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# Mathematical Reasoning · Mathematics · JEE Main

Start Practice## MCQ (Single Correct Answer)

JEE Main 2023 (Online) 15th April Morning Shift

Negation of $p \wedge(q \wedge \sim(p \wedge q))$ is :

JEE Main 2023 (Online) 13th April Evening Shift

The statement $$(p \wedge(\sim q)) \vee((\sim p) \wedge q) \vee((\sim p) \wedge(\sim q))$$ is equivalent to _________.

JEE Main 2023 (Online) 13th April Morning Shift

The negation of the statement $$((A \wedge(B \vee C)) \Rightarrow(A \vee B)) \Rightarrow A$$ is

JEE Main 2023 (Online) 12th April Morning Shift

Among the two statements
$$(\mathrm{S} 1):(p \Rightarrow q) \wedge(p \wedge(\sim q))$$ is a contradiction and
$$(\mathrm{S} 2):(p \wedge q) \vee((\sim...

JEE Main 2023 (Online) 11th April Evening Shift

The converse of $$((\sim p) \wedge q) \Rightarrow r$$ is

JEE Main 2023 (Online) 10th April Evening Shift

The statement $$\sim[p \vee(\sim(p \wedge q))]$$ is equivalent to :

JEE Main 2023 (Online) 10th April Morning Shift

The negation of the statement $$(p \vee q) \wedge (q \vee ( \sim r))$$ is :

JEE Main 2023 (Online) 8th April Evening Shift

The negation of $$(p \wedge(\sim q)) \vee(\sim p)$$ is equivalent to :

JEE Main 2023 (Online) 8th April Morning Shift

Negation of $$(p \Rightarrow q) \Rightarrow(q \Rightarrow p)$$ is :

JEE Main 2023 (Online) 6th April Evening Shift

Among the statements
(S1) : $$(p \Rightarrow q) \vee((\sim p) \wedge q)$$ is a tautology
(S2) : $$(q \Rightarrow p) \Rightarrow((\sim p) \wedge q)$$ i...

JEE Main 2023 (Online) 6th April Morning Shift

Statement $$\mathrm{(P \Rightarrow Q) \wedge(R \Rightarrow Q)}$$ is logically equivalent to :

JEE Main 2023 (Online) 1st February Evening Shift

Which of the following statements is a tautology?

JEE Main 2023 (Online) 1st February Morning Shift

The negation of the expression $$q \vee \left( {( \sim \,q) \wedge p} \right)$$ is equivalent to

JEE Main 2023 (Online) 31st January Evening Shift

The number of values of $\mathrm{r} \in\{\mathrm{p}, \mathrm{q}, \sim \mathrm{p}, \sim \mathrm{q}\}$ for which $((\mathrm{p} \wedge \mathrm{q}) \Right...

JEE Main 2023 (Online) 31st January Morning Shift

$$(\mathrm{S} 1)~(p \Rightarrow q) \vee(p \wedge(\sim q))$$ is a tautology
$$(\mathrm{S} 2)~((\sim p) \Rightarrow(\sim q)) \wedge((\sim p) \vee q)$$ i...

JEE Main 2023 (Online) 30th January Evening Shift

Consider the following statements:
P : I have fever
Q: I will not take medicine
$\mathrm{R}$ : I will take rest.
The statement "If I have fever, then ...

JEE Main 2023 (Online) 30th January Morning Shift

Among the statements :
$$(\mathrm{S} 1)~((\mathrm{p} \vee \mathrm{q}) \Rightarrow \mathrm{r}) \Leftrightarrow(\mathrm{p} \Rightarrow \mathrm{r})$$
$$(...

JEE Main 2023 (Online) 29th January Morning Shift

If $$p,q$$ and $$r$$ are three propositions, then which of the following combination of truth values of $$p,q$$ and $$r$$ makes the logical expression...

