$A$ coin is tossed three times. Consider the following events:
$A$: 'No head appears',
$B$: 'Exactly one head appears',
$C$: 'At least two heads appear'.
Do they form a set of mutually exclusive and exhaustive events?

(A) The sample space of the experiment is:
$S = \{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$
The events are defined as:
$A = \{TTT\}$
$B = \{HTT, THT, TTH\}$
$C = \{HHT, HTH, THH, HHH\}$
Check for exhaustive events:
$A \cup B \cup C = \{TTT, HTT, THT, TTH, HHT, HTH, THH, HHH\} = S$
Since the union of the events is the sample space $S$,the events are exhaustive.
Check for mutually exclusive events:
$A \cap B = \phi$
$A \cap C = \phi$
$B \cap C = \phi$
Since the intersection of any two events is the empty set $\phi$,the events are pair-wise disjoint,i.e.,mutually exclusive.
Conclusion:
Yes,$A, B,$ and $C$ form a set of mutually exclusive and exhaustive events.