The contrapositive of the statement ${ }^\alpha$ if $2^2=5$, then I get first class" is

If I do not get a first class, then $2^2=5$

If I get a first class, then $2^2=5$

The contrapositive of $(p \vee q) \Rightarrow r$ is

$\sim {r} \Rightarrow({p} \vee {q})$

${p} \Rightarrow({q} \vee {r})$

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JEE • MATHEMATICS • MATHEMATICAL REASONING

Mathematical reasoning practice · Medium

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Showing 1-30 of 58 questions

1 Medium

1 of 58

Understanding +4 | -1

Consider the following compound statements.
(i) Mumbai is the capital of Rajasthan or Maharashtra.
(ii)

\sqrt{3}

is a rational or an irrational number.
(iii) 125 is a multiple of 7 or 8 .
(iv) A rectangle is a quadrilateral or a regular hexagon.
Which of the above statements is not true?

View answer and explanation

Correct option: C

(iii)

The correct answer is (iii) because 125 is neither a multiple of 7 nor 8.

2 Medium

2 of 58

Understanding +4 | -1

Suppose

p

: A natural number

n

is odd and

q:

Natural number

n

is not divisible by 2 , then the biconditional statement

p \Leftrightarrow q

View answer and explanation

Correct option: B

a natural number

n

is odd, if and only if it is not divisible by 2

Option B (a natural number

n

is odd, if and only if it is not divisible by 2) correctly represents the biconditional statement

p \Leftrightarrow q

based on the given definitions of

p

and

q

3 Medium

3 of 58

Application +4 | -1

p, q

are true statements and

r, s

are false statements, then the truth value of

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

View answer and explanation

Correct option: B

Option B (0) is correct because by substituting the truth values of

p, q, r,

and

s

into the expression and simplifying, we find the truth value to be false (0).

4 Medium

4 of 58

Understanding +4 | -1

The negation of the statement, " if a quadrilateral is a square then it is a rhombus" is

View answer and explanation

Correct option: C

a quadrilateral is a square and it is not a rhombus

The correct answer is 'a quadrilateral is a square and it is not a rhombus' because it correctly represents the negation of a conditional statement (p -> q) as (p and not q).

5 Medium

5 of 58

Understanding +4 | -1

Consider the following statements.

I. A sentence is called a statement, if it is either true or false.

II. The sentence, "Today is a windy day", is not a statement.

Choose the correct option.

View answer and explanation

Correct option: A

Both I and II are true

Option A (Both I and II are true) is correct because both statements accurately describe statements and non-statements in mathematical logic.

6 Medium

6 of 58

Understanding +4 | -1

p

: It rains today,

q

: I go to school,

r:

I shall meet any friends and

s

: I shall go for a movie, then which of the following is the proposition: If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie.

View answer and explanation

Correct option: A

\sim(p \wedge q) \Rightarrow(r \wedge s)

The correct answer is

\sim(p \wedge q) \Rightarrow(r \wedge s)

because it correctly translates the given English sentence into a logical proposition.

7 Medium

7 of 58

Understanding +4 | -1

For two statements

p

and

q

p

: A quadrilateral is a parallelogram.

q

: The opposite sides are parallel.
Then, the compound proposition, 'A quadrilateral is a parallelogram if and only if the opposite sides are parallel', is represented by

View answer and explanation

Correct option: D

p \leftrightarrow q

Option D (

p \leftrightarrow q

) is correct because the \"if and only if\" statement represents a biconditional, which is denoted by the double arrow.

8 Medium

8 of 58

Analyzing +4 | -1

Statement - I - The Boolean expression

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

is equivalent to

p \vee q

Statement- II - The Boolean expression

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

is equivalent to

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

View answer and explanation

Correct option: C

Both statements are true

Option C (Both statements are true) is correct because both Boolean expressions simplify to the stated equivalences using Boolean algebra rules.

9 Medium

9 of 58

Analyzing +4 | -1

Which of the following statements is a tautology?