JEE Main 2023 (Online) 25th January Evening Shift

Let $$\Delta ,\nabla \in \{ \wedge , \vee \} $$ be such that $$\mathrm{(p \to q)\Delta (p\nabla q)}$$ is a tautology. Then

JEE Main 2023 (Online) 25th January Morning Shift

The statement $$\left( {p \wedge \left( { \sim q} \right)} \right) \Rightarrow \left( {p \Rightarrow \left( { \sim q} \right)} \right)$$ is

JEE Main 2023 (Online) 24th January Evening Shift

Let p and q be two statements. Then $$ \sim \left( {p \wedge (p \Rightarrow \, \sim q)} \right)$$ is equivalent to

JEE Main 2023 (Online) 24th January Morning Shift

The compound statement $$\left( { \sim (P \wedge Q)} \right) \vee \left( {( \sim P) \wedge Q} \right) \Rightarrow \left( {( \sim P) \wedge ( \sim Q)} ...

JEE Main 2022 (Online) 29th July Evening Shift

The statement $$(p \Rightarrow q) \vee(p \Rightarrow r)$$ is NOT equivalent to

JEE Main 2022 (Online) 29th July Morning Shift

The statement $$(p \wedge q) \Rightarrow(p \wedge r)$$ is equivalent to :

JEE Main 2022 (Online) 28th July Evening Shift

Let
$$\mathrm{p}$$ : Ramesh listens to music.
$$\mathrm{q}$$ : Ramesh is out of his village.
$$\mathrm{r}$$ : It is Sunday.
$$\mathrm{s}$$ : It is Sat...

JEE Main 2022 (Online) 28th July Morning Shift

Let the operations $$*, \odot \in\{\wedge, \vee\}$$. If $$(\mathrm{p} * \mathrm{q}) \odot(\mathrm{p}\, \odot \sim \mathrm{q})$$ is a tautology, then t...

JEE Main 2022 (Online) 27th July Evening Shift

If the truth value of the statement $$(P \wedge(\sim R)) \rightarrow((\sim R) \wedge Q)$$ is F, then the truth value of which of the following is $$\m...

JEE Main 2022 (Online) 27th July Morning Shift

$$(p \wedge r) \Leftrightarrow(p \wedge(\sim q))$$ is equivalent to $$(\sim p)$$ when $$r$$ is

JEE Main 2022 (Online) 26th July Evening Shift

Negation of the Boolean expression $$p \Leftrightarrow(q \Rightarrow p)$$ is

JEE Main 2022 (Online) 26th July Morning Shift

The statement $$(\sim(\mathrm{p} \Leftrightarrow \,\sim \mathrm{q})) \wedge \mathrm{q}$$ is :

JEE Main 2022 (Online) 25th July Evening Shift

Consider the following statements:
P : Ramu is intelligent.
Q : Ramu is rich.
R : Ramu is not honest.
The negation of the statement "Ramu is intellige...

JEE Main 2022 (Online) 25th July Morning Shift

Which of the following statements is a tautology ?

JEE Main 2022 (Online) 30th June Morning Shift

The conditional statement
$$((p \wedge q) \to (( \sim p) \vee r)) \vee ((( \sim p) \vee r) \to (p \wedge q))$$ is :

JEE Main 2022 (Online) 29th June Evening Shift

Negation of the Boolean statement (p $$\vee$$ q) $$\Rightarrow$$ (($$\sim$$ r) $$\vee$$ p) is equivalent to :

JEE Main 2022 (Online) 29th June Morning Shift

Let $$\Delta$$ $$\in$$ {$$\wedge$$, $$\vee$$, $$\Rightarrow$$, $$\Leftrightarrow$$} be such that (p $$\wedge$$ q) $$\Delta$$ ((p $$\vee$$ q) $$\Righta...

JEE Main 2022 (Online) 28th June Morning Shift

Let p, q, r be three logical statements. Consider the compound statements
$${S_1}:(( \sim p) \vee q) \vee (( \sim p) \vee r)$$ and
$${S_2}:p \to (q \v...

JEE Main 2022 (Online) 27th June Evening Shift

Which of the following statement is a tautology?

