The position vector$r$ of a particle of mass $m$ is given by,
the following equation, $\overrightarrow{r}\left(t\right)=\alpha t^3\hat{\mathrm{i}}+\beta t^2\hat{\mathrm{j}} \mathrm{m}$, where $\alpha =\frac{10}{3} \mathrm{m} {\mathrm{s}}^{-3}, \beta =5 \mathrm{m} {\mathrm{s}}^{-2}$ and $m=0.1 \mathrm{kg}.$ At, $t=1 \mathrm{s},$ which of the following statement (s) is are true about the particle?
 

Consider regular polygons with number of sides $n= 3,4,5 ....$ as shown in the figure. The centre of mass of all the polygons is at height $\mathrm{h}$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the centre of mass for each polygon is $∆$. Then $∆$ depends on $n$ and $h$ as

A wheel of radius $R$ and mass $M$ is placed at the bottom of a fixed step of height $R$ as shown in the figure. A constant force is continuously applied on the surface of the wheel so that it just climbs the step without slipping. Consider the torque $T$ about an axis normal to the plane of the paper passing through the point $Q$. Which of the following options is/are correct ?

In the List-I below, four different paths of a particle are given as functions of time. In these functions, $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are positive constants of appropriate dimensions and $\mathrm{\alpha }\ne \mathrm{\beta }$. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: p is the linear momentum, L is the angular momentum about the origin, $K$ is the kinetic energy, $U$ is the potential energy and $E$ is the total energy. Match each path in List-I with those quantities in List-II, which are conserved for that path.