View answer and explanation

Correct option: C

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

Option C is correct because the truth table or logical simplification shows that the statement is always true, regardless of the truth values of p and q.

10 Medium

10 of 58

Understanding +4 | -1

Assertion: The sentence "8 is less than 6" is a statement.

Reason: A sentence is called a statement, if it is either true or false but not both.

View answer and explanation

Correct option: A

Assertion is correct, reason is correct; reason is a correct explanation for assertion

Option A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because the assertion is a statement, the reason correctly defines a statement, and the reason explains why the assertion is a statement.

11 Medium

11 of 58

Understanding +4 | -1

Which of the following is true for the statements

p

and

q

View answer and explanation

Correct option: D

-(p \vee q)

is true only when both

p

and

q

are false

Option D (

-(p \vee q)

is true only when both

p

and

q

are false) is correct because the negation of a disjunction is true only when both components are false.

12 Medium

12 of 58

Understanding +4 | -1

The dual of the statement

(p \wedge q) \wedge \sim q

View answer and explanation

Correct option: B

(p \vee q) \vee-q

The correct answer is (p ∨ q) ∨ ~q because the dual is found by replacing AND with OR.

13 Medium

13 of 58

Analyzing +4 | -1

The Boolean expression

(\mathrm{p} \wedge \mathrm{q}) \vee \sim \mathrm{q} \vee(\mathrm{p} \rightarrow \mathrm{q})

is equivalent to

View answer and explanation

Correct option: D

\mathrm{p} \vee \sim \mathrm{p}

Option D (

\mathrm{p} \vee \sim \mathrm{p}

) is correct because the given Boolean expression simplifies to a tautology, which is always true.

14 Medium

14 of 58

Understanding +4 | -1

Negation of 'Ram is in class

X

or Rashmi is in class XII' is

View answer and explanation

Correct option: D

None of the above

The correct answer is None of the above because the negation of 'p or q' is 'not p and not q', which is not represented by any of the options.

15 Medium

15 of 58

Analyzing +4 | -1

Let

\oplus

and

\otimes

are two mathematical operators. If

p \oplus(p \otimes q)

is not a tautology, then

\oplus

and

\otimes

can be

View answer and explanation

Correct option: B

\Rightarrow

and

\wedge

respectively

Option B (

\Rightarrow

and

\wedge

respectively) is correct because

p \Rightarrow (p \wedge q)

is not a tautology, as it can be false when

p

is true and

q

is false.

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16 Medium

16 of 58

Understanding +4 | -1

The inverse of the statement

(p \wedge \sim q) \rightarrow r

View answer and explanation

Correct option: C

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

Option C ((

\sim p \vee q

)

\rightarrow \sim r

) is correct because the inverse of a conditional statement p -> q is ~p -> ~q. Applying this to the given statement (p ^ ~q) -> r results in ~(p ^ ~q) -> ~r which simplifies to (~p v q) -> ~r.

17 Medium

17 of 58

Understanding +4 | -1

Let

p:

I am well.

q:

I will not take rest

r

: I will not sleep properly, then

"If I am not well then I will not take rest and I will not sleep properly" is logically equivalent to

View answer and explanation

Correct option: B

\sim p \rightarrow(q \wedge r)

Option B (

\sim p \rightarrow(q \wedge r)

) is correct because it accurately translates the given English statement into a logical expression.

18 Medium

18 of 58

Analyzing +4 | -1

The statement

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

is equivalent to

View answer and explanation

Correct option: B

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

Option B,

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

, is correct because the given statement simplifies to this expression using logical equivalences.

19 Medium

19 of 58

Understanding +4 | -1

The negation of the following statement :

P

: Neha lives in Ludhiana or she lives in Gurudaspur.

View answer and explanation

Correct option: B

Neha does not live in Ludhiana and she does not live in Gurudaspur

Option B (Neha does not live in Ludhiana and she does not live in Gurudaspur) is correct because it represents the negation of the disjunction (OR) statement.

20 Medium

20 of 58

Analyzing +4 | -1

The logical statement

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

is a:

View answer and explanation

Correct option: A

Tautology

Option A (Tautology) is correct because the given logical statement simplifies to a statement that is always true, regardless of the truth values of p and q.

