A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to $\mathrm{v}\left(\mathrm{x}\right)$ $={\mathrm{\beta x}}^{-2\mathrm{n}}$ where $\mathrm{\beta }$ and \(\mathrm{n}\) are constants and \(\mathrm{x}\) is the position of the particle. The acceleration of the particle as a function of \(\mathrm{x}\) is given by:
1. $-2{\mathrm{n\beta }}^2{\mathrm{x}}^{-2\mathrm{n}-1}$
2. $-2{\mathrm{n\beta }}^2{\mathrm{x}}^{-4\mathrm{n}-1}$
3. $-2{\mathrm{\beta }}^2{\mathrm{x}}^{-2\mathrm{n}+1}$
4. $-2{\mathrm{n\beta }}^2{\mathrm{x}}^{-4\mathrm{n}+1}$

A particle moving along the x-axis has acceleration \(f,\) at time \(t,\) given by, \(f=f_0\left ( 1-\frac{t}{T} \right ),\)  where \(f_0\) and \(T\) are constants. The particle at \(t=0\) has zero velocity. In the time interval between \(t=0\) and the instant when \(f=0,\) the particle’s velocity \( \left ( v_x \right )\) is:
1. \(f_0T\)
2. \(\frac{1}{2}f_0T^{2}\)
3. \(f_0T^2\)
4. \(\frac{1}{2}f_0T\)

The displacement of a particle is given by $\mathrm{y}=\mathrm{a}+\mathrm{bt}+{\mathrm{ct}}^2-{\mathrm{dt}}^4$. The initial velocity and acceleration are, respectively:

The acceleration \(a\) (in ${\mathrm{ms}}^{-2}$) of a body, starting from rest varies with time \(t\) (in \(\mathrm{s}\)) as per the equation \(a=3t+4.\) The velocity of the body at time \(t=2\) \(\mathrm{s}\) will be:
1. \(10\) ${\mathrm{ms}}^{-1}$
2. \(18\) ${\mathrm{ms}}^{-1}$
3. \(14\) ${\mathrm{ms}}^{-1}$
4. \(26\) ${\mathrm{ms}}^{-1}$

A point moves in a straight line under the retardation a${\mathrm{v}}^2$. If the initial velocity is \(\mathrm{u},\) the distance covered in \(\mathrm{t}\) seconds is:
1. $\mathrm{aut}$
2. $\frac{1}{\mathrm{a}}\ln$ $\left(\mathrm{aut}\right)$
3. $\frac{1}{\mathrm{a}}\ln$ $\left(1+\mathrm{aut}\right)$
4. $\mathrm{a}$ $\ln$ $\left(\mathrm{aut}\right)$

For the given acceleration \(\left ( a \right )\) versus time \(\left ( t \right )\) graph of a body, the body is initially at rest.

       
From the following, the velocity \(\left ( v \right )\) versus time \(\left ( t \right )\) graph will be:

1.		2.
3.		4.