1

### 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 ?
A
P $\wedge$ r
B
(p $\vee$ r) $\to$ (p $\wedge$ r)
C
p $\vee$ r
D
(p $\wedge$ r) $\to$ (p $\vee$ r)

## Explanation

Given q is F and (p $\wedge$ q) $\leftrightarrow$ r is T

$\Rightarrow$  p $\wedge$ q is F which implies that r is F

$\Rightarrow$  q is F and r is F

$\Rightarrow$  (p $\wedge$ r) is always F

$\Rightarrow$  (p $\wedge$ r) $\to$ (p $\vee$ r) is tautology.
2

### 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 :
A
If the squares of two numbers are equal, then the numbers are not equal
B
If the squares of two numbers are equal, then the numbers are equal
C
If the squares of two numbers are not equal, then the numbers are equal
D
If the squares of two numbers are not equal, then the numbers are not equal

## Explanation

Let,

p : two numbers are not equal

q : squares of two numbers are not equal

Contrapositive of p $\to$ q is $\sim$q $\to$ $\sim$p.

$\therefore$ $\sim$q $\to$ $\sim$p means "If the squares of two numbers are equal, then the numbers are equal".
3

### 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 to :
A
p $\wedge$ q
B
p $\wedge$ ($\sim$ q)
C
p $\vee$ ($\sim$ q)
D
($\sim$ p) $\wedge$ ($\sim$ q)

## Explanation

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

$\equiv$ $\left( {\left( {\left( {p \vee \left( {p \vee \sim q} \right)} \right)} \right) \wedge \left( {q \vee \left( {p \vee \sim q} \right)} \right)} \right) \wedge \left( { \sim p \wedge \sim q} \right)$

$\equiv$ $\left( {\left( {p \vee \sim q} \right) \wedge \left( {q \vee \sim q \vee p} \right)} \right) \wedge \left( { \sim p \wedge \sim q} \right)$

$\equiv$ $\left( {\left( {p \vee \sim q} \right) \wedge \left( {t \vee p} \right)} \right) \wedge \left( { \sim p \wedge \sim q} \right)$

$\equiv$ $\left( {\left( {p \vee \sim q} \right) \wedge t} \right) \wedge \left( { \sim p \wedge \sim q} \right)$

$\equiv$ $\left( {p \vee \sim q} \right) \wedge \left( { \sim p \wedge \sim q} \right)$

$\equiv$ $\left( {p \wedge \sim p \wedge \sim q} \right) \vee \left( { \sim q \wedge \sim p \wedge \sim q} \right)$

$\equiv$ $\left( {f \wedge \sim q} \right) \vee \left( { \sim q \wedge \sim p} \right)$

$\equiv$ $f' \vee \left( { \sim q \wedge \sim p} \right)$

$\equiv$ $\left( { \sim q \wedge \sim p} \right)$
4

### JEE Main 2019 (Online) 12th January Evening Slot

The expression $\sim$ ($\sim$ p $\to$ q) is logically equivalent to :
A
p $\wedge$ q
B
$\sim$ p $\wedge$ $\sim$ q
C
p $\wedge$ $\sim$ q
D
$\sim$ p $\wedge$ q

## Explanation 