21 Medium

21 of 58

Analyzing +4 | -1

The conditional statement

\left( {\left( {p \wedge q} \right) \to \left( {\left( { \sim p} \right) \vee r} \right)} \right) \vee \left( {\left( {\left( { \sim p} \right) \vee r} \right) \to \left( {p \wedge q} \right)} \right)

is:

View answer and explanation

Correct option: A

A tautology

Option A (A tautology) is correct because the given statement is of the form (A -> B) v (B -> A), which is always true regardless of the truth values of A and B.

22 Medium

22 of 58

Understanding +4 | -1

Negation of the statement

(p \wedge r) \longrightarrow(r \vee q)

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because the negation of

p \rightarrow q

p \wedge \sim q

23 Medium

23 of 58

Analyzing +4 | -1

The following statement

(p \rightarrow q) \rightarrow[(\sim p \rightarrow q) \rightarrow q]

View answer and explanation

Correct option: D

A tautology

Option D (A tautology) is correct because the given statement simplifies to

(p \rightarrow q) \rightarrow (p \rightarrow q)

, which is always true regardless of the truth values of p and q.

24 Medium

24 of 58

Understanding +4 | -1

If a positive integer

\mathrm{n}

is divisible by 9 , then the sum of the digits of

n

is divisible by 9 . So which statement is it contrapositive.

View answer and explanation

Correct option: B

(sum of digits of

n

is not divisible by 9 )

\Rightarrow

(

\mathrm{n}

is not divisible by 9 )

The correct answer is (sum of digits of n is not divisible by 9) => (n is not divisible by 9) because it is the contrapositive of the given statement.

25 Medium

25 of 58

Understanding +4 | -1

Consider the statements
(i) Two plus three is five.
(ii) Every square is a rectangle.
(iii) Sun rises in the East.
(iv) The Earth is a star.
Which of the above statements having truth value

(\mathrm{T})

View answer and explanation

Correct option: D

All of these

The correct answer is 'All of these' because all the statements are true.

26 Medium

26 of 58

Understanding +4 | -1

The contrapositive of the statement

{ }^\alpha

2^2=5

, then I get first class" is

View answer and explanation

Correct option: B

If I do not get first class, then

2^2 \neq 5

Option B (If I do not get first class, then

2^2 \neq 5

) is correct because it is the contrapositive of the given statement, negating both the hypothesis and the conclusion and reversing the implication.

27 Medium

27 of 58

Analyzing +4 | -1

The number of values of

r \in\{p, q, \sim p, \sim q\}

for which

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

is a tautology, is

View answer and explanation

Correct option: C

The correct answer is 2 because only two values of r (

\sim p

and q) make the given expression a tautology.

28 Medium

28 of 58

Understanding +4 | -1

p

and

q

are two statements such that

p

: The question paper is easy.

q

: We shall pass.
Then, the symbolic statement

\sim p \rightarrow \sim q

means

View answer and explanation

Correct option: B

if the question paper is not easy, then we shall not pass

Option B (if the question paper is not easy, then we shall not pass) is the correct interpretation of the symbolic statement

\sim p \rightarrow \sim q

29 Medium

29 of 58

Analyzing +4 | -1

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

is false and

q

and

r

are both false, then

p

View answer and explanation

Correct option: A

TRUE

Option A (TRUE) is correct because if

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

is false, then

(p \wedge \sim r)

must be true and

(q \vee r)

must be false. Since

r

is false,

\sim r

is true, and for

(p \wedge \sim r)

to be true,

p

must be true.

30 Medium

30 of 58

Understanding +4 | -1

The contrapositive of

(p \vee q) \Rightarrow r

View answer and explanation

Correct option: C

\sim {r} \Rightarrow \sim {p} \wedge \sim {q}

Option C (

\sim {r} \Rightarrow \sim {p} \wedge \sim {q}

) is correct because it represents the contrapositive of the given implication.

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 58 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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