JEE Main 2022 (Online) 27th June Morning Shift

The boolean expression $$( \sim (p \wedge q)) \vee q$$ is equivalent to :

JEE Main 2022 (Online) 26th June Evening Shift

Let r $$\in$$ {p, q, $$\sim$$p, $$\sim$$q} be such that the logical statement
r $$\vee$$ ($$\sim$$p) $$\Rightarrow$$ (p $$\wedge$$ q) $$\vee$$ r
is a ...

JEE Main 2022 (Online) 26th June Morning Shift

Let $$\Delta$$, $$\nabla $$ $$\in$$ {$$\wedge$$, $$\vee$$} be such that p $$\nabla$$ q $$\Rightarrow$$ ((p $$\Delta$$ q) $$\nabla$$ r) is a tautology....

JEE Main 2022 (Online) 25th June Evening Shift

The negation of the Boolean expression (($$\sim$$ q) $$\wedge$$ p) $$\Rightarrow$$ (($$\sim$$ p) $$\vee$$ q) is logically equivalent to :

JEE Main 2022 (Online) 25th June Morning Shift

Consider the following two propositions:
$$P1: \sim (p \to \sim q)$$
$$P2:(p \wedge \sim q) \wedge (( \sim p) \vee q)$$
If the proposition $$p \to (...

JEE Main 2022 (Online) 24th June Evening Shift

Consider the following statements:
A : Rishi is a judge.
B : Rishi is honest.
C : Rishi is not arrogant.
The negation of the statement "if Rishi is a ...

JEE Main 2022 (Online) 24th June Morning Shift

The number of choices for $$\Delta \in \{ \wedge , \vee , \Rightarrow , \Leftrightarrow \} $$, such that $$(p\Delta q) \Rightarrow ((p\Delta \sim q...

JEE Main 2021 (Online) 1st September Evening Shift

Which of the following is equivalent to the Boolean expression p $$\wedge$$ $$\sim$$ q ?

JEE Main 2021 (Online) 31st August Evening Shift

Negation of the statement (p $$\vee$$ r) $$\Rightarrow$$ (q $$\vee$$ r) is :

JEE Main 2021 (Online) 31st August Morning Shift

Let *, ▢ $$\in$${$$\wedge$$, $$\vee$$} be such that the Boolean expression (p * $$\sim$$ q) $$\Rightarrow$$ (p ▢ q) is a tautology. Then :

JEE Main 2021 (Online) 27th August Evening Shift

The Boolean expression (p $$\wedge$$ q) $$\Rightarrow$$ ((r $$\wedge$$ q) $$\wedge$$ p) is equivalent to :

JEE Main 2021 (Online) 27th August Morning Shift

The statement (p $$ \wedge $$ (p $$\to$$ q) $$\wedge$$ (q $$\to$$ r)) $$\to$$ r is :

JEE Main 2021 (Online) 26th August Evening Shift

Consider the two statements :(S1) : (p $$\to$$ q) $$ \vee $$ ($$ \sim $$ q $$\to$$ p) is a tautology .(S2) : (p $$ \wedge $$ $$ \sim $$ q) $$ \wedge $...

JEE Main 2021 (Online) 26th August Morning Shift

If the truth value of the Boolean expression $$\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \rig...

JEE Main 2021 (Online) 27th July Evening Shift

Which of the following is the negation of the statement "for all M > 0, there exists x$$\in$$S such that x $$\ge$$ M" ?

JEE Main 2021 (Online) 27th July Morning Shift

The compound statement $$(P \vee Q) \wedge ( \sim P) \Rightarrow Q$$ is equivalent to :

JEE Main 2021 (Online) 25th July Evening Shift

Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following :

JEE Main 2021 (Online) 25th July Morning Shift

The Boolean expression $$(p \Rightarrow q) \wedge (q \Rightarrow \sim p)$$ is equivalent to :

JEE Main 2021 (Online) 22th July Evening Shift

Which of the following Boolean expressions is not a tautology?

