JEE Mathematics Mathematical Reasoning MCQs Set 03

Answer: (d) All prime numbers are odd numbers.

Explanation:
1. A statement must be definitely true or definitely false.
2. (a) is a command, (b) an exclamation and (c) a question, so none of them is a statement.
3. (d) is a sentence that is false (2 is an even prime), but it has a definite truth value, so it is a statement.

Teacher's Note:
1. A statement must be either true or false, but not both; questions, commands and exclamations are not statements.
2. A sentence whose truth depends on an unknown (such as 'tomorrow is Friday') is not a statement.

Answer: (c) Tomorrow is Friday.

Explanation:
1. (a), (b) and (d) are always true, so they are statements.
2. 'Tomorrow is Friday' is true on some days and false on others, so it has no definite truth value; it is not a statement.

Teacher's Note:
1. A statement must be either true or false, but not both; questions, commands and exclamations are not statements.
2. A sentence whose truth depends on an unknown (such as 'tomorrow is Friday') is not a statement.

Answer: (d) √2 is a complex number.

Explanation:
1. Negating a statement that already contains 'not' removes the 'not'.
2. The negation is: √2 is a complex number.

Teacher's Note:
1. Use De Morgan's laws: ~(p ∧ q) = ~p ∨ ~q and ~(p ∨ q) = ~p ∧ ~q.
2. The negation of p ⇒ q is p ∧ ~q, and the negation of p ⇔ q is (p ∧ ~q) ∨ (~p ∧ q).

Answer: (c) 100 is not divisible by 11

Explanation:
1. The compound statement is: (100 is divisible by 5) and (100 is divisible by 10) and (100 is divisible by 11).
2. '100 is not divisible by 11' is the negation of a component, not a component itself.

Teacher's Note:
1. A statement must be either true or false, but not both; questions, commands and exclamations are not statements.
2. A sentence whose truth depends on an unknown (such as 'tomorrow is Friday') is not a statement.

Answer: (c) The office is closed if it is a holiday or a Sunday.

Explanation:
1. In an inclusive 'or' both parts can be true together.
2. (a), (b) and (d) cannot have both parts true at once, so they are exclusive.
3. A day can be both a holiday and a Sunday, so (c) is inclusive.

Teacher's Note:
1. An inclusive 'or' allows both parts to be true; an exclusive 'or' allows exactly one.
2. Test whether both component statements could hold together to decide which type of 'or' is used.

Answer: (d) If Kota is in Rajasthan then Bhopal is the capital of Madhya Pradesh.

Explanation:
1. p ⇒ q is read 'if p then q'.
2. Substituting the statements gives option (d).

Teacher's Note:
1. 'p ⇒ q' can be read as: if p then q; p is sufficient for q; q is necessary for p; p only if q.
2. 'q only if p' and 'q is sufficient for p' describe q ⇒ p, which is the converse.

Answer: (d) p ⇔ q

Explanation:
1. 'If and only if' is the biconditional connective ⇔.
2. So the statement is p ⇔ q.

Teacher's Note:
1. 'p ⇒ q' can be read as: if p then q; p is sufficient for q; q is necessary for p; p only if q.
2. 'q only if p' and 'q is sufficient for p' describe q ⇒ p, which is the converse.

Answer: (b) If it stays warm for a month, then the banana trees will bloom.

Explanation:
1. 'B if A' means 'if A then B'.
2. So the statement means: if it stays warm for a month, then the banana trees will bloom.

Teacher's Note:
1. 'p ⇒ q' can be read as: if p then q; p is sufficient for q; q is necessary for p; p only if q.
2. 'q only if p' and 'q is sufficient for p' describe q ⇒ p, which is the converse.

Answer: (d) q only if p

Explanation:
1. p ⇒ q is expressed as: p is sufficient for q; q is necessary for p; p only if q.
2. 'q only if p' means q ⇒ p, which is the converse and is not equivalent.

Teacher's Note:
1. 'p ⇒ q' can be read as: if p then q; p is sufficient for q; q is necessary for p; p only if q.
2. 'q only if p' and 'q is sufficient for p' describe q ⇒ p, which is the converse.

