A small metal ball is given a negative charge, then brought near (i. e., within a few millimeters) to end A of the rod. What happens to end A of the rod when the ball approaches it closely this first time?

They will attract.

Explanation:

We know, in conductors electrons are free to move> in the case given both metal ball and rod are conductor.

Now, when conductor are brought to a charge all the opposite charges moves through the point of conductor which is close to the charge and attraction takes place.

In this case too, when metal ball is bought near rod, rod also develop opposite charge and attraction takes place.  

Hence, this is the required solution.

at right angles to each other