A true/false quiz consists of twenty questions. If a student answers the twenty questions by making random guesses, what is the probability of getting at least one correct answer?

For the second question, subtract from 1 the probability of betting none right in 20 attempts. That will leave you with the probability of getting at least 1 right.
1 - (1/2)^20 = 1.0

Is my answer correct?

Yes, but you need to calculate it to more decimal places... the answer should be 0.999999046

Just for kicks:
1-(1/2)^20=1-1/(1048576)=1048575/1048576=0.99999904632568359375 (exactly)