Test Level 3: Binary Logic - CAT MCQ

Let us look at the first statement of Carol - "Janice is the truth-teller". This statement implies that Carol is not the truth-teller as there is only one truth-teller, and if Janice is the truth-teller, then Carol cannot be the truth-teller. Thus, Carol can be the alternator or the liar.
Let us see the first statement of Janice - "Carol is not the alternator". If Janice is truth-teller, then Carol will be a liar, in which case Janice cannot be the truth-teller.
Let assume that Sarah is the truth-teller. Now, Janice can either be the liar or the alternator.
Case (i)
If Janice is the alternator, her first statement would be true and second statement would be false, and both the statements of Carol would be false. Hence, it is a possible case.
Case (ii)
If Janice is a liar, both her statements would be false, and the first statement of Carol would be false and her second statement would be true. As there are no contradictions, this is also a possible case. In this case, Carol would have failed in Economics, Sarah would have failed in Finance, and Janice would have failed in Data Analysis.
Case (i)

Case (ii)

Hence, it cannot be determined.

Let us look at the first statement of Carol - "Janice is the truth-teller". This statement implies that Carol is not the truth-teller as there is only one truth-teller, and if Janice is the truth-teller, then Carol cannot be the truth-teller. Thus, Carol can be the alternator or the liar.
Let us see the first statement of Janice - "Carol is not the alternator". If Janice is truth-teller, then Carol will be a liar, in which case Janice cannot be the truth-teller.
Let assume that Sarah is the truth-teller. Now, Janice can either be the liar or the alternator.
Case (i)
If Janice is the alternator, her first statement would be true and second statement would be false, and both the statements of Carol would be false. Hence, it is a possible case.
Case (ii)
If Janice is a liar, both her statements would be false, and the first statement of Carol would be false and her second statement would be true. As there are no contradictions, this is also a possible case. In this case, Carol would have failed in Economics, Sarah would have failed in Finance, and Janice would have failed in Data Analysis.
Case (i)

Case (ii)

Hence, it cannot be determined.

Let us look at the first statement of Carol - "Janice is the truth-teller". This statement implies that Carol is not the truth-teller as there is only one truth-teller, and if Janice is the truth-teller, then Carol cannot be the truth-teller. Thus, Carol can be the alternator or the liar.
Let us see the first statement of Janice - "Carol is not the alternator". If Janice is truth-teller, then Carol will be a liar, in which case Janice cannot be the truth-teller.
Let assume that Sarah is the truth-teller. Now, Janice can either be the liar or the alternator.
Case (i)
If Janice is the alternator, her first statement would be true and second statement would be false, and both the statements of Carol would be false. Hence, it is a possible case.
Case (ii)
If Janice is a liar, both her statements would be false, and the first statement of Carol would be false and her second statement would be true. As there are no contradictions, this is also a possible case. In this case, Carol would have failed in Economics, Sarah would have failed in Finance, and Janice would have failed in Data Analysis.
Case (i)

Case (ii)

Hence, Sarah always tells the truth.

Let us look at the first statement of Carol - "Janice is the truth-teller". This statement implies that Carol is not the truth-teller as there is only one truth-teller, and if Janice is the truth-teller, then Carol cannot be the truth-teller. Thus, Carol can be the alternator or the liar.
Let us see the first statement of Janice - "Carol is not the alternator". If Janice is truth-teller, then Carol will be a liar, in which case Janice cannot be the truth-teller.
Let assume that Sarah is the truth-teller. Now, Janice can either be the liar or the alternator.
Case (i)
If Janice is the alternator, her first statement would be true and second statement would be false, and both the statements of Carol would be false. Hence, it is a possible case.
Case (ii)
If Janice is a liar, both her statements would be false, and the first statement of Carol would be false and her second statement would be true. As there are no contradictions, this is also a possible case. In this case, Carol would have failed in Economics, Sarah would have failed in Finance, and Janice would have failed in Data Analysis.
Case (i)

Case (ii)

Norman's first statement is either a true statement of a truth-teller or a false statement of an alternator, irrespective of which Norman's first and third statement have to be true. This gives us the following result -
Duncan is going to Austria.
Keith is going to Sweden.
Norman is going to Romania.
Based on above, we can determine that Duncan's third statement is false and the second one is true. As the three of them have to be one of the three, i.e. truth-teller/liar/alternator, we can conclude that Duncan is an alternator and his first statement is hence, false (which means Norman is not an alternator). This leads us to the conclusion that Norman is a truth-teller; thus Norman's second statement can also be marked as true.
Hereafter, when we analyse Keith's statements, we realise they all are false. So, Keith must be a liar.
The following table tells us about the country they are travelling to:

Duncan or the alternator is travelling to Austria.

