The relation between time ' $t$ ' and distance ' $x$ ' is $t=$ $\alpha x^2+\beta x$, where $\alpha$ and $\beta$ are constants. The relation between acceleration $(a)$ and velocity $(v)$ is:

A particle moves along a straight line such that its displacement at any time $t$ is given by $s = (t^3 -3t^2 + 2)\,m$ The displacement when the acceleration becomes zero is........$m$

Starting from rest, acceleration of a particle is $a = 2(t - 1).$ The velocity of the particle at $t = 5\,s$ is.........$m/sec$

A body starts from the origin and moves along the $X-$axis such that the velocity at any instant is given by $(4{t^3} - 2t)$, where $t$ is in sec and velocity in$m/s$. What is the acceleration of the particle, when it is $2\, m$ from the origin..........$m/{s^2}$

Given below are two statements:

Statement $I:$ Area under velocity- time graph gives the distance travelled by the body in a given time.

Statement $II:$ Area under acceleration- time graph is equal to the change in velocity- in the given time.

In the light of given statements, choose the correct answer from the options given below.

The accompanying graph of position $x$ versus time $t$ represents the motion of a particle. If $p$ and $q$ are both positive constants, the expression that best describes the acceleration $a$ of the particle is