Multiple Choice Questions (MCQs) for JEE Mathematics: Mathematical Reasoning
Access targeted multiple-choice questions for Mathematical Reasoning designed to align with the latest JEE academic syllabus for JEE Mathematics. These objective practice sets help students evaluate their conceptual understanding and improve exam readiness.
Practice Mathematical Reasoning MCQs for JEE Mathematics
Access the complete set of multiple-choice questions for Mathematical Reasoning below. This focused format allows students to isolate specific topics for thorough review and uninterrupted practice alongside official JEE textbooks.
Multiple Choice Questions
(a) ~p ⇒ ~(q ∧ r)
(b) ~p ∨ (q ∧ r)
(c) (q ∧ r) ⇒ p
(d) p ∧ (~q ∨ ~r)
Show Answer & Explanation
Answer: (d) p ∧ (~q ∨ ~r)
Explanation:
1. ~(p ⇒ X) = p ∧ ~X with X = q ∧ r.
2. ~(q ∧ r) = ~q ∨ ~r, so the negation is p ∧ (~q ∨ ~r).
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).
(a) TTTT
(b) TFTF
(c) FFTT
(d) FTFT
Show Answer & Explanation
Answer: (a) TTTT
Explanation:
1. ~p ⇒ q = p ∨ q, so its negation is ~(p ∨ q); and p ∨ ~p is always true, so S1 is true.
2. S2 and S3 are the standard negation rules, so they are true.
3. p ⇔ q = (p ⇒ q) ∧ (q ⇒ p) = (~p ∨ q) ∧ (~q ∨ p), so S4 is true.
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).
(a) p ⇒ ~q
(b) ~p ∨ ~q
(c) ~(p ⇒ ~q)
(d) ~(~p ∧ ~q)
Show Answer & Explanation
Answer: (c) ~(p ⇒ ~q)
Explanation:
1. The statement is p ∧ q.
2. ~(p ⇒ ~q) = p ∧ ~(~q) = p ∧ q, so option (c) represents it.
3. (d) is p ∨ q and (b) is ~(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.
(a) (p ∧ r) ∧ (r ∨ q)
(b) (~p ∨ ~r) ∧ (r ∨ q)
(c) a tautology
(d) a fallacy
Show Answer & Explanation
Answer: (d) a fallacy
Explanation:
1. ~(X ⇒ Y) = X ∧ ~Y: (p ∧ r) ∧ ~(r ∨ q) = p ∧ r ∧ ~r ∧ ~q.
2. This contains r ∧ ~r, so it is always false: a fallacy.
3. Equivalently, (p ∧ r) ⇒ (r ∨ q) is always true (a tautology), so its negation 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.
(a) You watch television if and only if your mind is free.
(b) You watch television and your mind is free.
(c) You watch television or your mind is free.
(d) None of these
Show Answer & Explanation
Answer: (a) You watch television if and only if your mind is free.
Explanation:
1. The statement is (p ⇒ q) ∧ (q ⇒ p), which is p ⇔ q.
2. That is: you watch television if and only if your mind is free.
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.
(a) If x − 2y = 9 then x = 5 and y = −2
(b) If x − 2y ≠ 9 then x ≠ 5 and y ≠ −2
(c) If x − 2y ≠ 9 then x ≠ 5 or y ≠ −2
(d) If x − 2y ≠ 9 then either x ≠ 5 or y = −2
Show Answer & Explanation
Answer: (c) If x − 2y ≠ 9 then x ≠ 5 or y ≠ −2
Explanation:
1. The statement is (p ∧ q) ⇒ r with p : x = 5, q : y = −2, r : x − 2y = 9.
2. Its contrapositive is ~r ⇒ ~(p ∧ q) = ~r ⇒ (~p ∨ ~q).
3. So: if x − 2y ≠ 9 then x ≠ 5 or y ≠ −2.
Teacher's Note:
1. The contrapositive of p ⇒ q is ~q ⇒ ~p and it is always logically equivalent to p ⇒ q.
2. The converse q ⇒ p and the inverse ~p ⇒ ~q are equivalent to each other but not to the original statement.
(a) If the area of a square does not increase four times, then its side is not doubled.
(b) If the area of a square increases four times, then its side is not doubled.
(c) If the area of a square increases four times, then its side is doubled.
(d) If the side of a square is not doubled, then its area does not increase four times.