Answer: (a) 'Getting fine' is a necessary condition.

Explanation:
1. With p : driving over 100 and q : getting a fine, the statement is p ⇒ q.
2. Hence p is sufficient and q (getting a fine) is necessary.
3. (d) is the inverse, which is not equivalent to the original.

Teacher's Note:
1. 'p ⇒ q' can be read as: if p then q; p is sufficient for q; q is necessary for p; p only if q.
2. 'q only if p' and 'q is sufficient for p' describe q ⇒ p, which is the converse.

Answer: (b) p ⇒ q

Explanation:
1. With p = T and q = F: p ∨ q = T; p ⇒ q = F; p ∧ ~q = T; q ⇒ p = T.
2. Only p ⇒ q is not true.

Teacher's Note:
1. Substitute the given truth values step by step, innermost brackets first.
2. A conditional p ⇒ q is false only when p is true and q is false.

Answer: (d) T, F and F

Explanation:
1. (i) (T ∧ T) ⇒ T = T ⇒ T = T.
2. (ii) (T ∧ T) ⇒ ~T = T ⇒ F = F.
3. (iii) (T ∧ F) ∧ (T ⇒ T) = F ∧ T = F.
4. So the values are T, F and F.

Teacher's Note:
1. Substitute the given truth values step by step, innermost brackets first.
2. A conditional p ⇒ q is false only when p is true and q is false.

Answer: (d) If Ram is rich then he works hard.

Explanation:
1. The converse of p ⇒ q is q ⇒ p.
2. So the converse is: if Ram is rich then he works hard.

Teacher's Note:
1. The converse of p ⇒ q is q ⇒ p: just swap the hypothesis and the conclusion.
2. Do not confuse the converse with the contrapositive, which negates both parts as well as swapping them.

Answer: (a) F, T and F

Explanation:
1. The implication A ⇒ p is false only when A is true and p is false.
2. So p = F. Then ~p ∨ q = T, and A = T ∧ ~r is true only if r = F.
3. Hence it is false for p = F, r = F (q arbitrary): the option F, T, F.
4. In the other options p is true, so the implication is true.

Teacher's Note:
1. Substitute the given truth values step by step, innermost brackets first.
2. A conditional p ⇒ q is false only when p is true and q is false.

Answer: (a) p ⇒ (q ∨ r)

Explanation:
1. 'A only if B' means A ⇒ B.
2. Here A = p and B = (q ∨ r), so the proposition is p ⇒ (q ∨ r).

Teacher's Note:
1. 'p ⇒ q' can be read as: if p then q; p is sufficient for q; q is necessary for p; p only if q.
2. 'q only if p' and 'q is sufficient for p' describe q ⇒ p, which is the converse.

Answer: (a) (~p) ⇔ q

Explanation:
1. ~(p ⇔ q) is true exactly when p and q differ in truth value.
2. (~p) ⇔ q is also true exactly when p and q differ, so it is equivalent.
3. (~p) ⇔ (~q) is the same as p ⇔ q, and the other two options differ in the truth table.

Teacher's Note:
1. Use p ⇒ q = ~p ∨ q and p ⇔ q = (p ⇒ q) ∧ (q ⇒ p) to reduce every expression to ∧, ∨, ~.
2. Absorption laws p ∧ (p ∨ q) = p and p ∨ (p ∧ q) = p shorten many simplifications.

Answer: (b) (~p ⇒ q) ∧ (~q ⇒ p)

Explanation:
1. The statement is ~p ⇒ q, which is equivalent to p ∨ q.
2. (~p ⇒ q) ∧ (~q ⇒ p) = (p ∨ q) ∧ (q ∨ p) = p ∨ q, the same thing.
3. (a) and (d) are biconditionals and (c) is 'exactly one', which differ from p ∨ q.

Teacher's Note:
1. Use p ⇒ q = ~p ∨ q and p ⇔ q = (p ⇒ q) ∧ (q ⇒ p) to reduce every expression to ∧, ∨, ~.
2. Absorption laws p ∧ (p ∨ q) = p and p ∨ (p ∧ q) = p shorten many simplifications.

