Logic Questions and Answers

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.

Answer: D

The negation is Amit does not live in Delhi or Aman lives in Mumbai.

The statement has the form 'p and not q', where p is 'Amit lives in Delhi' and q is 'Aman lives in Mumbai'. By De Morgan's law the negation of 'p and not q' is 'not p or q'.

So the 'and' becomes 'or', the first part is negated, and the second part loses its 'not': Amit does not live in Delhi, or Aman lives in Mumbai.

The options keeping 'and' fail because negation swaps 'and' for 'or'. The option with 'Aman does not live in Mumbai' forgot to negate the second part.

Answer: B

The keyed reason is that contaminated Ganga water can spread contagious disease.

The Ganga receives large volumes of untreated sewage and industrial effluent, and millions of people bathe in it and use its water. Polluted river water spreads water-borne diseases such as cholera, typhoid and diarrhoea, which is the public-health reason for the clean-up programme (Namami Gange).

The river's religious importance and tourism are real but are not the rationale for pollution control. 'Groundwater cannot be used for drinking' is false as a general statement and is not why the Ganga is being cleaned.

Answer: B

The keyed reason is that hotel food is unhygienic and very spicy.

The statement says home food is always the better choice. The reason that supports this on health grounds is that restaurant food is often prepared with less control over hygiene and with more oil and spice than home cooking.

Expense alone is not a reason to call home food better for you. 'Tasty and full of nutrition' would argue for hotel food, not against it, so 'All of the above' cannot be right either.

Answer: C

True justified belief is often called Knowledge.

The classical definition of knowledge, going back to Plato's Theaetetus, is justified true belief: a person knows something if they believe it, it is true, and they have good reason for believing it. Edmund Gettier's examples of 1963 later questioned whether these three conditions are enough.

A hypothesis is a proposed explanation still to be tested, so it need not be true or justified. Intelligence is a mental capacity, not a kind of belief.

Answer: C

The distance between P and R is less than or equal to x + y.

Going from P to R through Q covers x + y km, and a direct route can never be longer than a route through an intermediate point. This is the triangle inequality: PR ≤ PQ + QR. Equality holds only when Q lies on the straight line between P and R.

The distance is not always exactly x + y, because Q need not lie between P and R, and it can never be greater than x + y. Since 'less than or equal to x + y' is always true, 'None of the above' is wrong too.