Mathematical Reasoning is a moderately important section in JEE Main paper. However, it has less weightage as compared to other topics in JEE Main Mathematics Syllabus having topics like Coordinate Geometry and Integral Calculus. Numericals and questions pertaining to this require less critical thinking and it is sufficiently easy to score in this section.

Mathematical Reasoning mainly deals with how to determine the truth/ truth values of the statements given in the question and principles of reasoning. This topic is covered under JEE Main and not JEE Advanced. It overall carries just 4 marks and is one of the easiest topics in JEE Main syllabus. Around a total of 1-2 questions are asked from this section which carries a whole of 4 marks. Therefore, the total weightage of this section on the entire JEE Main paper is a mere 1-2%.

The standard of difficulty of the questions is very easy to easy. If the candidate has understood the topic and solved the sample questions, then the candidate shall be able to attempt the questions on JEE Main question paper with a little bit of calculation. Some of the important topics that fall under this section are fallacy; tautology along with various mathematical reasoning laws.

Check JEE Main Mathematics Preparation

JEE Main Test Series Mathematical Reasoning: Previous Year Asked Questions and Solutions-

Ques. The negation of the statement, “If I become a teacher, then I will open a school” is which of the following?

1. Either I will not become a teacher, or I will not open a school.
2. I will not become a teacher or I will open a school.
3. I will become a teacher and I will not open a school.
4. Neither I will become a teacher, nor I will open a school. (AIEEE 2012)

Solution- c.); negation refers to the contradiction or denial of a particular fact/information etc. Therefore, option c will be the correct answer as it negates the statement correctly. Candidates might confuse b, c, and d as 3 of them appear to be as probable answers. But b and d can be eliminated on the basis of the word ‘or’ which divides the statement into parts in contrary to the original statement.

Ques. The statement ∼(p↔∼q) is parallel to which of the following-

1. Is parallel to ∼ p ↔ q
2. A fallacy
3. Is parallel to p ↔ q
4. A tautology (JEE 2014)

Solution- c.) Solving the bracket will give the following answer

Ques. Which of the following is the contrapositive of the following situation wherein two triangles are identical so that these are similar?

1. If two triangles are not identical, then these are similar
2. If two triangles are not similar, they are not identical
3. If two triangles are not similar, then these are identical
4. If two triangles are not identical, then these are not similar (JEE 2017)

Solution- b.)

Consider the following in order to determine the answer-

p: Two triangles are identical.

q: Two triangles are similar.

Clearly, the given statement in symbolic form is p ⇒ q.

Therefore, its contrapositive is given by ∼ q ⇒ ∼ p

Now,

or, ∼p: two triangles are not identical.

or, ∼q: two triangles are not similar.

Therefore, ~ q ⇒ ~ p

Ques. Given are the following statements-

P: Suman is brilliant

Q: Suman is rich

R: Suman is honest

Therefore, the negation of the statement can be, “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as which of the following- (JEE 2018)

1. ∼ Q ↔ P ^ R
2. ∼ [ Q ↔ (P ^ ∼ R)]
3. ∼P ^ (Q ↔ ∼ R)
4. ∼ P ^ (Q ↔ ∼ R)

Solution- b.)

"Suman is brilliant and dishonest" an be expressed as : P∧∼R

So “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as :

(P∧∼R)↔Q

Therefore, negation of this is = ∼[(P∧∼R)↔Q]

or, it is given that p↔q≡q↔p

So , ∼[(P∧∼R)↔Q]≡∼[Q↔(P∧∼R)]

Ques. If each of the following statements is true, then P ⇒ ~q, q ⇒ r, ~r

1. p is true
2. None of these
3. P is false
4. Q is true (JEE 2012)

Solution- c.)

Since ∼r is true, therefore, r is false. Also, q ⇒ r is true, therefore, q is false. (Therefore, a true statement cannot imply a false one) Also, p ⇒ q is true, therefore, p must be false.

Ques. What is the negation of the compound proposition? (JEE 2013)

If the examination is difficult, then I shall pass if I study hard.

1. If the examination is difficult, I shall not study to pass
2. If the examination is difficult, I shall not pass even if I study
3. The examination is difficult and I study hard but I shall not pass
4. The examination is difficult and I shall not study to pass

Solution- c.)

