How much of JEE Main comes from Mathematical Reasoning?

JEEGenie estimates about 1% of JEE Main marks come from Mathematical Reasoning, based on recent papers. The share varies from year to year.

Which formulas matter most in Mathematical Reasoning?

The Mathematical Reasoning sheet on this page has 13 formula entries, starting with Conjunction (AND), Disjunction (OR), Negation (NOT), Conditional (Implication) p → q, Biconditional p ↔ q. Each one comes with when to use it and the trap to avoid.

What topics does Mathematical Reasoning cover for JEE?

Mathematical Reasoning covers Statements, Logical Connectives, Quantifiers, Truth Tables, Validity of Statements, Tautology and Contradiction.

How should I revise Mathematical Reasoning before the exam?

Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Mathematical Reasoning questions on JEEGenie.

How many previous-year questions are there on Mathematical Reasoning?

JEEGenie has 134 previous-year questions on Mathematical Reasoning, and this page shows a few with worked explanations.

Is practice on Mathematical Reasoning free on JEEGenie?

Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.

Mathematical Reasoning for JEE Main & Advanced — formulas, what is asked, and traps

JEEGenie estimates about 1% of JEE Main marks come from Mathematical Reasoning. This page covers 13 formula entries, each with when to use it, what JEE asks and the traps. A common slip: Exclamatory, interrogative, or imperative sentences are not statements.

Key points to know

Statement (Proposition). A statement is a declarative sentence that is either true or false, but not both simultaneously.
Truth Value. Every statement has a truth value: T (true) or F (false).
Logical Connectives. Connectives (and, or, not, if-then, if and only if) combine statements to form compound statements.
Conjunction (AND). p ∧ q is true only when both p and q are true.
Disjunction (OR). p ∨ q is true if at least one of p or q is true.
Negation (NOT). ~p is true when p is false, and false when p is true.
Conditional (Implication) p → q. p → q is false only when p is true and q is false; otherwise true.
Biconditional p ↔ q. p ↔ q is true when p and q have the same truth value.
Tautology. A compound statement that is always true for all truth values of its components.
Contradiction (Fallacy). A compound statement that is always false for all truth values.
Contingency. A compound statement that is neither a tautology nor a contradiction; its truth value depends on components.
Logical Equivalence. Two statements are logically equivalent if they have the same truth value in all possible cases.
Converse, Inverse, Contrapositive. For p → q: converse is q → p, inverse is ~p → ~q, contrapositive is ~q → ~p. Only contrapositive is logically equivalent to original.
Quantifiers: Universal (∀) and Existential (∃). ∀ means 'for all' (every element), ∃ means 'there exists' (at least one).
Valid Arguments. An argument is valid if whenever all premises are true, the conclusion is also true.
Necessary and Sufficient Conditions. p is sufficient for q means p → q; p is necessary for q means q → p.

Formula sheet

Name	Formula	When to use
Conjunction (AND)	p ∧ q is T iff p = T and q = T	Combine two conditions that must both hold.
Disjunction (OR)	p ∨ q is F iff p = F and q = F	Combine conditions where at least one must be true.
Negation (NOT)	~p = T iff p = F	Form the opposite of a statement.
Conditional (Implication) p → q	p → q ≡ ~p ∨ q	Express 'if p then q' in logical reasoning.
Biconditional p ↔ q	p ↔ q ≡ (p → q) ∧ (q → p)	Express equivalence or necessary and sufficient conditions.
Logical Equivalence	p ≡ q means p ↔ q is a tautology	Simplify or replace statements with equivalent forms.
De Morgan's Laws	~(p ∧ q) ≡ ~p ∨ ~q ~(p ∨ q) ≡ ~p ∧ ~q	Negate compound statements with AND/OR.
Converse, Inverse, Contrapositive	p → q ≡ ~q → ~p	Rewrite implications for proofs or logical reasoning.
Quantifiers: Universal (∀) and Existential (∃)	∀x P(x): true if P(x) holds for every x in domain. ∃x P(x): true if P(x) holds for at least one x.	Express statements about all or some elements in a set.
Negation of Quantified Statements	~(∀x P(x)) ≡ ∃x ~P(x) ~(∃x P(x)) ≡ ∀x ~P(x)	Form the logical opposite of statements with quantifiers.
Rules of Inference	Modus Ponens: (p → q) ∧ p ⇒ q Modus Tollens: (p → q) ∧ ~q ⇒ ~p	Deduce conclusions from given premises.
Necessary and Sufficient Conditions	p is sufficient for q: p ⇒ q p is necessary for q: q ⇒ p	Express relationships between conditions in statements.
Simplifying Compound Statements	p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)	Reduce complex logical expressions to simpler forms.

