JEE Main > Mathematical Reasoning

Explore popular questions from Mathematical Reasoning for JEE Main. This collection covers Mathematical Reasoning previous year JEE Main questions hand picked by popular teachers.


Q 1.    

Correct4

Incorrect-1

The statement {tex} \mathrm { p } \rightarrow ( \mathrm { q } \rightarrow \mathrm { p } ) {/tex} is equivalent

A

{tex} p \rightarrow ( p \rightarrow q ) {/tex}

{tex} p \rightarrow ( p \vee q ) {/tex}

C

{tex} p \rightarrow ( p \wedge q ) {/tex}

D

{tex} p \rightarrow ( p \leftrightarrow q ) {/tex}

Explanation

Q 2.    

Correct4

Incorrect-1

Let p be the statement "{tex} x {/tex} is an irrational number", {tex} q {/tex} be the statement "{tex} y {/tex} is a trascendental number", and r be the statement "{tex} x {/tex} is a rational number if {tex} y {/tex} is a transcendental number"
{tex}\mathrm {Statement- 1 } {/tex}: {tex}\mathrm { r } {/tex} is equivalent to either {tex} \mathrm { q } {/tex} or {tex} \mathrm { p } {/tex}.
{tex}\mathrm {Statement- 2 } {/tex}: {tex}\mathrm { r } {/tex} is equivalent to {tex} \left( \mathrm { p } \leftrightarrow ^ {\sim } \mathrm { q } \right) {/tex}

{tex}\mathrm {Statement- 1 } {/tex} is false, Statement {tex} - 2 {/tex} is true

B

{tex}\mathrm {Statement-1} {/tex} is true, {tex}\mathrm {Statement-2} {/tex} is false

C

{tex}\mathrm {Statement-1} {/tex} is true, {tex}\mathrm {Statement-2} {/tex} is true;
{tex}\mathrm {Statement-2} {/tex} is a correct explanation for
{tex}\mathrm {Statement-1} {/tex}

D

{tex}\mathrm {Statement-1} {/tex} is true, {tex}\mathrm {Statement-2} {/tex} is true;
{tex}\mathrm {Statement-2} {/tex} is not a correct explanation for
{tex}\mathrm {Statement-1} {/tex}

Explanation

Q 3.    

Correct4

Incorrect-1

{tex}\mathrm {Statement-1:} \quad {/tex} {tex}\sim (p{/tex} {tex} \leftrightarrow - \sim q){/tex} is equivalent to {tex} \mathrm { p } \leftrightarrow \mathrm { q } {/tex}.
{tex}\mathrm {Statement- 2 } {/tex} : {tex}\sim ( p{/tex} {tex} \leftrightarrow - {/tex} {tex}\sim q{/tex}) is a tautology.

{tex}\mathrm {Statement-1} {/tex} is true, {tex}\mathrm {Statement-2} {/tex} is false.

B

{tex}\mathrm {Statement-1} {/tex} is false, {tex}\mathrm {Statement-2} {/tex} is true.

C

{tex}\mathrm {Statement-1} {/tex} is true; {tex}\mathrm {Statement-2} {/tex} is true;
{tex}\mathrm {Statement-2} {/tex} is a correct explanation for
{tex}\mathrm {Statement-1.} {/tex}

D

{tex}\mathrm {Statement-1} {/tex} is true, {tex}\mathrm {Statement-2} {/tex} is true;
{tex}\mathrm {Statement-2} {/tex} is not a correct explanation for
{tex}\mathrm {statement-1} {/tex}

Explanation

Q 4.    

Correct4

Incorrect-1

Let {tex}\mathrm S {/tex} be a non-empty subset of {tex}\mathrm R. {/tex} Consider the following statement:
p: There is a rational number {tex} x \in S {/tex} such that {tex} x > 0 {/tex} which of the following statements is the negation of the statement p?

A

There is a rational number {tex} x \in S {/tex} such that {tex} x \leq 0 {/tex}

B

There is no rational number {tex} x \in S {/tex} such that {tex} x \leq 0 {/tex}

Every rational number {tex} x \in S {/tex} satisfies {tex} x \leq 0 {/tex}

D

{tex} \mathrm { x } \in \mathrm { S } {/tex} and {tex} \mathrm { x } \leq 0 \Rightarrow \mathrm { x } {/tex} is not rational

Explanation

Q 5.    

