Logic Questions and Answers

Answer: C

The statement that is not necessarily true is If all students are standing, then the teacher is in the room.

The given statement has the form: teacher in room implies all students stand silently.

• Its contrapositive, 'if any student is not standing silently, the teacher is not in the room', is logically equivalent to it, so it must be true.
• 'All students are silent' and 'all students are standing' are each part of 'stand silently', so they follow whenever the teacher is in the room.
• 'If all students are standing, then the teacher is in the room' is the converse. The students could be standing for some other reason while the teacher is away, so it does not follow.

Answer: C

The statement that must be true is When peacocks are not dancing, it is not raining.

The sentence has the form 'if it is raining (R), peacocks dance (D)', that is R → D. A conditional is logically equivalent to its contrapositive, not-D → not-R: if the peacocks are not dancing, it cannot be raining, since rain would have made them dance.

'Peacocks dance only when it is raining' and 'When peacocks dance, it is raining' both state the converse, D → R: peacocks might also dance for other reasons. 'When it is not raining, peacocks do not dance' is the inverse, not-R → not-D, which is equivalent to the converse and equally unsupported.

Answer: C

The true statement is that the surface may be concave in some places and convex in other places.

A smooth closed object such as a peanut shell, a dumbbell or a bean bulges outwards in most places but curves inwards at its waist, so a surface can mix both kinds of curvature. Nothing about being smooth or 3-dimensional forces one kind everywhere.

'Convex everywhere' holds for a sphere or an egg but is not required, so 'must' makes it false. 'Concave everywhere' is impossible for a closed bounded object seen from outside, since its outermost points must bulge outwards. 'Edges, but no corners' contradicts smoothness: a smooth surface has neither edges nor corners.

Answer: B

The negation is Neither P marries Q nor X marries Y.

Read as 'P marries Q or X marries Y', the statement is true as soon as either marriage happens. Its negation must be true exactly when the statement is false, which is when neither happens: P does not marry Q and X does not marry Y.

The other options each assert that one of the marriages does take place, and so are compatible with the original statement rather than contradicting it. 'P marries Q and X marries Y' even makes the original statement true.

Answer: A

S owns 0 cars.

Test each possibility against "only one of P, Q and R is right".

• 0 cars: P (at least 3) is wrong, Q (less than 3) is right, R (at least one) is wrong. Exactly one is right.
• 1 or 2 cars: Q and R are both right, which is two.
• 3 or more: P and R are both right, again two.

Only 0 satisfies the condition, so the number is determined, ruling out "cannot be determined"; 1 and 3 each make two people right.

Answer: D

The statement that fails for some \(x\) is \(fl(x) \lt ce(x)\).

When \(x\) is an integer, both the ceiling and the floor equal \(x\) itself, so \(fl(x) = ce(x)\) and the strict inequality is false. For example, \(fl(3) = ce(3) = 3\).

The other three hold for every real \(x\): by definition the ceiling is at least \(x\) and the floor is at most \(x\), and so \(ce(x) \ge x \ge fl(x)\).

Answer: D

The statement that must be true is There exists at least one mango in the basket that is not good.

'All the mangoes are good' is false exactly when at least one mango is not good; that is its logical negation, so it is necessarily true.

The other options say more than the negation allows. 'All the mangoes are not good' and 'No mango is good' claim every mango is bad, but the statement is also false when just one mango is bad. 'Some are good and some are not' requires at least one good mango, which is not guaranteed: every mango could be bad.

Answer: C

The necessarily correct option is (S1) and (S3) together imply (S2).

If the average is 25, the four numbers add to 100. If each is at least 20, any three of them add to at least 60, so the fourth is at most \(100 - 60 = 40\). That is (S2).

The others fail. (S1) and (S2) allow 40, 40, 20, 0, which breaks (S3). (S2) and (S3) allow 20, 20, 20, 20, whose average is 20, not 25. And (S1) alone allows 100, 0, 0, 0, so it does not imply (S3).

Answer: D

The answer is neither (i) nor (ii).

The passage says only that coding regions, which are read in triplets to make amino acids, are about 2% of human DNA. It does not say what the other 98% does. Non-coding DNA could still take part in protein synthesis in other ways, for example by regulating which genes are expressed, so (i), that the majority has no role in protein synthesis, does not follow.

Statement (ii) is about the state of scientific understanding, which the passage never discusses. Being non-coding is not the same as having an unknown function. So neither can be inferred with certainty from the passage alone, which rules out 'only (i)', 'only (ii)' and 'both'.

Answer: B

Only two statements are correct.

Statement 1 is correct: 'taller than' is transitive, so A > B > C gives A > C.

Statement 2 is correct: X equals Y and Y is smaller than Z, so X, being the same as Y, is smaller than Z.

Statement 3 is wrong: P > Q > R gives P > R, the opposite of what it says. So statements 1 and 2 hold and 3 reverses the conclusion.