Increase in velocity by loss of mass? 
A trolley of mass 300kg carrying a sand bag of 25kg is moving uniformly with speed of 27km/h on a frictionless track. After a while, sand starts leaking out of a hole on the floor of the trolley at the rate of 0.05 kg/s.
What is the speed of the trolley after the entire sand bag is empty?

I was so surprised when I read this question. It doesn't make sense to me. I can't comprehend how the loss of sand creates an external unbalanced force on the trolley such that it affects its velocity.
Maybe I haven't analyzed the question enough but I find this a bit conceptually challenging for me. Maybe I have to consider how the sand particles affect the back wheels of the trolley or maybe consider the sand to be a propellant?
 A: The answer is...  27 km/h.
It is a trick question, the net force on the trolley is always zero.
People might be tricked into blindly applying momentum conservation to find an increase in velocity but this would be incorrect.  As the sandbag decreases in weight the momentum carried by the trolley-sandbag system decreases.
A: This is no trick question. It is simply an application of the fact that the velocity of the Center of Mass (COM) of a body is unaffected by internal forces. You would have been correct in saying the velocity of COM cannot change in absence of external forces. But this does not apply to the trolley alone.
So what's going on here? Initially the COM of the trolley + sandbag system is moving at $27$ kmph. Now the sand is falling out of the sandbag backwards by some means of an internal force. Since the velocity of the COM must, must be constant in the absence of external forces, the trolley speeds up so the net momentum of the system is conserved.
With some formulas:
$v_c$ = $(m_tv_t + m_sv_s)$ $/$ $(m_t + m_s)$
..where $v_c$ is the velocity of the COM. This will not change even after all the sand falls out.
This is basically just another way of writing linear conservation of momentum, but I want to bring your attention to the significance of the fact that it's the velocity of the COM which dictates the velocity of the sand and the trolley. 
To give you another example of where exactly the same happens, think of rocket propulsion. How does the rocket keep gaining speed by simply ejecting fuel at a constant velocity? Despite what you said in the comments, your interpretation is actually correct :)