JEE Main 2021 (Online) 20th July Evening Shift

Consider the following three statements :(A) If 3 + 3 = 7 then 4 + 3 = 8(B) If 5 + 3 = 8 then earth is flat.(C) If both (A) and (B) are true then 5 + ...

JEE Main 2021 (Online) 20th July Morning Shift

The Boolean expression $$(p \wedge \sim q) \Rightarrow (q \vee \sim p)$$ is equivalent to :

JEE Main 2021 (Online) 18th March Evening Shift

If P and Q are two statements, then which of the following compound statement is a tautology?

JEE Main 2021 (Online) 17th March Evening Shift

If the Boolean expression $$(p \wedge q) \odot (p \otimes q)$$ is a tautology, then $$ \odot $$ and $$ \otimes $$ are respectively given by :

JEE Main 2021 (Online) 17th March Morning Shift

If the Boolean expression (p $$ \Rightarrow $$ q) $$ \Leftrightarrow $$ (q * ($$ \sim $$p) is a tautology, then the boolean expression (p * ($$ \sim $...

JEE Main 2021 (Online) 16th March Morning Shift

Which of the following Boolean expression is a tautology?

JEE Main 2021 (Online) 26th February Evening Shift

Let F1(A, B, C) = (A $$ \wedge $$ $$ \sim $$ B) $$ \vee $$ [$$\sim$$C $$\wedge$$ (A $$\vee$$ B)] $$\vee$$ $$\sim$$ A and F2(A, B) = (A $$\vee$$ B) $$\...

JEE Main 2021 (Online) 25th February Evening Shift

The contrapositive of the statement "If you will work, you will earn money" is :

JEE Main 2021 (Online) 25th February Morning Shift

The statement A $$ \to $$ (B $$ \to $$ A) is equivalent to :

JEE Main 2021 (Online) 24th February Evening Shift

The negation of the statement $$ \sim p \wedge (p \vee q)$$ is :

JEE Main 2021 (Online) 24th February Evening Shift

For the statements p and q, consider the following compound statements :(a) $$( \sim q \wedge (p \to q)) \to \sim p$$(b) $$((p \vee q) \wedge \sim p...

JEE Main 2021 (Online) 24th February Morning Shift

The statement among the following that is a tautology is :

JEE Main 2020 (Online) 6th September Evening Slot

Consider the statement : ‘‘For an integer n, if n3 – 1 is even, then n is odd.’’ The contrapositive statement of this statement is :
...

JEE Main 2020 (Online) 6th September Morning Slot

The negation of the Boolean expression p $$ \vee $$ (~p $$ \wedge $$ q) is equivalent to :

JEE Main 2020 (Online) 5th September Evening Slot

The statement
$$\left( {p \to \left( {q \to p} \right)} \right) \to \left( {p \to \left( {p \vee q} \right)} \right)$$ is :

JEE Main 2020 (Online) 5th September Morning Slot

The negation of the Boolean expression x $$ \leftrightarrow $$ ~ y is equivalent to :

JEE Main 2020 (Online) 4th September Evening Slot

Contrapositive of the statement :
‘If a function f is differentiable at a, then it is also continuous at a’, is:

JEE Main 2020 (Online) 4th September Morning Slot

Given the following two statements:
$$\left( {{S_1}} \right):\left( {q \vee p} \right) \to \left( {p \leftrightarrow \sim q} \right)$$ is a tautology...

JEE Main 2020 (Online) 3rd September Evening Slot

Let p, q, r be three statements such that the
truth value of (p $$ \wedge $$ q) $$ \to $$ ($$ \sim $$q $$ \vee $$ r) is F. Then the
truth values of p,...

JEE Main 2020 (Online) 3rd September Morning Slot

The proposition p
$$ \to $$ ~ (p
$$ \wedge $$ ~q) is equivalent
to :

JEE Main 2020 (Online) 2nd September Evening Slot

Which of the following is a tautology ?