Norman's first statement is either a true statement of a truth-teller or a false statement of an alternator, irrespective of which Norman's first and third statement have to be true. This gives us the following result -
Duncan is going to Austria.
Keith is going to Sweden.
Norman is going to Romania.
Based on above, we can determine that Duncan's third statement is false and the second one is true. As the three of them have to be one of the three, i.e. truth-teller/liar/alternator, we can conclude that Duncan is an alternator and his first statement is hence, false (which means Norman is not an alternator). This leads us to the conclusion that Norman is a truth-teller; thus Norman's second statement can also be marked as true.
Hereafter, when we analyse Keith's statements, we realise they all are false. So, Keith must be a liar.
The following table tells us about the country they are travelling to:

Hence, Norman is a truth-teller.

Norman's first statement is either a true statement of a truth-teller or a false statement of an alternator, irrespective of which Norman's first and third statement have to be true. This gives us the following result -
Duncan is going to Austria.
Keith is going to Sweden.
Norman is going to Romania.
Based on above, we can determine that Duncan's third statement is false and the second one is true. As the three of them have to be one of the three, i.e. truth-teller/liar/alternator, we can conclude that Duncan is an alternator and his first statement is hence, false (which means Norman is not an alternator). This leads us to the conclusion that Norman is a truth-teller; thus Norman's second statement can also be marked as true.
Hereafter, when we analyse Keith's statements, we realise they all are false. So, Keith must be a liar.
The following table tells us about the country they are travelling to:

Hence, Keith, who is going to Sweden, is a liar.

Norman's first statement is either a true statement of a truth-teller or a false statement of an alternator, irrespective of which Norman's first and third statement have to be true. This gives us the following result -
Duncan is going to Austria.
Keith is going to Sweden.
Norman is going to Romania.
Based on above, we can determine that Duncan's third statement is false and the second one is true. As the three of them have to be one of the three, i.e. truth-teller/liar/alternator, we can conclude that Duncan is an alternator and his first statement is hence, false (which means Norman is not an alternator). This leads us to the conclusion that Norman is a truth-teller; thus Norman's second statement can also be marked as true.
Hereafter, when we analyse Keith's statements, we realise they all are false. So, Keith must be a liar.
The following table tells us about the country they are travelling to:

It is known that Bus-4 is going to City-D, which means that the 2nd statement made by Bus-2 must be true, whereas the 2nd statement made by Bus-4 must be false. As the board on Bus-1 says that Bus-1 and Bus-3 are both going to City-A, Bus-1 cannot be the bus having both the boards with correct information, as each bus is going to exactly one city. This means that only Bus-3 can have both the boards with correct information, as there must be at least one bus having both boards that give correct information.
We can take this as the basis and get the arrangement as given below:

Hence, Bus-2 is going to City-A.

It is known that Bus-4 is going to City-D, which means that the 2nd statement made by Bus-2 must be true, whereas the 2nd statement made by Bus-4 must be false. As the board on Bus-1 says that Bus-1 and Bus-3 are both going to City-A, Bus-1 cannot be the bus having both the boards with correct information, as each bus is going to exactly one city. This means that only Bus-3 can have both the boards with correct information, as there must be at least one bus having both boards that give correct information.
We can take this as the basis and get the arrangement as given below:

Hence, on Bus-1, both the boards are giving false information.

It is known that Bus-4 is going to City-D, which means that the 2nd statement made by Bus-2 must be true, whereas the 2nd statement made by Bus-4 must be false. As the board on Bus-1 says that Bus-1 and Bus-3 are both going to City-A, Bus-1 cannot be the bus having both the boards with correct information, as each bus is going to exactly one city. This means that only Bus-3 can have both the boards with correct information, as there must be at least one bus having both boards that give correct information.
We can take this as the basis and get the arrangement as given below:

Hence, option (4) is the answer.

Given that every person spoke exactly one true statement. Let the first statement of Josh be true. Hence, the other two must be false. Therefore, Josh is not from L-1. Frank did not vote for C-3 and Philip is not from the L-3.
Frank's second statement is, therefore, true. Hence, his first and third statements must be false. Hence, Frank did not vote for C-2 and Josh did not vote for C-3. Frank must have voted for C-1. Josh should have voted for C-2. Philip must have voted to C-3.
Hence, Philip's last statement is true. But Philip's first statement is also true. Since this is not possible, Josh' first statement cannot be true.
Let the second statement of Josh be true. Therefore, Frank voted for C-3. From his other two statements we can say that Josh is from L-1 and Philip is from L-2. Hence, Frank is from L-3. Frank's second statement is true. His first and third statements are false.
Philip's second statement is true. His first and third statements must be false. Therefore, Josh must have voted for C-1 and Philip must have voted for C-2. This is one possible case.
Let Josh's third statement be true. Philip must be from L-3. From his first statement, we get that Josh must be from L-1. Therefore, Frank must be from L-2. Frank did not vote for C-3.
Frank's second statement and Philip's second statement are false. The only case possible in which both Josh and Philip tell one true statement each is when Frank's first statement and Philip's third statement are true. Hence, Frank voted for C-2, Philip voted for C-3 and Josh voted for C-1. This is another possible case.
The possible case:

The person who voted for C-2 is from L-2.

