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JEE Advanced > Algebra

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

Q 1.

Correct4

Incorrect-1

If {tex} x , y {/tex} and {tex} z {/tex} are {tex} p {/tex} th {tex} , q {/tex} th and {tex} r {/tex} th terms respectively of an A.P. and also of a G.P, then {tex} x ^ { y - z } y ^ { z- x } z ^ { x - y } {/tex} is equal to:

A

{tex} x y z {/tex}

B

0

1

D

None of these

Explanation

Q 2.

Correct4

Incorrect-1

The third term of a geometric progression is {tex} 4 . {/tex} The product of the first five terms is

A

{tex} 4 ^ { 3 } {/tex}

{tex} 4 ^ { 5 } {/tex}

C

{tex} 4 ^ { 4 } {/tex}

D

none of these

Explanation

Q 3.

Correct4

Incorrect-1

The rational number, which equals the number {tex}2.\overline{357}{/tex} with recurring decimal is

A

{tex} \frac { 2355 } { 1001 } {/tex}

B

{tex} \frac { 2379 } { 997 } {/tex}

{tex} \frac { 2355 } { 999 } {/tex}

D

none of these

Explanation

Q 4.

Correct4

Incorrect-1

If {tex} a , b , c {/tex} are in G.P., then the equations {tex} a x ^ { 2 } + 2 b x + c = 0 {/tex} and {tex} d x ^ { 2 } + 2 e x + f = 0 {/tex} have a common root if {tex} \frac { d } { a } , \frac { e } { b } , \frac { f } { c } {/tex} are in {tex} - {/tex}

{tex} \mathrm { A }. \mathrm { P }.{/tex}

B

{tex} \mathrm { G.P }. {/tex}

C

{tex} \mathrm { H. } \mathrm { P. } {/tex}

D

none of these

Explanation



Q 5.

Correct4

Incorrect-1

Sum of the first {tex} n {/tex} terms of the series {tex} \frac { 1 } { 2 } + \frac { 3 } { 4 } + \frac { 7 } { 8 } + \frac { 15 } { 16 } + \ldots \ldots \ldots \ldots {/tex} is equal to

A

{tex} 2 ^ { n } - n - 1 {/tex}

B

{tex} 1 - 2 ^ { - n } {/tex}

{tex} n + 2 ^ { - n } - 1 {/tex}

D

{tex} 2 ^ { n } + 1 {/tex}

Explanation

Q 6.

Correct4

Incorrect-1

The number {tex} \log _ { 2 } 7 {/tex} is

A

an integer

B

a rational number

an irrational number

D

a prime number

Explanation

Q 7.

Correct4

Incorrect-1

If {tex} \ln ( a + c ) , \ln ( a - c ) , \ln ( a - 2 b + c ) {/tex} are in A.P., then

A

{tex} a , b , c {/tex} are in {tex} \mathrm { } \mathrm { } {/tex}A.P

B

{tex} a ^ { 2 } , b ^ { 2 } , c ^ { 2 } {/tex} are in A.P

C

{tex} a , b , c {/tex} are in G.P.

{tex} a , b , c {/tex} are in H.P.

Explanation

Q 8.

Correct4

Incorrect-1

Let {tex} a _ { 1 } , a _ { 2 } , \ldots . a _ { 10 } {/tex} be in {tex} A , P , {/tex} and {tex} h _ { 1 } , h _ { 2 } , \ldots h _ { 10 } {/tex} be in H.P. If {tex} a _ { 1 } = h _ { 1 } = 2 {/tex} and {tex} a _ { 10 } = h _ { 10 } = 3 , {/tex} then {tex} a _ { 4 } h _ { 7 } {/tex} is

A

2

B

3

C

5

6

Explanation

Q 9.

Correct4

Incorrect-1

The harmonic mean of the roots of the equation
{tex} ( 5 + \sqrt { 2 } ) x ^ { 2 } - ( 4 + \sqrt { 5 } ) x + 8 + 2 \sqrt { 5 } = 0 {/tex} is

A

2

4

C

6

D

8

Explanation

Q 10.