Open the Mathematical Reasoning formula page

What JEE asks

Statement (Proposition). JEE often asks to pick statements from a list of sentences, testing the true/false dichotomy.
Truth Value. JEE may ask to assign truth values to given statements, often in compound form.
Logical Connectives. JEE tests identification of connectives in given sentences.
Conjunction (AND). JEE may ask truth table for p ∧ q.
Disjunction (OR). JEE tests truth table for p ∨ q and its distinction from XOR.
Negation (NOT). JEE often asks negation of statements with quantifiers.
Conditional (Implication) p → q. JEE tests truth table and equivalent forms like ~p ∨ q.
Biconditional p ↔ q. JEE tests truth table and equivalence to (p→q)∧(q→p).
Truth Tables. JEE may ask to complete a truth table or identify a tautology.
Tautology. JEE often asks to identify tautologies from given options.
Contradiction (Fallacy). JEE may ask to identify contradictions.
Contingency. JEE tests classification of statements into tautology/contradiction/contingency.
Logical Equivalence. JEE often asks to identify equivalent statements or apply De Morgan's laws.
De Morgan's Laws. JEE directly tests De Morgan's laws in negation problems.
Converse, Inverse, Contrapositive. JEE asks to identify contrapositive or check equivalence.
Quantifiers: Universal (∀) and Existential (∃). JEE tests negation of statements with quantifiers, e.g., 'All students are tall' negated as 'Some students are not tall'.
Negation of Quantified Statements. JEE frequently asks for negation of statements like 'Every student passed'.
Valid Arguments. JEE may ask to identify valid arguments from given statements.
Rules of Inference. JEE tests identification of which rule is used in a given argument.
Fallacies. JEE may ask to spot fallacies in given reasoning.
Necessary and Sufficient Conditions. JEE asks to identify necessary/sufficient conditions from given implications.
Using Truth Tables for Equivalence. JEE may ask to find which statement is equivalent to a given one.
Simplifying Compound Statements. JEE may ask to simplify statements using logical identities.

Common traps

Exclamatory, interrogative, or imperative sentences are not statements. Statement (Proposition)
Do not confuse truth value with the statement itself. Truth Value
Do not confuse 'or' (inclusive) with everyday exclusive 'or'. Logical Connectives
Conjunction is commutative: p ∧ q ≡ q ∧ p. Conjunction (AND)
Inclusive OR: both true gives true. Disjunction (OR)
Negation of a compound statement requires De Morgan's laws. Negation (NOT)
Do not confuse with causality; it's logical implication. Conditional (Implication) p → q
It is equivalent to 'if and only if'. Biconditional p ↔ q
For n statements, there are 2^n rows. Truth Tables
A tautology is not a single statement but a logical truth. Tautology
Negation of a tautology is a contradiction. Contradiction (Fallacy)
Most statements are contingencies. Contingency
Equivalence is not the same as implication. Logical Equivalence
Do not forget to negate both components. De Morgan's Laws

Topics in Mathematical Reasoning

Statements
Logical Connectives
Quantifiers
Truth Tables
Validity of Statements
Tautology and Contradiction

Previous-year questions on Mathematical Reasoning

JEEGenie has 134 previous-year questions on Mathematical Reasoning. A few worked examples follow.

Year	Questions
2023	23
2022	22
2021	25
2020	16
2019	16
2018	5
2017	5
2016	3
2015	2
2014	4
2013	2
2012	6
2011	1
2010	1
2009	1
2008	2

Question 1 easy

Which of the following statements is a tautology ?

$((\sim p) \lor q) \Rightarrow p$
$p \Rightarrow ((\sim p) \lor q)$
$((\sim p) \lor q) \Rightarrow q$
$q \Rightarrow ((\sim p) \lor q)$

Answer: D

Explanation. A tautology is a statement that is always true regardless of truth values. Option D, q

\Rightarrow

(∼p ∨ q), is logically equivalent to ∼q ∨ (∼p ∨ q) which simplifies to (∼q ∨ q) ∨ ∼p ≡ T ∨ ∼p ≡ T. Thus it is always true, making it a tautology.