Given that every person spoke exactly one true statement. Let the first statement of Josh be true. Hence, the other two must be false. Therefore, Josh is not from L-1. Frank did not vote for C-3 and Philip is not from the L-3.
Frank's second statement is, therefore, true. Hence, his first and third statements must be false. Hence, Frank did not vote for C-2 and Josh did not vote for C-3. Frank must have voted for C-1. Josh should have voted for C-2. Philip must have voted to C-3.
Hence, Philip's last statement is true. But Philip's first statement is also true. Since this is not possible, Josh' first statement cannot be true.
Let the second statement of Josh be true. Therefore, Frank voted for C-3. From his other two statements we can say that Josh is from L-1 and Philip is from L-2. Hence, Frank is from L-3. Frank's second statement is true. His first and third statements are false.
Philip's second statement is true. His first and third statements must be false. Therefore, Josh must have voted for C-1 and Philip must have voted for C-2. This is one possible case.
Let Josh's third statement be true. Philip must be from L-3. From his first statement, we get that Josh must be from L-1. Therefore, Frank must be from L-2. Frank did not vote for C-3.
Frank's second statement and Philip's second statement are false. The only case possible in which both Josh and Philip tell one true statement each is when Frank's first statement and Philip's third statement are true. Hence, Frank voted for C-2, Philip voted for C-3 and Josh voted for C-1. This is another possible case.
The possible case:

The person from L-3, voted for C-3.

Given that every person spoke exactly one true statement. Let the first statement of Josh be true. Hence, the other two must be false. Therefore, Josh is not from L-1. Frank did not vote for C-3 and Philip is not from the L-3.
Frank's second statement is, therefore, true. Hence, his first and third statements must be false. Hence, Frank did not vote for C-2 and Josh did not vote for C-3. Frank must have voted for C-1. Josh should have voted for C-2. Philip must have voted to C-3.
Hence, Philip's last statement is true. But Philip's first statement is also true. Since this is not possible, Josh' first statement cannot be true.
Let the second statement of Josh be true. Therefore, Frank voted for C-3. From his other two statements we can say that Josh is from L-1 and Philip is from L-2. Hence, Frank is from L-3. Frank's second statement is true. His first and third statements are false.
Philip's second statement is true. His first and third statements must be false. Therefore, Josh must have voted for C-1 and Philip must have voted for C-2. This is one possible case.
Let Josh's third statement be true. Philip must be from L-3. From his first statement, we get that Josh must be from L-1. Therefore, Frank must be from L-2. Frank did not vote for C-3.
Frank's second statement and Philip's second statement are false. The only case possible in which both Josh and Philip tell one true statement each is when Frank's first statement and Philip's third statement are true. Hence, Frank voted for C-2, Philip voted for C-3 and Josh voted for C-1. This is another possible case.
The possible case:

Frank can be from L-3 or L-2.

Given that every person spoke exactly one true statement. Let the first statement of Josh be true. Hence, the other two must be false. Therefore, Josh is not from L-1. Frank did not vote for C-3 and Philip is not from the L-3.
Frank's second statement is, therefore, true. Hence, his first and third statements must be false. Hence, Frank did not vote for C-2 and Josh did not vote for C-3. Frank must have voted for C-1. Josh should have voted for C-2. Philip must have voted to C-3.
Hence, Philip's last statement is true. But Philip's first statement is also true. Since this is not possible, Josh' first statement cannot be true.
Let the second statement of Josh be true. Therefore, Frank voted for C-3. From his other two statements we can say that Josh is from L-1 and Philip is from L-2. Hence, Frank is from L-3. Frank's second statement is true. His first and third statements are false.
Philip's second statement is true. His first and third statements must be false. Therefore, Josh must have voted for C-1 and Philip must have voted for C-2. This is one possible case.
Let Josh's third statement be true. Philip must be from L-3. From his first statement, we get that Josh must be from L-1. Therefore, Frank must be from L-2. Frank did not vote for C-3.
Frank's second statement and Philip's second statement are false. The only case possible in which both Josh and Philip tell one true statement each is when Frank's first statement and Philip's third statement are true. Hence, Frank voted for C-2, Philip voted for C-3 and Josh voted for C-1. This is another possible case.
The possible case:

Only statements (1) and (3) are definitely true.