JEE Main > Algebra

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


Q 1.    

Correct4

Incorrect-1

If the difference between the roots of the equation {tex} x ^ { 2 } + a x {/tex} {tex} + 1 = 0 {/tex} is less than {tex} \sqrt { 5 } {/tex} , then the set of possible values of {tex} a {/tex} is

{tex} ( - 3,3 ) {/tex}

B

{tex} ( - 3 , \infty ) {/tex}

C

{tex} ( 3 , \infty ) {/tex}

D

{tex} ( - \infty , - 3 ) {/tex}

Explanation

Q 2.    

Correct4

Incorrect-1

The quadratic equations {tex} x ^ { 2 } - 6 x + a = 0 {/tex} and {tex} x ^ { 2 } - cx + 6 = 0 {/tex} have one root in common. The other roots of the first and second equations are integers in the ratio {tex} 4 : 3 . {/tex} Then the common root is

A

1

B

4

C

3

2

Explanation

Q 3.    

Correct4

Incorrect-1

If the roots of the equation {tex} b x ^ { 2 } + c x + a = 0 {/tex} are imaginary, then for all real values of {tex} x , {/tex} the expression {tex} 3 b ^ { 2 } x ^ { 2 } + 6 b c x + 2 c ^ { 2 } {/tex} is

A

Greater than 4{tex} a b {/tex}

B

Less than 4{tex} a b {/tex}

Greater than {tex} - 4 a b {/tex}

D

Less than {tex} - 4 a b {/tex}

Explanation

Q 4.    

Correct4

Incorrect-1

{tex}If \overrightarrow{u},\overrightarrow{v},\overrightarrow{w}{/tex} are non coplanar vectors and p,q are real numbers, then the equality {tex}[3\overrightarrow{u}\space p\overrightarrow{v}\space p\overrightarrow{w}]-[p\overrightarrow{v}\space\overrightarrow{w}\space q\overrightarrow{u}\space]-[2\overrightarrow{w}\space q\overrightarrow{v}\space q\overrightarrow{u}]=0{/tex} holds for

exactly one value of {tex} ( p , q ) {/tex}

B

exactly two values of {tex} ( p , q ) {/tex}

C

more than two but not all values of {tex} ( p , q ) {/tex}

D

all values of {tex} ( p , q ) {/tex}

Explanation

Q 5.    

Correct4

Incorrect-1

The equation {tex} e ^ { \sin x } - e ^ { - \sin x } - 4 = 0 {/tex} has

A

infinite number of real roots

no real roots

C

exactly one real root

D

exactly four real roots

Explanation

Q 6.    

Correct4

Incorrect-1

If the equations {tex} x ^ { 2 } + 2 x + 3 = 0 {/tex} and {tex} a x ^ { 2 } + b x + c = 0 , a , b , c \in R , {/tex} have a common root, then {tex} a : b \cdot c {/tex} is

A

{tex} 3 : 2 :1 {/tex}

B

{tex} 1 : 3 : 2 {/tex}

C

{tex} 3 : 1 : 2 {/tex}

{tex} 1 : 2 : 3 {/tex}

Explanation

Q 7.    

Correct4

Incorrect-1

Let {tex} \alpha {/tex} and {tex} \beta {/tex} be the roots of the equation {tex} p x ^ { 2 } + q x + r = 0 , p \neq 0 {/tex} . If {tex} p , q , r {/tex} are in {tex} A P {/tex} and {tex} \frac { 1 } { \alpha } + \frac { 1 } { \beta } = 4 , {/tex} then the value of {tex} | \alpha - \beta | {/tex} is

A

{tex} - \sqrt { 34} /9{/tex}

{tex} 2 \sqrt { 13 }/9 {/tex}

C

{tex} \sqrt { 61 }/9 {/tex}

D

{tex}2 \sqrt { 17 } /9{/tex}

Explanation

Q 8.    

Correct4

Incorrect-1

If equations {tex}ax^{2}+bx+c =0(a,b,c\epsilon R,a\neq0) {/tex} and {tex} 2x^{2}+3x+4=0 {/tex}
have a common root then a:b:c equals

A

{tex} 1 : 2 : 3 {/tex}

{tex} 2 : 3 : 4 {/tex}

C

{tex} 4 : 3 : 2 {/tex}

D

{tex} 3 : 2 : 1 {/tex}

Explanation

Q 9.    