Correct4

Incorrect-1

Consider the following statements
{tex} \mathrm { p }: {/tex} Suman is brilliant
{tex} \mathrm { q }: {/tex} Suman is rich
{tex} \mathrm { r }: {/tex} 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

{tex} \sim q \leftrightarrow \sim p \wedge r {/tex}

B

{tex} \sim ( p \wedge \sim r ) \leftrightarrow q {/tex}

C

{tex} ( \mathrm { q } \leftrightarrow ( \mathrm { p } \wedge \sim \mathrm { r } ) ) {/tex}

Both 2 and 3

Explanation

Q 6.    

Correct4

Incorrect-1

The only statement among the followings that is
a tautology is :

A

{tex} \mathrm { q } \rightarrow [ \mathrm { p } \wedge ( \mathrm { p } \rightarrow \mathrm { q } ) ] {/tex}

B

{tex} p \wedge ( p \vee q ) {/tex}

C

{tex} \mathrm { p } \vee ( \mathrm { p } \wedge \mathrm { q } ) {/tex}

{tex} [ p \wedge ( p \rightarrow q ) ] \rightarrow q {/tex}

Explanation

Q 7.    

Correct4

Incorrect-1

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

A

I will not become a teacher or I will open a school.

I will become a teacher and I will not open a school.

C

Either I will not become a teacher or I will not open a school.

D

Neither I will become a teacher nor I will open a school.

Explanation

Q 8.    

Correct4

Incorrect-1

The inverse of the statement ( {tex}^\sim {\mathrm p} \left. \wedge ^\sim \mathrm { q } \right) \rightarrow \mathrm { r } {/tex} is-

A

{tex} ^\sim ( \mathrm {p \vee ^\sim q ) \rightarrow ^\sim r} {/tex}

B

{tex} \left( ^ { ^\sim } \mathrm { \mathrm {p } \wedge \mathrm { q} } \right) \rightarrow ^\sim \mathrm { \mathrm {r} } {/tex}

{tex} \left( ^ { ^\sim } \mathrm {p \vee q} \right) \rightarrow ^\sim { \mathrm {r} } {/tex}

D

None of these

Explanation

Q 9.    

Correct4

Incorrect-1

{tex} \left( ^ { ^\sim } \mathrm { p } \vee ^ { ^\sim } \mathrm { q } \right) {/tex} is logically equivalent to-

A

{tex} \mathrm { p } \wedge \mathrm { q } {/tex}

B

{tex} ^\sim \mathrm {p \rightarrow q} {/tex}

{tex} \mathrm {p \rightarrow ^\sim q} {/tex}

D

{tex} ^\sim \mathrm {p \rightarrow ^\sim q} {/tex}

Explanation

Q 10.    

Correct4

Incorrect-1

The equivalent statement of {tex} ( \mathrm {p \leftrightarrow q} ) {/tex} is-

A

{tex} ( \mathrm {p \wedge q} ) \vee ( \mathrm {p \vee q} ) {/tex}

B

{tex} ( \mathrm {p \rightarrow q} ) \vee ( \mathrm {q \rightarrow p} ) {/tex}

C

{tex} \left( ^\sim { \mathrm { p } } \vee \mathrm { q } \right) \vee \left( \mathrm { p } \vee ^\sim { \mathrm { q } } \right) {/tex}

{tex} \left( ^\sim { \mathrm { p } } \vee \mathrm { q } \right) \wedge \left( \mathrm { p} \vee ^\sim { \mathrm { q } } \right) {/tex}

Explanation

Q 11.    

Correct4

Incorrect-1

If the compound statement {tex}\mathrm {p} \rightarrow \left( ^\sim { \mathrm { p } } \vee \mathrm q \right) {/tex} is false then the truth value of {tex} \mathrm {p} {/tex} and {tex} \mathrm {q} {/tex} are respectively-

A

{tex} \mathrm { T } , \mathrm { T } {/tex}

{tex} \mathrm { T } , \mathrm { F } {/tex}

C

{tex} \mathrm { F } , \mathrm { T } {/tex}

D

{tex} \mathrm { F } , \mathrm { F } {/tex}

Explanation

Q 12.    