JEE Main 2020 (Online) 2nd September Morning Slot

The contrapositive of the statement "If I reach the
station in time, then I will catch the train" is :

JEE Main 2020 (Online) 9th January Evening Slot

If p $$ \to $$ (p $$ \wedge $$ ~q) is false, then the truth values
of p and q are respectively :

JEE Main 2020 (Online) 9th January Morning Slot

Negation of the statement :
$$\sqrt 5 $$ is an integer or 5 is an irrational is :

JEE Main 2020 (Online) 8th January Evening Slot

Which of the following statements is a tautology?

JEE Main 2020 (Online) 8th January Morning Slot

Which one of the following is a tautology?

JEE Main 2020 (Online) 7th January Evening Slot

Let A, B, C and D be four non-empty sets. The contrapositive statement of "If A $$ \subseteq $$ B and B $$ \subseteq $$ D,
then A $$ \subseteq $$ C" i...

JEE Main 2020 (Online) 7th January Morning Slot

The logical statement (p $$ \Rightarrow $$ q) $$\Lambda $$ ( q $$ \Rightarrow $$ ~p) is equivalent to :

JEE Main 2019 (Online) 12th April Evening Slot

The Boolean expression ~(p $$ \Rightarrow $$ (~q)) is equivalent to :

JEE Main 2019 (Online) 12th April Morning Slot

If the truth value of the statement p $$ \to $$ (~q $$ \vee $$ r) is false (F), then the truth values of the statements p, q, r are
respectively :

JEE Main 2019 (Online) 10th April Evening Slot

The negation of the Boolean expression ~ s $$ \vee $$ (~r $$ \wedge $$ s) is equivalent to :

JEE Main 2019 (Online) 10th April Morning Slot

Which one of the following Boolean expressions is a tautology?

JEE Main 2019 (Online) 9th April Evening Slot

If p $$ \Rightarrow $$ (q $$ \vee $$ r) is false, then the truth values of p,
q, r are respectively :-

JEE Main 2019 (Online) 9th April Morning Slot

For any two statements p and q, the negation of
the expression
p $$ \vee $$ (~p $$ \wedge $$ q) is :

JEE Main 2019 (Online) 8th April Evening Slot

Which one of the following statements is not a
tautology?

JEE Main 2019 (Online) 8th April Morning Slot

The contrapositive of the statement "If you are
born in India, then you are a citizen of India", is :

JEE Main 2019 (Online) 12th January Evening Slot

The expression $$ \sim $$ ($$ \sim $$ p $$ \to $$ q) is logically equivalent to :

JEE Main 2019 (Online) 12th January Morning Slot

The Boolean expression ((p $$ \wedge $$ q) $$ \vee $$ (p $$ \vee $$ $$ \sim $$ q)) $$ \wedge $$ ($$ \sim $$ p $$ \wedge $$ $$ \sim $$ q) is equivalent...

JEE Main 2019 (Online) 11th January Evening Slot

Contrapositive of the statement " If two numbers are not equal, then their squares are not equal." is :

JEE Main 2019 (Online) 11th January Morning Slot

If q is false and p $$ \wedge $$ q $$ \leftrightarrow $$ r is true, then which one of the following statements is a tautology ?

JEE Main 2019 (Online) 10th January Evening Slot

Consider the following three statements :
P : 5 is a prime number
Q : 7 is a factor of 192
R : L.C.M. of 5 and 7 is 35
Then the truth value of which o...

JEE Main 2019 (Online) 10th January Morning Slot

Consider the statement : "P(n) : n2 – n + 41 is prime". Then which one of the following is true ?

JEE Main 2019 (Online) 9th January Evening Slot

The logical statement
[ $$ \sim $$ ( $$ \sim $$ p $$ \vee $$ q) $$ \vee $$ (p $$ \wedge $$ r)] $$ \wedge $$ ($$ \sim $$ q $$ \wedge $$ r) is equivalen...