Correct4

Incorrect-1

If {tex} \frac { 1 } { \sqrt { \alpha } } {/tex} and {tex} \frac { 1 } { \sqrt { \beta } } {/tex} are the roots of the equation {tex} a x ^ { 2 } + b x + 1 = 0 {/tex} {tex} ( a \neq 0 , a , b \in R ) , {/tex} then the equation {tex} x \left( x + b ^ { 3 } \right) + \left( a ^ { 3 } - 3 a b x \right) = 0 {/tex} has roots

{tex} \alpha ^ { 3 / 2 } {/tex} and {tex} \beta ^ { 3 / 2 } {/tex}

B

{tex} \alpha \beta ^ { 1 / 2 } {/tex} and {tex} \alpha ^ { 1 / 2 } \beta {/tex}

C

{tex} \sqrt { \alpha \beta } {/tex} and {tex} \alpha \beta {/tex}

D

{tex} \alpha ^ { - 3 / 2 } {/tex} and {tex} \beta ^ { - 3 / 2 } {/tex}

Explanation

Q 10.    

Correct4

Incorrect-1

If {tex} f ( x ) = \left( \frac { 3 } { 5 } \right) ^ { x } + \left( \frac { 4 } { 5 } \right) ^ { x } - 1 , {/tex} where {tex} x \in R , {/tex} then the equation {tex} f ( x ) {/tex} {tex} = 0 {/tex} has

A

no solution

one solution

C

two solutions

D

more than two solutions

Explanation

Q 11.    

Correct4

Incorrect-1

If {tex} \alpha {/tex} and {tex} \beta {/tex} are roots of the equation {tex} x ^ { 2 } - 4 \sqrt { 2 } k x + 2 e ^ { 4 \ln k } - 1 = 0 {/tex} for some {tex} k {/tex} and {tex} \alpha ^ { 2 } + \beta ^ { 2 } = 66 , {/tex} then {tex} \alpha ^ { 3 } + \beta ^ { 3 } {/tex} is equal to

A

248{tex} \sqrt { 2 } {/tex}

280{tex} \sqrt { 2 } {/tex}

C

{tex} - 32 \sqrt { 2 } {/tex}

D

{tex} - 280 \sqrt { 2 } {/tex}

Explanation

Q 12.    

Correct4

Incorrect-1

The sum of the roots of the equation {tex} x ^ { 2 } + | 2 x - 3 | - 4 = 0 {/tex} is

A

{tex} 2 {/tex}

B

{tex} -2 {/tex}

{tex} \sqrt { 2 } {/tex}

D

{tex} - \sqrt { 2 } {/tex}

Explanation

Q 13.    

Correct4

Incorrect-1

The equation {tex} \sqrt { 3 x ^ { 2 } + x + 5 } = x - 3 , {/tex} where {tex} x {/tex} is real, has

no solution

B

exactly one solution

C

exactly two solutions

D

exactly four solutions

Explanation

Q 14.    

Correct4

Incorrect-1

If non-zero real numbers {tex} b {/tex} and {tex} c {/tex} are such that {tex} \min f ( x ) > \max {/tex} {tex} g ( x ) , {/tex} where {tex} f ( x ) = x ^ { 2 } + 2 b x + 2 c ^ { 2 } {/tex} and {tex} g ( x ) = - x ^ { 2 } - 2 c x + b ^ { 2 } ( x \in R ) {/tex} , then {tex} \left| \frac { c } { b } \right| {/tex} lies in the interval

A

{tex} \left( 0 , \frac { 1 } { 2 } \right) {/tex}

B

{tex} \left[ \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right] {/tex}

C

{tex} \left[ \frac { 1 } { \sqrt { 2 } } , \sqrt { 2 } \right] {/tex}

{tex} ( \sqrt { 2 } , \infty ) {/tex}

Explanation

Q 15.    

Correct4

Incorrect-1

The sum of coefficients of integral powers of {tex} x {/tex} in the binomial expansion of {tex} ( 1 - 2 \sqrt { x } ) ^ { 50 } {/tex} is

A

{tex} \frac { 1 } { 2 } \left( 3 ^ { 50 } \right) {/tex}

B

{tex} \frac { 1 } { 2 } \left( 3 ^ { 50 } - 1 \right) {/tex}

C

{tex} \frac { 1 } { 2 } \left( 2 ^ { 50 } + 1 \right) {/tex}

{tex} \frac { 1 } { 2 } \left( 3 ^ { 50 } + 1 \right) {/tex}

Explanation

Q 16.    

Correct4

Incorrect-1

Let {tex} \alpha {/tex} and {tex} \beta {/tex} be the roots of equation {tex} x ^ { 2 } - 6 x - 2 = 0 . {/tex} If {tex} a _ { n } = {/tex} {tex} a ^ { n } - \beta ^ { n } , {/tex} for {tex} n \geq 1 , {/tex} then the value of {tex} \frac { a _ { 10 } - 2 a _ { 8 } } { 2 a _ { 9 } } {/tex} is equal to

A

- 6

3

C

- 3

D

6

Explanation

Q 17.    