Answer: (b) a contradiction

Explanation:
1. p ⇒ ~p is equivalent to ~p, and ~p ⇒ p is equivalent to p.
2. So the proposition is ~p ∧ p, which is always false: a contradiction.

Teacher's Note:
1. A tautology is true for every combination of truth values and a contradiction (fallacy) is false for every combination.
2. Simplify with the laws of algebra of statements, or build a short truth table to be certain.

Answer: (d) ~(~p ⇔ ~q) ⇔ ~(~p ⇔ q)

Explanation:
1. (a) p ⇒ q = ~p ∨ q = q ∨ ~p, so (a) is a tautology.
2. (b) ~(~p ∧ q) ∧ (p ∨ q) = (p ∨ ~q) ∧ (p ∨ q) = p, so (b) is a tautology.
3. (c) ~p ⇔ q and p ⇔ ~q are both true when p, q differ, so (c) is a tautology.
4. (d) ~(~p ⇔ ~q) = ~(p ⇔ q) and ~(~p ⇔ q) = p ⇔ q; the two sides always have opposite truth values, so (d) is a fallacy.

Teacher's Note:
1. A tautology is true for every combination of truth values and a contradiction (fallacy) is false for every combination.
2. Simplify with the laws of algebra of statements, or build a short truth table to be certain.

Answer: (a) p ∧ p = p

Explanation:
1. p ∧ p = p is the idempotent law and is true.
2. p ∨ ~p is always true (a tautology), and p ∧ ~p is always false, so (b), (c) and (d) are wrong.

Teacher's Note:
1. A tautology is true for every combination of truth values and a contradiction (fallacy) is false for every combination.
2. Simplify with the laws of algebra of statements, or build a short truth table to be certain.

Answer: (b) Either the sand does not heat up quickly in the sun or it cools down fast at night.

Explanation:
1. With p : the sand heats up quickly and q : it cools down fast at night, the statement is p ∧ ~q.
2. ~(p ∧ ~q) = ~p ∨ q.
3. So: either it does not heat up quickly or it cools down fast at night.

Teacher's Note:
1. Use De Morgan's laws: ~(p ∧ q) = ~p ∨ ~q and ~(p ∨ q) = ~p ∧ ~q.
2. The negation of p ⇒ q is p ∧ ~q, and the negation of p ⇔ q is (p ∧ ~q) ∨ (~p ∧ q).

Answer: (c) A quadrilateral is a square and it is not a rhombus.

Explanation:
1. ~(p ⇒ q) = p ∧ ~q.
2. So the negation is: a quadrilateral is a square and it is not a rhombus.

Teacher's Note:
1. Use De Morgan's laws: ~(p ∧ q) = ~p ∨ ~q and ~(p ∨ q) = ~p ∧ ~q.
2. The negation of p ⇒ q is p ∧ ~q, and the negation of p ⇔ q is (p ∧ ~q) ∨ (~p ∧ q).

Answer: (b) India beats Australia and India does not qualify for the world cup.

Explanation:
1. ~(p ⇒ q) = p ∧ ~q.
2. So: India beats Australia and does not qualify.

Teacher's Note:
1. Use De Morgan's laws: ~(p ∧ q) = ~p ∨ ~q and ~(p ∨ q) = ~p ∧ ~q.
2. The negation of p ⇒ q is p ∧ ~q, and the negation of p ⇔ q is (p ∧ ~q) ∨ (~p ∧ q).

Answer: (d) Demonetisation is a successful step if and only if Modi Ji is not the prime minister.

Explanation:
1. With p : demonetisation is successful, q : Modi Ji is the prime minister, the statement is p ⇔ q.
2. ~(p ⇔ q) is true exactly when p and q have opposite truth values, which is p ⇔ ~q.
3. (c) is (~p ∧ ~q) ∨ (p ∧ q) = p ⇔ q itself, so it is not the negation.

Teacher's Note:
1. Use De Morgan's laws: ~(p ∧ q) = ~p ∨ ~q and ~(p ∨ q) = ~p ∧ ~q.
2. The negation of p ⇒ q is p ∧ ~q, and the negation of p ⇔ q is (p ∧ ~q) ∨ (~p ∧ q).