A particle is moving in $X Y$-plane as $\mathbf{x}=\left(4 t+t^2\right) \hat{\mathbf{i}}$, $\mathbf{y}=\left(2 t+\frac{t^2}{2}\right) \hat{\mathbf{j}}$, where $\mathbf{x}$ and $\mathbf{y}$ are displacements measured along $X$ and $Y$-axes respectively, in metres and $t$ in seconds, What is the velocity of the particle?

The surface of a hill inclined at an angle $30^{\circ}$ to the horizontal. A stone is thrown from the summit of the hill (point $A$ ) at an initial speed $10 \mathrm{~m} / \mathrm{s}$ at angle $60^{\circ}$ to the vertical. If the stone strikes the hill at point $B$ as shown in the figure, the distance between $A$ and $B$ is (take, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

Statement I An object subjected to velocities $\mathbf{v}_1$ and $\mathbf{v}_2$ has a resultant velocity with magnitude $|\mathbf{v}|=\left|\mathbf{v}_1\right|+\left|\mathbf{v}_2\right|$.

Statement II The magnitude of displacement is either less or equal to the path length of an object between two points.

Statement III The instantaeous acceleration is the limiting value of the average acceleration as the time interval approaches zero.

Which of the following is correct?

For a projectile, if $\alpha$ is the angle of projection, $R$ is the range, $h$ is the maximum height, $t$ is the time of flight then