From a Grammar perspective, every complete sentence has a subject and a predicate. The predicate is the part of the sentence that contains the full, complete verb and gives information about the subject of the sentence.

However, I am going to answer as though you are working from a logic perspective and in that case, a predicate is a statement that affirms or denies something about the subject.
1 is a predicate because it is affirms that the subject is a murder suspect.
2 and 3 are problem cases because at first glance it seems to affirm something about the subject, but upon closer examination it is noticed that these two sentences state that something is required, but does not affirm that the statement is in fact true for the subject. So they are not really predicates.
4 is a predicate, because it affirms that the suspect is expected to be found..
5 is a predicate because it affirms that the subject is believed to be less than the number seven.
6 is a predicate because it affirms that the subject "four" is less than seven.
7 is a predicate because it affirms something that is true about the subject because of the statement in number 5.
8 is not a predicate
9 is a predicate because is denies that Tom has met the subject
10 is a predicate because it affirms that the Pope gave his blessing to the subject.

Hope this helps, it is based on my best judgment and is certainly open to debate.
Source(s):
http://www.merriam-webster.com/dictionary/predicate
http://dictionary.reference.com/browse/predicate

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