Question

If an object is accelerating to the right , then net force on the object must be directed towards the right ! True Or False ?

Answer 1

Not so fast.

I think you're using 'accelerating' to mean 'speeding up', but you really need
to be more careful with it. "Acceleration" means ANY change in speed OR
direction.

If an object's speed to the left is decreasing, or its speed to the right is
increasing, then the net force on the object must be directed towards
the right.

If an object is moving with constant speed in a circular path, then it's
constantly accelerating, because its direction is constantly changing.
The force on it is always directed towards the center of the circle, so
there's one point on the path where the force is directed straight to the right.

Answer 2

This is correct. Just think about it. If you push an object to the right then it will accelerate to the right.