Show Answer & Explanation
Answer: (a) If the area of a square does not increase four times, then its side is not doubled.
Explanation:
1. Contrapositive of p ⇒ q is ~q ⇒ ~p.
2. Here: if the area does not increase four times, then the side is not doubled.
Teacher's Note:
1. The contrapositive of p ⇒ q is ~q ⇒ ~p and it is always logically equivalent to p ⇒ q.
2. The converse q ⇒ p and the inverse ~p ⇒ ~q are equivalent to each other but not to the original statement.
(a) If I will come, then it is raining
(b) If I will not come, then it is not raining
(c) If I will not come, then it is raining
(d) If I will come, then it is not raining
Show Answer & Explanation
Answer: (d) If I will come, then it is not raining
Explanation:
1. With p : it is raining and q : I will not come, the statement is p ⇒ q.
2. The contrapositive is ~q ⇒ ~p: if I will come, then it is not raining.
Teacher's Note:
1. The contrapositive of p ⇒ q is ~q ⇒ ~p and it is always logically equivalent to p ⇒ q.
2. The converse q ⇒ p and the inverse ~p ⇒ ~q are equivalent to each other but not to the original statement.
(a) ~p implies ~q
(b) q implies p
(c) ~q implies ~p
(d) p only if q
Show Answer & Explanation
Answer: (c) ~q implies ~p
Explanation:
1. The contrapositive of p ⇒ q is ~q ⇒ ~p.
Teacher's Note:
1. The contrapositive of p ⇒ q is ~q ⇒ ~p and it is always logically equivalent to p ⇒ q.
2. The converse q ⇒ p and the inverse ~p ⇒ ~q are equivalent to each other but not to the original statement.
(a) ~r ⇒ (p ∨ q)
(b) r ⇒ (p ∨ q)
(c) ~r ⇒ (~p ∨ ~q)
(d) p ⇒ (q ∨ r)
Show Answer & Explanation
Answer: (c) ~r ⇒ (~p ∨ ~q)
Explanation:
1. Contrapositive: ~r ⇒ ~(p ∧ q).
2. By De Morgan, ~(p ∧ q) = ~p ∨ ~q.
3. So the contrapositive is ~r ⇒ (~p ∨ ~q).
Teacher's Note:
1. The contrapositive of p ⇒ q is ~q ⇒ ~p and it is always logically equivalent to p ⇒ q.
2. The converse q ⇒ p and the inverse ~p ⇒ ~q are equivalent to each other but not to the original statement.
(a) (~q ∧ r) ⇒ ~p
(b) (q ∧ ~r) ⇒ p
(c) p ⇒ (~r ∨ q)
(d) p ∧ (q ∨ r)
Show Answer & Explanation
Answer: (a) (~q ∧ r) ⇒ ~p
Explanation:
1. Contrapositive: ~(~q ⇒ ~r) ⇒ ~p.
2. ~(~q ⇒ ~r) = ~q ∧ r.
3. So the contrapositive is (~q ∧ r) ⇒ ~p.
Teacher's Note:
1. The contrapositive of p ⇒ q is ~q ⇒ ~p and it is always logically equivalent to p ⇒ q.
2. The converse q ⇒ p and the inverse ~p ⇒ ~q are equivalent to each other but not to the original statement.
(a) Not everyone in Germany speaks German.
(b) No one in Germany speaks German.
(c) There are persons in Germany who do not speak German.
(d) There is at least one person in Germany who does not speak German.
Show Answer & Explanation
Answer: (b) No one in Germany speaks German.
Explanation:
1. The negation of 'everyone speaks German' is 'at least one person does not speak German'.
2. (a), (c) and (d) all say this.
3. 'No one speaks German' is much stronger than the negation, so it is not a negation of p.
Teacher's Note:
1. The negation of 'for all' is 'there exists ... not', and the negation of 'there exists' is 'for all ... not'.
2. Negate the property inside the quantifier as well.
(a) p ⇒ q is true
(b) p ⇒ q is false
(c) ~q ⇒ ~p is false
(d) p ∧ q is true
Show Answer & Explanation
Answer: (a) p ⇒ q is true
Explanation:
1. p is false and q is false.
2. A conditional with a false hypothesis is true, so p ⇒ q is true.
3. The contrapositive ~q ⇒ ~p = T ⇒ T is also 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.
(a) If it is a national holiday, then kids go to picnic with their parents.
(b) Kids go to picnic with their parents and it is a national holiday.
