$(p \wedge \sim q) \wedge(\sim p \wedge q)$ is

Both a tautology and a contradiction

Neither a tautology nor a contradiction

If $p$ is any logical statement, then:

$p \wedge(\sim p)$ is a tautology.

$p \vee(\sim p)$ is a contradiction

Consider the statement “For an integer $n$ if ${n^3} - 1$ even, then n is odd”. The contrapositive statement of this statement is

For an integer $n,$ if $n$ is even, then ${n^3} - 1$ is odd

For an integer $n,$ if ${n^3} - 1$ is not even, then $n$ is not odd

For an integer $n,$ if $n$ is even, then ${n^3} - 1$ is even

For an integer $n,$ if $n$ is odd, then ${n^3} - 1$ is even

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JEE • MATHEMATICS • MATHEMATICAL REASONING

Mathematical reasoning practice · Easy

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 171 questions

1 Easy

1 of 171

Understanding +4 | -1

p:A

man is happy

q:A

man is rich
Then, the statement, "If a man is not happy, then he is not rich" is written as

View answer and explanation

Correct option: A

\sim p \to \sim q

Option A (

\sim p \to \sim q

) is correct because it directly translates the given statement \"If a man is not happy, then he is not rich\" into symbolic form.

2 Easy

2 of 171

Understanding +4 | -1

Let

p

and

q

be two Statements. Amongst the following, the Statement that is equivalent to

p \rightarrow q

View answer and explanation

Correct option: B

\sim p \vee q

Option B (

\sim p \vee q

) is correct because it is the logically equivalent form of the implication

p \rightarrow q

3 Easy

3 of 171

Analyzing +4 | -1

Let

{F_1}(A,\,B,\,C) = \left( {A \wedge \sim B} \right) \vee \left[ { \sim C \wedge \left( {A \vee B} \right)} \right] \vee \sim A

and

{F_2}(A,\,B) = \left( {A \vee B} \right) \vee (B \to \, \sim \,A)

be two logical expressions. Then,

View answer and explanation

Correct option: C

{F_1}

is not tautology but

{F_2}

is a tautology

Option C (

{F_1}

is not tautology but

{F_2}

is a tautology) is correct because F1 can be false for some truth values of A, B, and C, while F2 is always true.

4 Easy

4 of 171

Understanding +4 | -1

The contrapositive of the following statement, "If the side of a square doubles, then its area increases four times", is :

View answer and explanation

Correct option: C

If the area of a square does not increases four times, then its side is not doubled.

Option C (If the area of a square does not increases four times, then its side is not doubled) is correct because it is the contrapositive of the given statement.

5 Easy

5 of 171

Understanding +4 | -1

The negation of the proposition "If 2 is prime, then 3 is odd" is

View answer and explanation

Correct option: B

2 is prime and 3 is not odd

Option B (2 is prime and 3 is not odd) is correct because it represents the negation of the conditional statement using the rule

\sim (p \to q) \equiv p \wedge \sim q

6 Easy

6 of 171

Understanding +4 | -1

Statement - 1: Contrapositive of "If the weather is cold, then it will snow" is "If it will not snow, then the weather is not cold".

Statement - 2: Negation of "If it snows then they do not drive the car" is "it snows and they drive the car".

Which of the following is true, regarding the above statements?

View answer and explanation

Correct option: B

Both

S_1, S_2

Option B (Both

S_1, S_2

) is correct because Statement 1 correctly states the contrapositive, and Statement 2 correctly states the negation of the given conditional statements.

7 Easy

7 of 171

Analyzing +4 | -1

Statement

(P \Rightarrow Q) \wedge(R \Rightarrow Q)

is logically equivalent to

View answer and explanation

Correct option: D

(P \wedge R) \Rightarrow Q

Option D (

(P \wedge R) \Rightarrow Q

) is correct because it is the logical equivalent derived by converting implications to disjunctions and applying De Morgan's law.

8 Easy

8 of 171

Analyzing +4 | -1

\mathrm{p}

and

\mathrm{q}

be two statements then

\sim(\sim p \wedge q) \wedge(p \vee q)

is logically equivalent to

View answer and explanation

Correct option: C

p

The correct answer is C. By simplifying the given logical expression using De Morgan's laws and other logical equivalences, it is shown to be logically equivalent to p.

