At t = 0 , the velocity of a particle is zero. Which of the...

At $t = 0$, $v = 0$.

For the first statement, the acceleration may or may not be zero. For example, a particle at its highest point in vertical motion has $v = 0$ but $a = g \neq 0$. If a particle is permanently at rest, $v = 0$ and $a = 0$. Thus, acceleration may be zero, making the first statement correct.

For the second statement, if speed is zero from $t = 0$ to $t = 10 \text{ s}$, the velocity is constant and equal to zero. Therefore, $a = \dfrac{dv}{dt} = 0$, making the second statement correct.

For the third statement, if acceleration is zero from $t = 0$ to $t = 10 \text{ s}$, the velocity is constant. Since $v = 0$ at $t = 0$, the velocity remains zero throughout the interval, meaning the speed is zero. The third statement is correct.

For the fourth statement, as shown previously, acceleration does not have to be zero when $v = 0$. The fourth statement is incorrect.