Select the correct statements for a particle travelling on a...

For statement 1: If velocity $v = 0$ for a time interval, it means velocity is constant during that interval. Therefore, acceleration $a = \dfrac{dv}{dt} = 0$ at any instant within that interval. This statement is correct.

For statement 2: If velocity is zero at a specific instant, acceleration need not be zero. For example, at the highest point of a vertically projected body, $v = 0$ but $a = -g \neq 0$. This statement is incorrect.

For statement 3: If velocity and acceleration have opposite signs, the acceleration acts in the direction opposite to the motion. This causes the speed of the object to decrease, meaning it is slowing down. This statement is correct.

For statement 4: If position $x$ and velocity $v$ have opposite signs, then either $x > 0$ and $v < 0$ (the particle is on the positive side and moving in the negative direction) or $x < 0$ and $v > 0$ (the particle is on the negative side and moving in the positive direction). In both cases, the distance from the origin decreases, so the particle is travelling towards the origin. This statement is correct.