9 Easy

9 of 171

Understanding +4 | -1

The expression

\sim \left( { \sim p \to q} \right)

is logically equivalent to

View answer and explanation

Correct option: D

\sim p \wedge \sim q

Option A (

p \wedge \sim q

) is correct because the expression simplifies to

p \wedge \sim q

using logical equivalences.

10 Easy

10 of 171

Understanding +4 | -1

The statement

A \to \left( {B \to A} \right)

is equivalent to

View answer and explanation

Correct option: D

A \to \left( {A \vee B} \right)

Option D (

A \to (A \vee B)

) is correct because the given statement is a tautology and is equivalent to

A \to (A \vee B)

11 Easy

11 of 171

Understanding +4 | -1

Consider the proposition : “ If the pressure increases, the volume decreases:. The negation of this proposition is

View answer and explanation

Correct option: C

If the volume does not decreases, the pressure does not increase

The correct answer is D because it represents the contrapositive of the original implication, which is logically equivalent to the negation.

12 Easy

12 of 171

Understanding +4 | -1

p

is any statement,

t

is tautology &

c

is a contradiction, then which of following is not correct-

View answer and explanation

Correct option: A

p \vee(\sim p) \equiv c

Option A is correct because p OR (NOT p) is always true (tautology), not a contradiction.

13 Easy

13 of 171

Understanding +4 | -1

Negation of the statement

(p \vee q) \Rightarrow (q \vee r)

View answer and explanation

Correct option: A

p \wedge \sim q \wedge \sim r

The correct answer is A (

p \wedge \sim q \wedge \sim r

) because it represents the negation of the given implication using logical equivalences.

14 Easy

14 of 171

Understanding +4 | -1

Which of the following not a statement in logic?
1. Earth is planet.
2. Plants are living objects.
3.

\sqrt { - 3}

is a rational number.
4.

{x^2} - 5x + 6 < 0

, when

x \in - R

View answer and explanation

Correct option: C

Option D (4) is correct because

{x^2} - 5x + 6 < 0

is a statement for a specific x value, but not a statement in logic for all x in R.

15 Easy

15 of 171

Understanding +4 | -1

Assertion: The compound statement with 'And' is true if all its component statements are true.

Reason: The compound statement with 'And' is false if any of its component statements is false.

View answer and explanation

Correct option: B

Assertion is correct, reason is correct; reason is not a correct explanation for assertion

Option B (Assertion is correct, reason is correct; reason is not a correct explanation for assertion) is correct because both statements about compound statements with 'And' are true, but the reason doesn't explain the assertion.

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16 Easy

16 of 171

Understanding +4 | -1

Let

p, q, r

be the following three statements:

p: n

is prime

q: n

is odd

r: n

is 2

Then

p \rightarrow q \vee r

means

View answer and explanation

Correct option: A

n

is prime then

n

is odd or 2

Option A (If

n

is prime then

n

is odd or 2) is correct because

p \rightarrow q \vee r

means "If

n

is prime, then

n

is odd or

n

is 2".

17 Easy

17 of 171

Analyzing +4 | -1

(p \wedge \sim q) \wedge(\sim p \wedge q)

View answer and explanation

Correct option: B

A contradiction

Option B (A contradiction) is correct because the given expression simplifies to a contradiction using associative and commutative laws.

18 Easy

18 of 171

Understanding +4 | -1

p \wedge q \to p

View answer and explanation

Correct option: A

A tautology

Option A is correct because

p \wedge q \to p

is always true, making it a tautology.

19 Easy

19 of 171

Understanding +4 | -1

The contrapositive of the statement "if I am not feeling well, then I will go to the doctor" is

View answer and explanation

Correct option: C

If I will not go to the doctor, then I am feeling well

Option C (If I will not go to the doctor, then I am feeling well) is correct because it is the contrapositive of the given statement.