From NTA/IIT papers: JEE Main 2022 · 25 Jul Shift 1

Question 2 medium

If the truth value of the Boolean expression

((p \lor q) \land (q \to r) \land (\sim r)) \to (p \land q)

is false, then the truth values of the statements p, q, r respectively can be :

T F T
F F T
T F F
F T F

Answer: C

Explanation. We need the whole implication to be false. An implication

A \to B

is false only when

A

is true and

B

is false. Here

A = (p \lor q) \land (q \to r) \land (\sim r)

and

B = p \land q

. For

B

false:

p \land q

false means at least one of

p, q

is false. For

A

true: -

\sim r

true

\Rightarrow r = F

. -

q \to r

true with

r = F

forces

q = F

. -

p \lor q

true with

q = F

forces

p = T

. Thus

p = T

q = F

r = F

makes

A

true and

B

false, so implication false. Hence the truth values are T F F. The answer is option C.

From NTA/IIT papers: JEE Main 2021 · 26 Aug Shift 1

Question 3 easy

Which one of the following is a tautology?

P $\land$ (P $\lor$ Q)
P $\lor$ (P $\land$ Q)
$Q \to (P \land (P \to Q))$
$(P \land (P \to Q)) \to Q$

Answer: D

Explanation. A tautology is a statement true for every truth assignment. The cleanest test is to try to make each option false. For (D):

(P \land (P \to Q)) \to Q

. This is **Modus Ponens**. Assume the antecedent

P \land (P \to Q)

is true. Then

P

is true and

P \to Q

is true. Since

P

is true,

Q

must be true. So the consequent

Q

is true. Hence the implication can never have true

\to

false, so it is always true. Check others quickly: - (A)

P \land (P \lor Q)

: if

P = F

, expression is false. - (B)

P \lor (P \land Q)

: if

P = F

, expression is false. - (C)

Q \to (P \land (P \to Q))

: take

Q = T

P = F

. Then

P \to Q

is true, but

P \land (P \to Q)

is false, so

T \to F

is false. Thus only (D) is true under all truth values. Answer: (D)

From NTA/IIT papers: JEE Main 2020 · 8 Jan Shift 1

Question 4 medium

Which one of the following statements is not a tautology?

$(p \land q) \to (p) \lor q$
(p $\land$ q) $\to$ p
$(p \lor q) \to (p \lor (q))$
$p \to (p \lor q)$

Answer: C

Explanation. We check each option by truth values. **Option A:**

(p \land q) \to (\sim p \lor q)

When

p = T, q = T

: LHS

= T

, RHS

= F \lor T = T

, so

T \to T = T

. When

p = T, q = F

: LHS

= F

, so implication is

T

. When

p = F, q = T

: LHS

= F

, so

T

. When

p = F, q = F

: LHS

= F

, so

T

. Thus A is a tautology. **Option B:**

(p \land q) \to p

p \land q

is true, then

p

is true, so implication is always true. Tautology. **Option C:**

(p \lor q) \to (p \lor \sim q)

Check

p = F, q = T

: LHS

= T

, RHS

= F \lor F = F

, so

T \to F = F

. Hence not a tautology. **Option D:**

p \to (p \lor q)

p

is true,

p \lor q

is true, so always true. Tautology. Therefore, the statement that is **not** a tautology is:

C

From NTA/IIT papers: JEE Main 2019 · 8 Apr Shift 2

Frequently asked questions

How much of JEE Main comes from Mathematical Reasoning?: JEEGenie estimates about 1% of JEE Main marks come from Mathematical Reasoning, based on recent papers. The share varies from year to year.
Which formulas matter most in Mathematical Reasoning?: The Mathematical Reasoning sheet on this page has 13 formula entries, starting with Conjunction (AND), Disjunction (OR), Negation (NOT), Conditional (Implication) p → q, Biconditional p ↔ q. Each one comes with when to use it and the trap to avoid.
What topics does Mathematical Reasoning cover for JEE?: Mathematical Reasoning covers Statements, Logical Connectives, Quantifiers, Truth Tables, Validity of Statements, Tautology and Contradiction.
How should I revise Mathematical Reasoning before the exam?: Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Mathematical Reasoning questions on JEEGenie.
How many previous-year questions are there on Mathematical Reasoning?: JEEGenie has 134 previous-year questions on Mathematical Reasoning, and this page shows a few with worked explanations.
Is practice on Mathematical Reasoning free on JEEGenie?: Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.