Correct4

Incorrect-1

Consider an infinite geometric series with first term a and common ratio {tex} r . {/tex} If its sum is 4 and the second term is {tex} 3 / 4 , {/tex} then

A

{tex} a = \frac { 4 } { 7 } , r = \frac { 3 } { 7 } {/tex}

B

{tex} a = 2 , r = \frac { 3 } { 8 } {/tex}

C

{tex} a = \frac { 3 } { 2 } , r = \frac { 1 } { 2 } {/tex}

{tex} a = 3 , r = \frac { 1 } { 4 } {/tex}

Explanation

Q 11.

Correct4

Incorrect-1

Let {tex} \alpha , \beta {/tex} be the roots of {tex} x ^ { 2 } - x + p = 0 {/tex} and {tex} \gamma , \delta {/tex} be the roots of {tex} x ^ { 2 } - 4 x + q = 0 . {/tex} If {tex} \alpha , \beta , \gamma , \delta {/tex} are in G.P., then the integral values of {tex} p {/tex} and {tex} q {/tex} respectively, are

{tex} - 2 , - 32 {/tex}

B

{tex} - 2,3 {/tex}

C

{tex} - 6,3 {/tex}

D

{tex} - 6 , - 32 {/tex}

Explanation



Q 12.

Correct4

Incorrect-1

Let the positive numbers {tex} a , b , c , d {/tex} be in A.P. Then {tex} a b c , a b d , {/tex} {tex} a c d , b c d {/tex} are

A

NOT in A.P/G.P/H.P.

B

in A.P.

C

in G.P

in H.P

Explanation

Q 13.

Correct4

Incorrect-1

If the sum of the first {tex} 2 n {/tex} terms of the A.P. {tex} 2,5,8 , \ldots , {/tex} is equal to the sum of the first {tex} n {/tex} terms of the A.P. {tex} 57,59,61 , \ldots , {/tex} then {tex} n {/tex} equals

A

10

B

12

11

D

13

Explanation

Q 14.

Correct4

Incorrect-1

Suppose {tex} a , b , c {/tex} are in A.P. and {tex} a ^ { 2 } , b ^ { 2 } , c ^ { 2 } {/tex} are in G.P. if {tex} a < b < c {/tex} and {tex} a + b + c = \frac { 3 } { 2 } , {/tex} then the value of {tex} a {/tex} is

A

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

B

{tex} \frac { 1 } { 2 \sqrt { 3 } } {/tex}

C

{tex} \frac { 1 } { 2 } - \frac { 1 } { \sqrt { 3 } } {/tex}

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

Explanation


Q 15.

Correct4

Incorrect-1

An infinite G.P. has first term '{tex} x {/tex}'and sum '5', then {tex} x {/tex} belongs to

A

{tex} x < - 10 {/tex}

B

{tex} - 10 < x < 0 {/tex}

{tex} 0 < x < 10 {/tex}

D

{tex} x > 10 {/tex}

Explanation

Q 16.

Correct4

Incorrect-1

In the quadratic equation {tex} a x ^ { 2 } + b x + c = 0 , \Delta = b ^ { 2 } - 4 a c {/tex} and {tex} \alpha + \beta , \alpha ^ { 2 } + \beta ^ { 2 } , \alpha ^ { 3 } + \beta ^ { 3 } , {/tex} are in G.P. where {tex} \alpha , \beta {/tex} are the root of {tex} a x ^ { 2 } + b x + c = 0 , {/tex} then

A

{tex} \Delta \neq 0 {/tex}

B

{tex} b \Delta = 0 {/tex}

{tex} c \Delta = 0 {/tex}

D

{tex} \Delta = 0 {/tex}

Explanation



Q 17.

Correct4

Incorrect-1

In the sum of first {tex} n {/tex} terms of an A.P. is {tex} c n ^ { 2 } {/tex}, the sum of squares of these {tex} n {/tex} terms is

A

{tex} \frac { n \left( 4 n ^ { 2 } - 1 \right) c ^ { 2 } } { 6 } {/tex}

B

{tex} \frac { n \left( 4 n ^ { 2 } + 1 \right) c ^ { 2 } } { 3 } {/tex}

{tex} \frac { n \left( 4 n ^ { 2 } - 1 \right) c ^ { 2 } } { 3 } {/tex}

D

{tex} \frac { n \left( 4 n ^ { 2 } + 1 \right) c ^ { 2 } } { 6 } {/tex}

Explanation


Q 18.

