Find the angle between the two hands of a clock at $30$ $minutes$ past $4$ $O$'clock. (in $^{\circ}$)

How often between $11$ $O$'clock and $12$ $O$'clock are the hands of a clock in integral number of $minutes$ apart (in $\times$)?

How many times in a day are the hands of a clock at a right angle?

$A$ watch which gains uniformly is $6$ minutes slow at $4$ $p.m.$ on a Sunday and $10 \frac{2}{3}$ minutes fast on the following Sunday at $8$ $p.m.$ During this period,when was the watch correct?

$A$ girl was born on September $6, 1970$,which happened to be a Sunday. Her birthday would have fallen again on a Sunday in:

My watch gains $5$ seconds in $3$ minutes. It was set right at $7$ am. In the afternoon of the same day,when the watch indicates quarter past $4$ o'clock,what is the true time?