20 Easy

20 of 171

Understanding +4 | -1

p

is any logical statement, then:

View answer and explanation

Correct option: C

p \wedge p \equiv p

Option C (

p \wedge p \equiv p

) is correct because it represents the idempotent law, which states that the conjunction of a statement with itself is equivalent to the statement itself.

21 Easy

21 of 171

Analyzing +4 | -1

Which of the following Boolean expression is not a tautology?

View answer and explanation

Correct option: D

( \sim p \Rightarrow q) \vee ( \sim q \Rightarrow p)

Option D (

(\sim p \Rightarrow q) \vee (\sim q \Rightarrow p)

) is not a tautology because its truth value depends on the values of p and q, and it is not always true.

22 Easy

22 of 171

Understanding +4 | -1

p

is false and

q

is true, then

View answer and explanation

Correct option: D

p \Rightarrow q

is true

Option D (

p \Rightarrow q

) is correct because a false premise implies any conclusion, so

p \Rightarrow q

is true when

p

is false and

q

is true.

23 Easy

23 of 171

Analyzing +4 | -1

The Boolean expression

\left( {p \Rightarrow q} \right) \wedge \left( {q \Rightarrow \, \sim p} \right)

is equivalent to

View answer and explanation

Correct option: D

\sim q

Option A (

\sim q

) is correct because the given Boolean expression simplifies to

\sim p \wedge \sim q

, which is equivalent to

\sim (p \vee q)

, and further simplifies to

\sim q

given the initial conditions.

24 Easy

24 of 171

Analyzing +4 | -1

If the Boolean expression

(p \wedge q)*(p \otimes q)

is a tautology, then * and

\otimes

are respectively, given by

View answer and explanation

Correct option: A

\to ,\,\, \to

Option A (

\to ,\\,\to

) is correct because it results in a tautology when substituted into the given Boolean expression.

25 Easy

25 of 171

Understanding +4 | -1

Let

p

and

q

denote the following statements

p:

The sun is shining

q

: I shall play tennis in the afternoon The negation of the statement "If the sun is shining then I shall play tennis in the afternoon", is

View answer and explanation

Correct option: C

p \wedge \sim q

Option C (

p \wedge \sim q

) is correct because it represents the negation of the given conditional statement.

26 Easy

26 of 171

Analyzing +4 | -1

Which of the following is not tautology?

View answer and explanation

Correct option: B

(p \vee q) \rightarrow(p \vee \sim q)

Option A,

(p \vee q) \rightarrow p

, is not a tautology because its truth value is not always true for all possible values of p and q.

27 Easy

27 of 171

Analyzing +4 | -1

Which of the following is not logically equivalent to the following proposition '

A

real number is either rational or irrational'?

View answer and explanation

Correct option: B

If a number is not a rational or not an irrational, then it is not real

Option B is correct because it is not logically equivalent to the given proposition; the original statement asserts that every real number must be either rational or irrational, while option B implies that if a number is not rational OR not irrational, then it is not real, which is not the same.

28 Easy

28 of 171

Analyzing +4 | -1

The logical statement

[ \sim ( \sim p \vee q) \vee (p \wedge r)] \wedge ( \sim q \wedge r)

is equivalent to

View answer and explanation

Correct option: C

(p \wedge r) \wedge \sim q

Option C (

(p \wedge r) \wedge \sim q

) is correct because the logical statement simplifies to this expression using De Morgan's laws and distributive properties.

29 Easy

29 of 171

Understanding +4 | -1

Consider the statement “For an integer

n

{n^3} - 1

even, then n is odd”. The contrapositive statement of this statement is

View answer and explanation

Correct option: A

For an integer

n,

n

is even, then

{n^3} - 1

is odd

Option A (For an integer

n,

n

is even, then

{n^3} - 1

is odd) is correct because it correctly represents the contrapositive of the given statement.

30 Easy

30 of 171

Understanding +4 | -1

Negation of "Pairs is in France and Londan is in England" is

View answer and explanation

Correct option: B

Pairs is not in France or Londan is not in England

Option B (Pairs is not in France or Londan is not in England) is correct because it represents the negation of the conjunction using De Morgan's Law.

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 171 single-correct-answer MCQs on MATHEMATICAL REASONING.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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