-1}e$
3. ${\tan }^{-1}\left(1/e\right)$
4. None of these

Q. At the moment $t=0$ particle leaves origin and moves in the positive direction of $x$-axis. Its velocity as a function of time is given as $v=(5-t)\text{m/s}$. $x$-coordinate of particle at $t=2\text{s}$ is

1. 8 m
2. 6 m
3. 10 m
4. 12 m

Q. At the moment t = 0 particle leaves origin and moves in the positive direction of x-axis. Its velocity as a function of time is given as v = (5-t) m/s. x-coordinate of particle at t = 2 sec is

1. 8 m
2. 6 m
3. 12 m
4. 10 m

Q. The given graph shows the variation of $\sqrt{f}$ vs $Z$ for characteristics X-rays. Lines 1, 2, 3, 4 shown in the graph corresponds to any one of $K_{\alpha },K_{\beta },L_{\alpha },L_{\beta },$ then $L_{\beta }$ is represented by $(f=$ frequency, $Z=$ Atomic number$)$

1. Line 1
2. Line 2
3. Line 3
4. Line 4

Q. A particle moves along a straight line and its velocity depends on time as $v=4t-t^2.$ Then for first $5\text{s}$

1. Average velocity is $\frac{25}{3}\text{m}$
2. Average speed is $10{\text{ms}}^{-1}$
3. Average velocity is $\frac{5}{3}{\text{ms}}^{-1}$
4. Acceleration is $4{\text{ms}}^{-2}\text{at}t=0$