JEE Main 2019 (Online) 9th January Morning Slot

If the Boolean expression
(p $$ \oplus $$ q) $$\wedge$$ (~ p $$ \odot $$ q) is equivalent
to p $$\wedge$$ q, where $$ \oplus , \odot \in \left\{ ...

JEE Main 2018 (Online) 16th April Morning Slot

If p $$ \to $$ ($$ \sim $$ p$$ \vee $$ $$ \sim $$ q) is false, then the truth values of p and q are respectively :

JEE Main 2018 (Offline)

The Boolean expression
$$ \sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$$ is equvalent to :

JEE Main 2018 (Online) 15th April Evening Slot

Consider the following two statements :
Statement p :
The value of sin 120o can be derived by taking $$\theta = {240^o}$$ in the equation
2sin$${\...

JEE Main 2018 (Online) 15th April Morning Slot

If (p $$ \wedge $$ $$ \sim $$ q) $$ \wedge $$ (p $$ \wedge $$ r) $$ \to $$ $$ \sim $$ p $$ \vee $$ q is false, then the truth values of $$p, q$$ and $...

JEE Main 2017 (Online) 9th April Morning Slot

Contrapositive of the statement
‘If two numbers are not equal, then their squares are not equal’, is :

JEE Main 2017 (Online) 8th April Morning Slot

The proposition $$\left( { \sim p} \right) \vee \left( {p \wedge \sim q} \right)$$ is equivalent to :

JEE Main 2017 (Offline)

The following statement
$$\left( {p \to q} \right) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$$ is :

JEE Main 2016 (Online) 10th April Morning Slot

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

JEE Main 2016 (Online) 9th April Morning Slot

Consider the following two statements :
P : If 7 is an odd number, then 7 is
divisible by 2.
Q : If 7 is a prime...

JEE Main 2016 (Offline)

The Boolean expression
$$\left( {p \wedge \sim q} \right) \vee q \vee \left( { \sim p \wedge q} \right)$$ is equivalent to :

JEE Main 2015 (Offline)

The negation of $$ \sim s \vee \left( { \sim r \wedge s} \right)$$ is equivalent to :

JEE Main 2014 (Offline)

The statement $$ \sim \left( {p \leftrightarrow \sim q} \right)$$ is :

JEE Main 2013 (Offline)

Consider :
Statement − I : $$\left( {p \wedge \sim q} \right) \wedge \left( { \sim p \wedge q} \right)$$ is a fallacy.
Statement − II :$$\left( {p \t...

AIEEE 2012

The negation of the statement “If I become a teacher, then I will open a school” is :

AIEEE 2011

Consider the following statements
P : Suman is brilliant
Q : Suman is rich
R : Suman is honest
The negation of the statement,
“Suman is brilliant ...

AIEEE 2010

Let S be a non-empty subset of R. Consider the following statement:
P : There is a rational number x ∈ S such that x > 0.
Which of the following st...

AIEEE 2009

Statement-1 : $$ \sim \left( {p \leftrightarrow \sim q} \right)$$ is equivalent to $${p \leftrightarrow q}$$.
Statement-2 : $$ \sim \left( {p \leftri...

AIEEE 2008

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”,
and r be the statement “x is a rational number ...

AIEEE 2008

The statement $$p \to \left( {q \to p} \right)$$ is equivalent to

## Numerical

JEE Main 2023 (Online) 11th April Morning Shift

The number of ordered triplets of the truth values of $$p, q$$ and $$r$$ such that the truth value of the statement $$(p \vee q) \wedge(p \vee r) \Rig...

JEE Main 2022 (Online) 28th June Evening Shift

The maximum number of compound propositions, out of p$$\vee$$r$$\vee$$s, p$$\vee$$r$$\vee$$$$\sim$$s, p$$\vee$$$$\sim$$q$$\vee$$s, $$\sim$$p$$\vee$$$$...

## MCQ (More than One Correct Answer)

JEE Main 2023 (Online) 29th January Evening Shift

The statement $$B \Rightarrow \left( {\left( { \sim A} \right) \vee B} \right)$$ is equivalent to :