Logic Questions and Answers

Answer: C

The inferences that follow are (ii) and (iv) only.

The proposal says the SITs are to cater to the technological and industrial needs of a developing country. That is (iv) directly, and it is a national purpose of their own, which is (ii).

Statements (i) and (iii) are not supported. Being 'in line with' IITs means modelled on them, not that IIT awards SIT degrees or must be consulted before such institutions are set up. Any option including (i) or (iii) adds a claim the sentence does not make.

Answer: D

The only certain inference is Some Fields medalists have dropped out of college.

June Huh is a Fields medalist who dropped out of college, so at least one Fields medalist dropped out, which is all that “some” claims.

The other statements go beyond the one example. “Every Fields medalist has won an Olympiad medal” is contradicted by Huh himself, who won none. “Everyone who has dropped out of college has won the Fields medal” reverses the relationship and generalises from one person to all dropouts. “All Fields medalists are part-time poets” generalises from one medalist, who in any case was a poet when younger, not necessarily now.

Answer: B

The match is P - 3; Q - 4; R - 1; S - 2.

Each response uses a phrasal verb with 'up' that answers its statement:

• P, a messy house: 'let me clear it up' (tidy it), 3.
• Q, unhappy with marks: 'I will take it up with your teacher' (raise the matter), 4.
• R, avoiding politics: 'I won't bring it up' (mention it), 1.
• S, an unknown word: 'you can easily look it up' (in a dictionary), 2.

Option A pairs the mess with 'look it up' and the marks with 'clear it up', and the other two options likewise give replies that do not answer the statements.

Answer: B

The correct statement is \(a_{1} \lt 50 \lt a_{2}\).

The weight function rises strictly up to 50 years and falls strictly after it. On the rising part, different ages give different weights, and the same holds on the falling part. So two different ages can give the same weight only if one is on the rising side and the other on the falling side: the younger brother is under 50 and the elder over 50.

If both ages were below 50, or both above 50, strict monotonicity would make their weights different. Neither age can be exactly 50, since the weight at 50 is the maximum and is reached at no other age. That rules out the other three options.

Answer: A

The statement that is always true is There are at least 10 students who belong to the same standard.

This is the pigeonhole principle. If every one of the 10 standards had at most 9 students, the school could hold at most \(10 \times 9 = 90\) students. There are 100, so at least one standard must have 10 or more.

The other statements can fail. All 100 students could be in one standard, so a standard may be empty, and the 10th standard may hold more than 10. Equally, the 6th to 10th standards could hold all 100 students, leaving fewer than 50 in the 1st to 5th standards.

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.