Let’s consider the following-

If p: Examination is difficult

q: I shall pass

r: I study hard

Given result is P ⇒ (r ⇒ q)

Now, ∼ (r ⇒ q) = r ∧ ∼q ∼(p ⇒ (r ⇒ q)) = p ∧ (r ∧ ∼q)

The examination is difficult and I study hard but I shall not pass.

Ques. If p is true and q is false, then which among the following given below is not true- ( JEE 2019)

1. p ∧ ( ~q)
2. p ∨ q
3. p ⇒ p
4. p ⇒ q 

Solution- d.)

When p is true and q is false, then p∨q is true, q ⇒ p is true and p ∧ (∼q) is true. (Therefore, both p and ∼q both are true)

Here, p ⇒ q is not true as a true statement.

Ques. If p and q are two statements, then (p ⇒ q) ⇔ ( ~q ⇒ ~p) is a ___________. (JEE 2016)

1. Reasoning principle
2. Tautology
3. Symbolic fallacy
4. Contrapositive variables following a reasoning principle

Solution- b.)

Therefore, the given proposition is a tautology

Ques. Which of the following is rationally parallel to ~( ~ p ⇒ q)? (JEE 2014)

1. p ∧ ~q
2. ~p ∧ ~q
3. p ∧ q
4. ~p ∧ q

Solution- b.); It is clear from the truth table given below that- ∼ (∼p ⇒ q) is equivalent to ∼p ∧ ∼q.

Ques. The Boolean Expression (p ∧ ~q) ∨ q ∨( ~p ∧ q) is equivalent to ___________.

Choose from the following-

1. p <=q
2. q ^p/any Boolean expression divisible by zero
3. ~p >q
4. p ∨ q (JEE 2018)

Solution- d.)

[(p ∧ ∼q) ∨ q] ∨ (∼p ∧ q) = (p ∨ q) ∧ (∼q ∨ q) ∨ (∼p ∧ q)

= (p ∨ q) ∧ [t ∨ (∼p ∧ q)]

= (p ∨ q) ∧ t

= p ∨ q

Check JEE Main Sample Papers

JEE Main Test Series Mathematical Reasoning: Quick Formulas

There are eight truth tables which need to be adhered to while solving these questions. The following is given below-

1. Truth table 1, wherein (p ∨ q, q ∨ p)

T= true;

F= false

Rule: p ∧ q is true only when p and q are true

2. Truth table 2 wherein,(p v q, q v p)

Rule: p ∨ q is false only when both p and q are false.

3. Truth table 3, wherein (~p)

Rule: ~ is true only when p is false

4. Truth table 4, wherein(p → q, q → p)

Rule: p → q is false only when p is true and q is false.

5. Truth table 5, wherein (p ↔ q, q ↔ p)

Rule: p ↔ q is true only when both p and q have the same truth value.

6. Truth table 6, wherein(p→ q)

These compound statements which are true having a truth value for their components are called tautologies.

7. Truth table 7, wherein

The contradiction or the negation of a tautology is called a fallacy.

8. Truth table 8, wherein (p ∧ ~p)

Some Other Important Laws

• Associative Laws- Provided p, q and r are three statements, then the relation between them shall be held as the following - p ∨ (q ∨ r) = (p ∨ q) ∨ r and p ∧ (q ∧ r) = (p ∧ q) ∧ r
• Distributive Laws- Provided p, q and r are three statements then the relation between them shall be held as the following- p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r) and p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r)
• Complement Laws- Given t as tautology, c as contradiction, then p is any statement as stated here- p ∨ t = t, p ∧ t = p, p ∨ c = p and p ∧ c = c
• Identity Laws- If p is any statement, t is tautology, c is contradiction then p ∨ t = t, p ∧ t = p, p ∨ c = p and p ∧ c = c
• Commutative Laws- Provided p and q are any two statements, then then p ∨ q = q ∨ p and p ∧ q = q ∧ p
• Idempotent Laws- If p is any statement, then p ∨ p = p and p ∧ p = p
• De- Morgan’s Law- Provided p and q are two statements then ~(p ∨ q) = (~p) ∧ (~q) and ~(p ∧ q) = (~p) ∨ (~q)
• Involution Law- Provided p is any statement, then ~(~p) = p

Check JEE Main Study Notes on Mathematical Reasoning