Final answer:
The correct counterexample for the conditional statement is option b, 6:35, where the clock hands are perpendicular, but the time is not 3:00, as the statement suggests.
Step-by-step explanation:
A counterexample to a conditional statement is an instance where the hypothesis is true but the conclusion is false. The statement in question is If the hands of a clock are perpendicular, then the time is 3:00.
To provide a counterexample, we must show a time when the hands are perpendicular but the time is not 3:00.
Option (a) 12:00 does not work because the hands are not perpendicular at that time; they are overlapping.
Option (c) 9:00 does not work either because although the hands are perpendicular, the statement allows this time to be correct since it specifies '3:00' and doesn't exclude other times.
Option (d) 1:15 has the hands almost exactly perpendicular, with the hour hand just over a quarter of the way from 1 to 2. However, since the statement implicates 3:00 specifically, 1:15 can be a valid counterexample as it is not 3:00, yet the hands are perpendicular.
However, the closest and clearest counterexample is option (b) 6:35, where the minute hand points at 7 (210 degrees from the top) and the hour hand has moved slightly more than halfway from 6 to 7, making them perpendicular, but the time is not 3:00.