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:

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. $-2{\mathrm{n\beta }}^2{\mathrm{x}}^{-4\mathrm{n}+1}$

The acceleration of a particle starting from rest varies with time according to the relation A = – aω²sinωt. The displacement of this particle at a time t will be:

1. $-\frac{1}{2}\left({\mathrm{a\omega }}^2\mathrm{sin\omega t}\right){\mathrm{t}}^2$

2. $\mathrm{a\omega sin\omega t}$

3. $\mathrm{a\omega cos\omega t}$

4. $\mathrm{asin\omega t}$

A particle is moving along the x-axis such that its velocity varies with time as per the equation $\mathrm{v}=20\left(1-\frac{\mathrm{t}}{2}\right)$. At t=0 particle is at the origin. From the following, select the correct position (x) - time (t) plot for the particle:

1.		2.
3.		4.