A particle, moving horizontally, collides perpendicularly at one end of a rod having equal mass and placed on a smooth horizontal surface

a)	Particle cones to rest if collision is perfectly elastic and center of rod starts to move with the same velocity
b)	Particle continues to move along the same direction, whatever is the value of
c)	Particle may get rebound back
d)	Velocity of mid-point of the rod will be less than if the particle gets stuck

(b,d)

Whenever two particles having equal mass collide head on elastically, their velocities get interchanged. Therefore, if the particle collides at midpoint of the rod, then velocities would get interchanged. In that case option (a) would be correct. But now the particle strikes at an end of the rod, hence particle head-on collision does not take place. Therefore, option (a) is incorrect

It the particle gets rebound back to its original path, then its final momentum will become negative. Since mass of the particle and rod is equal, therefore law of conservation of momentum will be satisfied only when velocity of centre of mass of the rod is greater than original velocity of the particle. Hence, kinetic energy of the system (just after collision) will become greater than that (just before the collision), which is impossible. Hence, the particle cannot rebound back or it will continue but option (c) is incorrect. If the particle gets stick to the rod, centre of mass of the system will lie a distance  from the end at which the particle sticks. According to the law of conservation of momentum, centre of mass will start to move with velocity .  But the particle will simultaneously start to rotate about the centre of mass in an anticlockwise direction.Angular velocity  of that rotational motion can be calculated by applying law of conservation of angular momentum. Hence, the resultant velocity of the midpoint of the rod will be equal to , which is obviously .Therefore option (d) is correct