Correct4

Incorrect-1

The statement {tex} ( \mathrm {p \rightarrow ^\sim p} ) \wedge ( ^\sim \mathrm {p \rightarrow p} ) {/tex} is -

A

a tautology

a contradiction

C

neither a tautology nor a contradiction

D

None of these

Explanation

Q 13.    

Correct4

Incorrect-1

Negation of the statement (p {tex} \wedge \mathrm { r } ) \rightarrow ( \mathrm { r } \vee \mathrm { q } ) {/tex} is -

A

{tex} ^\sim ( \mathrm {p \wedge r} ) \rightarrow ^\sim ( \mathrm {r \vee q} ) {/tex}

B

{tex} ( ^\sim \mathrm p \vee ^\sim \mathrm r ) \vee ( \mathrm r \vee \mathrm q ) {/tex}

C

{tex} ( \mathrm {p \wedge r} ) \wedge ( \mathrm {r \wedge q} ) {/tex}

{tex} ( \mathrm {p \wedge r} ) \wedge \left( ^\sim { \mathrm {r \wedge } ^\sim \mathrm q} \right) {/tex}

Explanation

Q 14.    

Correct4

Incorrect-1

The dual of the statement {tex} ^\sim \mathrm { p } \wedge [ ^{\sim} q \wedge (p \vee q) \wedge ^\sim r] {/tex} is -

A

{tex} ^\sim \mathrm {p \vee [ ^\sim q \vee ( p \vee q ) \vee ^\sim r} ] {/tex}

B

{tex} \mathrm {p} \vee \left[ \mathrm {q} \vee \left( ^\sim { \mathrm {p} \wedge } ^\sim { \mathrm {q} } \right) \vee \mathrm {r} \right] {/tex}

{tex} ^\sim \mathrm {p} \vee [ ^ {\sim }\mathrm {q} \vee ( \mathrm {p \wedge q} ) \vee ^\sim { \mathrm {r} } {/tex}

D

{tex} ^\sim \mathrm {p} \vee \left[ ^ { ^\sim } \mathrm {q} \wedge ( \mathrm {p \wedge q} ) \wedge ^\sim \mathrm {r} \right] {/tex}

Explanation

Q 15.    

Correct4

Incorrect-1

Which of the following is correct-

A

{tex} \left( ^\sim _ { \mathrm { p } } \vee ^\sim _ { \mathrm { q } } \right) \equiv ( \mathrm { p } \wedge \mathrm { q } ) {/tex}

{tex} ( \mathrm {p \rightarrow q} ) \equiv ( ^\sim \mathrm {q \rightarrow ^\sim p} ) {/tex}

C

{tex} ^\sim ( \mathrm {p \rightarrow ^\sim q} ) \equiv ( \mathrm {p \wedge ^\sim q} ) {/tex}

D

{tex} ( \mathrm {p \leftrightarrow q ) \equiv ( p \rightarrow q ) \vee ( q \rightarrow p} ) {/tex}

Explanation

Q 16.    

Correct4

Incorrect-1

The contrapositive of {tex} \mathrm {p} \rightarrow \left( ^\sim { \mathrm {q } \rightarrow ^\sim { r} } \right) {/tex} is-

{tex} \left( ^ { ^\sim } \mathrm { q } \wedge \mathrm { r } \right) \rightarrow ^\sim { \mathrm { p } } {/tex}

B

{tex} ( \mathrm {q \rightarrow r} ) \rightarrow ^\sim { \mathrm {p} } {/tex}

C

{tex} (\mathrm q \vee ^\sim \mathrm r) \rightarrow ^\sim \mathrm p {/tex}

D

None of these

Explanation

Q 17.    

Correct4

Incorrect-1

The converse of {tex} \mathrm {p \rightarrow ( q \rightarrow r} ) {/tex} is -

{tex} ( \mathrm {q \wedge ^\sim r ) \vee p} {/tex}

B

{tex} \mathrm { \left( ^ { ^\sim } q \vee r \right) \vee p} {/tex}

C

{tex} ( \mathrm {q \wedge ^\sim r ) \wedge ^\sim p} {/tex}

D

{tex} ( \mathrm {q \wedge ^\sim r ) \wedge p} {/tex}

Explanation

Q 18.    