Correct4

Incorrect-1

If {tex} 2 + 3 i {/tex} is one of the roots of the equation {tex} 2 x ^ { 3 } - 9 x ^ { 2 } + k x - 13 {/tex} {tex} = 0 , k \in R , {/tex} then the real root of this equation

A

does not exist

exists and is equal to {tex} \frac { 1 } { 2 } {/tex}

C

exists and is equal to {tex} - \frac { 1 } { 2 } {/tex}

D

exists and is equal to 1

Explanation

Q 18.    

Correct4

Incorrect-1

If the two roots of the equation {tex} ( a - 1 ) \left( x ^ { 4 } + x ^ { 2 } + 1 \right) + ( a + 1 ) {/tex} {tex} \left( x ^ { 2 } + x + 1 \right) ^ { 2 } = 0 {/tex} are real and distinct, then the set of all values of {tex} a {/tex} is

A

{tex} \left( - \frac { 1 } { 2 } , 0 \right) {/tex}

B

{tex} ( - \infty , - 2 ) \cup ( 2 , \infty ) {/tex}

{tex} \left( - \frac { 1 } { 2 } , 0 \right) \cup \left( 0 , \frac { 1 } { 2 } \right) {/tex}

D

{tex}\left(0, \frac { 1 } { 2 }\right){/tex}

Explanation

Q 19.    

Correct4

Incorrect-1

The sum of all real values of {tex} x {/tex} satisfying the equation {tex} \left( x ^ { 2 } - 5 x + 5 \right) ^ { x ^ { 2 } + 4 x - 60 } = 1 {/tex} is

A

{tex}5 {/tex}

{tex}3 {/tex}

C

{tex} - 4 {/tex}

D

{tex} 6 {/tex}

Explanation

Q 20.    

Correct4

Incorrect-1

If the equations {tex} x ^ { 2 } + b x - 1 = 0 {/tex} and {tex} x ^ { 2 } + x + b = 0 {/tex} have a common root different from {tex} - 1 {/tex} , then {tex} | b | {/tex} is equal to

A

{tex}-2 {/tex}

B

{tex}3 {/tex}

{tex} \sqrt { 3 } {/tex}

D

{tex} \sqrt { 2 } {/tex}

Explanation

Q 21.    

Correct4

Incorrect-1

Let {tex} x , y , z {/tex} be positive real numbers such that {tex} x + y + z = 12 {/tex} and {tex} x ^ { 3 } y ^ { 4 } z ^ { 5 } = ( 0.1 ) ( 600 ) ^ { 3 } . {/tex} Then {tex} x ^ { 3 } + y ^ { 3 } + z ^ { 3 } {/tex} is equal to

A

342

216

C

258

D

270

Explanation

Q 22.    

Correct4

Incorrect-1

If {tex} x {/tex} is a solution of the equation {tex} \sqrt { 2 x + 1 } - \sqrt { 2 x - 1 } = 1 , \left( x \geq \frac { 1 } { 2 } \right) {/tex} then {tex} \sqrt { 4 x ^ { 2 } - 1 } {/tex} is equal to

{tex} \frac { 3 } { 4 } {/tex}

B

{tex} \frac { 1 } { 2 } {/tex}

C

2{tex} \sqrt { 2 } {/tex}

D

{tex}2 {/tex}

Explanation

Q 23.    

Correct4

Incorrect-1

If {tex} ( 27 ) ^ { 999 } {/tex} is divided by {tex} 7 , {/tex} then the remainder is

6

B

1

C

1

D

3

Explanation

Q 24.    

Correct4

Incorrect-1

The coefficient of {tex} x ^ { - 5 } {/tex} in the binomial expansion of
{tex} \left( \frac { x + 1 } { x ^ { \frac { 2 } { 3 } } - x ^ { \frac { 1 } { 3 } } + 1 } - \frac { x - 1 } { x - x ^ { \frac { 1 } { 2 } } } \right) ^ { 10 } , {/tex} where {tex} x \neq 0,1 , {/tex} is

1

B

-4

C

{tex} - 1 {/tex}

D

4

Explanation

Q 25.    

Correct4

Incorrect-1

{tex} z {/tex} and {tex}\omega {/tex} are two nonzero complex number such that {tex} | z | = | \omega | {/tex} and {tex} \operatorname { Arg } z + \operatorname { Arg } \omega = \pi {/tex} then {tex} z {/tex} equals

A

{tex} \overline { \mathrm { \omega} } {/tex}

{tex} - \bar { \omega } {/tex}

C

{tex} \omega {/tex}

D

{tex} - \omega {/tex}

Explanation