$A$ cyclist traversed half the distance of a linear track with a velocity $10 \ m \ s^{-1}$. The remaining part of the track was covered with a velocity $v_1$ for half the time and a velocity $v_2$ for the other half of the time. If $v_1+v_2=20 \ m \ s^{-1}$, then the average velocity of the cyclist during the completion of the journey through the track is (in $m \ s^{-1}$)

Two stones are thrown up simultaneously from the edge of a cliff $240 \ m$ high with initial speeds of $10 \ m/s$ and $40 \ m/s$ respectively. Which of the following graphs best represents the time variation of the relative position of the second stone with respect to the first? (Assume stones do not rebound after hitting the ground and neglect air resistance,take $g = 10 \ m/s^2$)

The relation between time $t$ and distance $x$ is given by $t = \alpha x^2 + \beta x$,where $\alpha$ and $\beta$ are constants. The retardation of the particle is:

The velocity of a particle moving along the $x$-axis is given by $V = x^2 - 5x + 4$ (in $\text{m/s}$), where $x$ denotes the $x$-coordinate of the particle in metres. The magnitude of the acceleration of the particle when the velocity is zero is:

Which of the following four statements is false?

Statement $(I)$: An object subjected to velocities $\overrightarrow{v_1}$ and $\overrightarrow{v_2}$ has a resultant velocity with magnitude $|\vec{v}| = |\overrightarrow{v_1}| + |\overrightarrow{v_2}|$.
Statement $(II)$: The magnitude of displacement is either less than or equal to the path length of an object between two points.
Statement $(III)$: The instantaneous acceleration is the limiting value of the average acceleration as the time interval approaches zero.
Which of the following is correct?