Correct4

Incorrect-1

If {tex} \mathrm { p } {/tex} and {tex} \mathrm { q } {/tex} are two statement then {tex} ( \mathrm { p } \leftrightarrow ^\sim \mathrm { q } ) {/tex} is true when-

A

{tex} \mathrm { p } {/tex} and {tex} \mathrm { q } {/tex} both are true

B

{tex} \mathrm { p } {/tex} and {tex} \mathrm { q } {/tex} both are false

{tex} \mathrm { \mathrm {p} } {/tex} is false and {tex} \mathrm { \mathrm {q} } {/tex} is true

D

None of these

Explanation

Q 19.    

Correct4

Incorrect-1

Statement {tex} ( \mathrm { p } \wedge \mathrm { q } ) \rightarrow \mathrm { p } {/tex} is-

a tautology

B

a contradiction

C

neither {tex} { (1) } {/tex} nor {tex} { ( 2 ) } {/tex}

D

None of these

Explanation

Q 20.    

Correct4

Incorrect-1

If statements {tex} \mathrm {p, q, r} {/tex} have truth values {tex} \mathrm {T, F, T} {/tex} respectively then which of the following statement

A

{tex} ( \mathrm {p \rightarrow q ) \wedge r} {/tex}

B

{tex} ( \mathrm {p \rightarrow q ) \vee ^\sim r} {/tex}

C

{tex} ( \mathrm {p \wedge q ) \vee ( q \wedge r} ) {/tex}

{tex} ( \mathrm {p \rightarrow q ) \rightarrow r} {/tex}

Explanation

Q 21.    

Correct4

Incorrect-1

If statement {tex} \mathrm {p} \rightarrow ( \mathrm {q \vee r} ) {/tex} is true then the truth values of statements {tex} \mathrm {p, q, r} {/tex} respectively-

A

{tex} \mathrm { T } , \mathrm { F } , \mathrm { T } {/tex}

B

{tex} \mathrm { F } , \mathrm { T } , \mathrm { F } {/tex}

C

{tex} \mathrm { F } , \mathrm { F } , \mathrm { F } {/tex}

All of these

Explanation

Q 22.    

Correct4

Incorrect-1

Which of the following statement is a contradiction-

{tex} ( \mathrm {p \wedge q ) \wedge(\mathrm{^\sim ( p \vee q} )} ){/tex}

B

{tex} \mathrm {p} \vee ( ^\sim \mathrm {p \wedge q} ) {/tex}

C

{tex} ( \mathrm {p \rightarrow q} ) \rightarrow \mathrm {p} {/tex}

D

{tex} ^\sim \mathrm {p \vee ^\sim q} {/tex}

Explanation

Q 23.    

Correct4

Incorrect-1

The negative of the statement "If a number is divisible by {tex} { 15 } {/tex} then it is divisible by {tex} { 5 } {/tex} or {tex} { 3 " } {/tex}

A

If a number is divisible by {tex} 15 {/tex} then it is not divisible by {tex} 5 {/tex} and {tex}3 {/tex}

B

A number is divisible by {tex} 15 {/tex} and it is not divisible by {tex} 5 {/tex} or {tex} 3 {/tex}

C

A number is divisible by {tex} 15 {/tex} or it is not divisible by {tex} 5 {/tex} and {tex}3 {/tex}

A number is divisible by {tex} 15 {/tex} and it is not divisible by {tex} 5 {/tex} and {tex} 3 {/tex}

Explanation

Q 24.    

Correct4

Incorrect-1

Which of the following is a statement-

A

Open the door

B

Do your home work

C

Hurrah! we have won the match

Two plus two is five

Explanation

Q 25.    

Correct4

Incorrect-1

The negation of the statement "{tex} 2 + 3 = 5 {/tex} and {tex} 8< 10 {/tex}" is -

A

{tex} 2 + 3 \neq 5 {/tex} and {tex} 8 \nless 10 {/tex}

B

{tex} 2 + 3 \neq 5 {/tex} or {tex} 8 > 10 {/tex}

{tex} 2 + 3 \neq 5 {/tex} or {tex} 8 \geq 10 {/tex}

D

None of these

Explanation