2018
JEE Mains
MCQ
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 :
A.
F, F
B.
T, F
C.
F, T
D.
T, T
2018
JEE Mains
MCQ
JEE Main 2018 (Offline)
The Boolean expression
$ \sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$ is equvalent to :
$ \sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)$ is equvalent to :
A.
${ \sim q}$
B.
${ \sim p}$
C.
p
D.
q
2018
JEE Mains
MCQ
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${\theta \over 2} = \sqrt {1 + \sin \theta } - \sqrt {1 - \sin \theta } $
Statement q :
The angles A, B, C and D of any quadrilateral ABCD satisfy the equation
cos$\left( {{1 \over 2}\left( {A + C} \right)} \right) + \cos \left( {{1 \over 2}\left( {B + D} \right)} \right) = 0$
Then the truth values of p and q are respectively :
Statement p :
The value of sin 120o can be derived by taking $\theta = {240^o}$ in the equation
2sin${\theta \over 2} = \sqrt {1 + \sin \theta } - \sqrt {1 - \sin \theta } $
Statement q :
The angles A, B, C and D of any quadrilateral ABCD satisfy the equation
cos$\left( {{1 \over 2}\left( {A + C} \right)} \right) + \cos \left( {{1 \over 2}\left( {B + D} \right)} \right) = 0$
Then the truth values of p and q are respectively :
A.
F, T
B.
T, F
C.
T, T
D.
F, F
2018
JEE Mains
MCQ
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 $r$ are, respectively :
A.
F, T, F
B.
T, F, T
C.
T, T, T
D.
F, F, F
2017
JEE Mains
MCQ
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 :
‘If two numbers are not equal, then their squares are not equal’, is :
A.
If the squares of two numbers are equal, then the numbers are equal.
B.
If the squares of two numbers are equal, then the numbers are not equal.
C.
If the squares of two numbers are not equal, then the numbers are not equal.
D.
If the squares of two numbers are not equal, then the numbers are equal.
2017
JEE Mains
MCQ
JEE Main 2017 (Online) 8th April Morning Slot
The proposition $\left( { \sim p} \right) \vee \left( {p \wedge \sim q} \right)$ is equivalent to :
A.
p $ \vee $ ~ q
B.
p $ \to $ ~ q
C.
p $ \wedge $ ~ q
D.
q $ \to $ p
2017
JEE Mains
MCQ
JEE Main 2017 (Offline)
The following statement
$\left( {p \to q} \right) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$ is :
$\left( {p \to q} \right) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$ is :
A.
equivalent to ${ \sim p \to q}$
B.
equivalent to ${p \to \sim q}$
C.
a fallacy
D.
a tautology
2016
JEE Mains
MCQ
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 :
“If the side of a square doubles, then its area increases four times”, is :
A.
If the side of a square is not doubled, then its area does not increase four times.
B.
If the area of a square increases four times, then its side is doubled.
C.
If the area of a square increases four times, then its side is not doubled.
D.
If the area of a square does not increase four times, then its side is not doubled.
2016
JEE Mains
MCQ
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 number, then 7 is an odd number
If V1 is the truth value of the contrapositive of P and V2 is the truth value of contrapositive of Q, then the ordered pair (V1 , V2) equals :
P : If 7 is an odd number, then 7 is divisible by 2.
Q : If 7 is a prime number, then 7 is an odd number
If V1 is the truth value of the contrapositive of P and V2 is the truth value of contrapositive of Q, then the ordered pair (V1 , V2) equals :
A.
(T, T)
B.
(T, F)
C.
(F, T)
D.
(F, F)
2016
JEE Mains
MCQ
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 :
$\left( {p \wedge \sim q} \right) \vee q \vee \left( { \sim p \wedge q} \right)$ is equivalent to :
A.
${ \sim p \wedge q}$
B.
${p \wedge q}$
C.
$p \vee q$
D.
$p \vee \sim q$
2015
JEE Mains
MCQ
JEE Main 2015 (Offline)
The negation of $ \sim s \vee \left( { \sim r \wedge s} \right)$ is equivalent to :
A.
$s \vee \left( {r \vee \sim s} \right)$
B.
$s \wedge r$
C.
$s \wedge \sim r$
D.
$s \wedge \left( {r \wedge \sim s} \right)$
2014
JEE Mains
MCQ
JEE Main 2014 (Offline)
The statement $ \sim \left( {p \leftrightarrow \sim q} \right)$ is :
A.
equivalent to ${ \sim p \leftrightarrow q}$
B.
a tautology
C.
a fallacy
D.
equivalent to ${p \leftrightarrow q}$
2013
JEE Mains
MCQ
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 :$\left( {p \to q} \right) \leftrightarrow \left( { \sim q \to \sim p} \right)$ is a tautology.
Statement − I : $\left( {p \wedge \sim q} \right) \wedge \left( { \sim p \wedge q} \right)$ is a fallacy.
Statement − II :$\left( {p \to q} \right) \leftrightarrow \left( { \sim q \to \sim p} \right)$ is a tautology.
A.
Statement - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I
B.
Statement -I is True; Statement -II is False.
C.
Statement -I is False; Statement -II is True
D.
Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I
2012
JEE Mains
MCQ
AIEEE 2012
The negation of the statement “If I become a teacher, then I will open a school” is :
A.
I will become a teacher and I will not open a school
B.