(c) If kids go to picnic with their parents, then it is a national holiday.
(d) Kids go to picnic with their parents or it is a national holiday.
Show Answer & Explanation
Answer: (c) If kids go to picnic with their parents, then it is a national holiday.
Explanation:
1. The converse of p ⇒ q is q ⇒ p.
2. So: if kids go to picnic with their parents then it is a national holiday.
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.
(a) Having clouds in the sky is sufficient to have rain.
(b) It is not necessary to have rain if there are clouds in the sky.
(c) If it is raining then there are clouds in the sky.
(d) There are clouds in the sky implies it will not rain.
Show Answer & Explanation
Answer: (a) Having clouds in the sky is sufficient to have rain.
Explanation:
1. The statement is p ⇒ q with p : clouds and q : rain.
2. p ⇒ q means p is sufficient for q, so (a) is correct.
3. (c) is the converse, and (b), (d) contradict the statement.
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.
(a) It is not true that the traders do not reduce the prices and the government does not take action against them.
(b) It is true that the traders do not reduce the prices and the government does not take action against them.
(c) It is not true that the traders do not reduce the prices and the government takes action against them.
(d) It is not true that the traders do not reduce the prices or the government takes action against them.
Show Answer & Explanation
Answer: (a) It is not true that the traders do not reduce the prices and the government does not take action against them.
Explanation:
1. With p : traders reduce the price and q : government takes action, the statement is ~p ⇒ q = p ∨ q.
2. (a) is ~(~p ∧ ~q) = p ∨ q, which matches.
3. (b) is the negation of the statement, and (c), (d) simplify to different expressions.
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.
(a) p ∨ q
(b) p ∧ q
(c) ~p ∧ q
(d) ~p ∨ q
Show Answer & Explanation
Answer: (b) p ∧ q
Explanation:
1. p ∧ (~p ∨ q) = (p ∧ ~p) ∨ (p ∧ q) = F ∨ (p ∧ q) = 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.
(a) Tautology
(b) Fallacy
(c) ~p ∧ ~q
(d) p ∨ q
Show Answer & Explanation
Answer: (a) Tautology
Explanation:
1. p ∨ ~p is always true, and T ∨ (anything) = T.
2. So the statement is a tautology.
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.
(a) Fallacy
(b) Tautology
(c) ~p ∧ ~q
(d) p ∨ q
Show Answer & Explanation
Answer: (b) Tautology
Explanation:
1. The antecedent contains r as a conjunct: whenever it is true, r is true.
2. So the implication can never be false: a tautology.
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.
(a) p
(b) q
(c) ~p
(d) ~q
Show Answer & Explanation
Answer: (c) ~p
Explanation:
1. ~(p ∨ ~q) = ~p ∧ q and ~(p ∨ q) = ~p ∧ ~q.
2. (~p ∧ q) ∨ (~p ∧ ~q) = ~p ∧ (q ∨ ~q) = ~p.
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.
(a) a tautology
(b) a fallacy
(c) ~p ∧ q
(d) ~p ∨ q
Show Answer & Explanation
Answer: (b) a fallacy
Explanation:
1. p ∨ (~p ∨ q) = (p ∨ ~p) ∨ q = T, a tautology.
2. The negation of a tautology 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.
(a) △ABC is right angled at B and AB² + BC² ≠ AC².
(b) △ABC is right angled at B then AB² + BC² ≠ AC².
(c) △ABC is right angled at B or AB² + BC² ≠ AC².
(d) △ABC is right angled at B and AB² + BC² = AC².
Show Answer & Explanation
Answer: (a) △ABC is right angled at B and AB² + BC² ≠ AC².
Explanation:
1. ~(p ⇒ q) = p ∧ ~q.
2. So: △ABC is right angled at B and AB² + BC² ≠ AC².
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).
(a) If you want to succeed then you cannot find a way.
(b) If you do not want to succeed then you will find a way.
(c) You will not succeed and you find a way.
(d) You will not succeed and you do not find a way.
Show Answer & Explanation
Answer: (b) If you do not want to succeed then you will find a way.
Explanation:
1. The negation of ~(p ∨ q) is ~(~(p ∨ q)) = p ∨ q.
2. p ∨ q is equivalent to ~p ⇒ q: if you do not want to succeed then you will find a way.
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.
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Mathematical Reasoning Objective Questions & Solutions for JEE Mathematics
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FAQs
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