Either I will not become a teacher or I will not open a school
C.
Neither I will become a teacher nor I will open a school
D.
I will not become a teacher or I will open a school
2011
JEE Mains
MCQ
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 and dishonest if and only if Suman is rich” can be expressed as :
P : Suman is brilliant
Q : Suman is rich
R : Suman is honest
The negation of the statement,
“Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as :
A.
$ \sim \left[ {Q \leftrightarrow \left( {P \wedge \sim R} \right)} \right]$
B.
$ \sim Q \leftrightarrow P \wedge R$
C.
$ \sim \left( {P \wedge \sim R} \right) \leftrightarrow Q$
D.
$ \sim P \wedge \left( {Q \leftrightarrow \sim R} \right)$
2010
JEE Mains
MCQ
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 statements is the negation of the statement P?
P : There is a rational number x ∈ S such that x > 0.
Which of the following statements is the negation of the statement P?
A.
There is no rational number x ∈ S such that x ≤ 0
B.
Every rational number x ∈ S satisfies x ≤ 0
C.
x ∈ S and x ≤ 0 $ \Rightarrow $ x is not rational
D.
There is a rational number x ∈ S such that x ≤ 0
2009
JEE Mains
MCQ
AIEEE 2009
Statement-1 : $ \sim \left( {p \leftrightarrow \sim q} \right)$ is equivalent to ${p \leftrightarrow q}$.
Statement-2 : $ \sim \left( {p \leftrightarrow \sim q} \right)$ is a tautology.
Statement-2 : $ \sim \left( {p \leftrightarrow \sim q} \right)$ is a tautology.
A.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
B.
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1
C.
Statement-1 is true, Statement-2 is false
D.
Statement-1 is false, Statement-2 is true
2008
JEE Mains
MCQ
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 iff y is a transcendental number”.
Statement –1: r is equivalent to either q or p.
Statement –2: r is equivalent to $ \sim \left( {p \leftrightarrow \sim q} \right)$
Statement –1: r is equivalent to either q or p.
Statement –2: r is equivalent to $ \sim \left( {p \leftrightarrow \sim q} \right)$
A.
Statement − 1 is false, Statement − 2 is false
B.
Statement −1 is false, Statement −2 is true
C.
Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1
D.
Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for
Statement −1
2008
JEE Mains
MCQ
AIEEE 2008
The statement $p \to \left( {q \to p} \right)$ is equivalent to
A.
$p \to \left( {p \leftrightarrow q} \right)$
B.
$p \to \left( {p \to q} \right)$
C.
$p \to \left( {p \vee q} \right)$
D.
$p \to \left( {p \wedge q} \right)$
2023
JEE Mains
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 $(p \vee q) \wedge(p \vee r) \Rightarrow(q \vee r)$ is True, is equal to ___________.
Correct Answer: 7
Explanation:
$
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline \boldsymbol{p} & \boldsymbol{q} & \boldsymbol{r} & \boldsymbol{p} \vee \boldsymbol{q} & \boldsymbol{p} \vee \boldsymbol{r} & \begin{array}{c}
(\boldsymbol{p} \vee \boldsymbol{q}) \wedge \\
(\boldsymbol{p} \vee \boldsymbol{r})
\end{array} & \boldsymbol{q} \vee \boldsymbol{r} & \begin{array}{c}
(\boldsymbol{p} \vee \boldsymbol{q}) \wedge(\boldsymbol{p} \vee \boldsymbol{r}) \\
\Rightarrow(\boldsymbol{q} \vee \boldsymbol{r})
\end{array} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\
\hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\
\hline
\end{array}
$
Hence, the total number of ordered triplets are 7.
Hence, the total number of ordered triplets are 7.
2022
JEE Mains
Numerical
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$$\sim$r$\vee$s, $\sim$p$\vee$$\sim$r$\vee$$\sim$s, $\sim$p$\vee$q$\vee$$\sim$s, q$\vee$r$\vee$$\sim$s, q$\vee$$\sim$r$\vee$$\sim$s, $\sim$p$\vee$$\sim$q$\vee$$\sim$s that can be made simultaneously true by an assignment of the truth values to p, q, r and s, is equal to __________.
Correct Answer: 9
Explanation:
There are total 9 compound propositions, out of which 6 contain $\sim s$. So if we assign $s$ as false, these 6 propositions will be true.
In remaining 3 compound propositions, two contain $p$ and the third contains $\sim r$. So if we assign $p$ and $r$ as true and false respectively, these 3 propositions will also be true.
Hence maximum number of propositions that can be true are 9.
In remaining 3 compound propositions, two contain $p$ and the third contains $\sim r$. So if we assign $p$ and $r$ as true and false respectively, these 3 propositions will also be true.
Hence maximum number of propositions that can be true are 9.
2023
JEE Mains
MSQ
JEE Main 2023 (Online) 29th January Evening Shift
The statement $B \Rightarrow \left( {\left( { \sim A} \right) \vee B} \right)$ is equivalent to :
A.
$B \Rightarrow \left( {\left( { \sim A} \right) \Rightarrow B} \right)$
B.
$A \Rightarrow \left( {A \Leftrightarrow B} \right)$
C.
$A \Rightarrow \left( {\left( { \sim A} \right) \Rightarrow B} \right)$
D.
$B \Rightarrow \left( {A \Rightarrow B} \right)$