Correct4

Incorrect-1

Let {tex} a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots . {/tex} be in harmonic progression with {tex} a _ { 1 } = 5 {/tex} and {tex} a _ { 20 } = 25 . {/tex} The least positive integer {tex} n {/tex} for which {tex} a _ { n } < 0 {/tex} is

A

22

B

23

C

24

25

Explanation


Q 19.

Correct4

Incorrect-1

The inequality {tex} | z - 4 | < | z - 2 | {/tex} represents the region given by

A

{tex} \operatorname { Re } ( z ) \geq 0 {/tex}

B

{tex} \operatorname { Re } ( z ) < 0 {/tex}

C

{tex} \operatorname { Re } ( z ) > 0 {/tex}

none of these

Explanation

Q 20.

Correct4

Incorrect-1

Let {tex} z {/tex} and {tex} \omega {/tex} be two non zero complex numbers such that {tex} | z | = | \omega | {/tex} and {tex} \operatorname { Arg } z + {/tex} Arg {tex} \omega = \pi , {/tex} then {tex} z {/tex} equals

A

{tex}\omega{/tex}

B

{tex}\mathrm - \omega {/tex}

C

{tex}\overline\omega{/tex}

{tex} - \overline { \omega } {/tex}

Explanation

Q 21.

Correct4

Incorrect-1

Let {tex} z {/tex} and {tex} \omega {/tex} be two complex numbers such that {tex} | z | \leq 1, {/tex} {tex} | \omega | \leq 1 {/tex} and {tex} | z + i \omega | = | z - i \bar { \omega } | = 2 {/tex} then z equals

A

1 or {tex} i {/tex}

B

{tex} i {/tex} or {tex} - i {/tex}

1 or -1

D

{tex} i {/tex} or - 1

Explanation

Q 22.

Correct4

Incorrect-1

For positive integers {tex} n _ { 1 } , n _ { 2 } {/tex} the value of the expression
{tex} ( 1 + i ) ^ { n _ { 1 } } + \left( 1 + i ^ { 3 } \right) ^ { n _ { 1 } } + \left( 1 + i ^ { 5 } \right) ^ { n _ { 2 } } + \left( 1 + i ^ { 7 } \right) ^ { n _ { 2 } } , {/tex} where {tex} i = \sqrt { - 1 } {/tex} is a real number if and only if

A

{tex} n _ { 1 } = n _ { 2 } + 1 {/tex}

B

{tex} n _ { 1 } = n _ { 2 } - 1 {/tex}

C

{tex} n _ { 1 } = n _ { 2 } {/tex}

{tex} n _ { 1 } > 0 , n _ { 2 } > 0 {/tex}

Explanation


Q 23.

Correct4

Incorrect-1

If {tex} i = \sqrt { - 1 } , {/tex} then {tex} 4 + 5 \left( - \frac { 1 } { 2 } + \frac { i \sqrt { 3 } } { 2 } \right) ^ { 334 } + 3 \left( - \frac { 1 } { 2 } + \frac { i \sqrt { 3 } } { 2 } \right) ^ { 365 } {/tex} is equal to

A

{tex} 1 - i \sqrt { 3 } {/tex}

B

{tex} - 1 + i \sqrt { 3 } {/tex}

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

D

{tex} - i \sqrt { 3 } {/tex}

Explanation

Q 24.

Correct4

Incorrect-1

If {tex} \arg ( z ) < 0 , {/tex} then {tex} \arg ( - z ) - \arg ( z ) = {/tex}

{tex} \pi {/tex}

B

{tex} - \pi {/tex}

C

{tex} - \frac { \pi } { 2 } {/tex}

D

{tex} \frac { \pi } { 2 } {/tex}

Explanation

Q 25.

Correct4

Incorrect-1

If {tex} z _ { 1 } , z _ { 2 } {/tex} and {tex} z _ { 3 } {/tex} are complex numbers such that

{tex} \left| z _ { 1 } \right| = \left| z _ { 2 } \right| = \left| z _ { 3 } \right| = \left| \frac { 1 } { z _ { 1 } } + \frac { 1 } { z _ { 2 } } + \frac { 1 } { z _ { 3 } } \right| = 1 , \quad {/tex} then {tex} \quad \left| z _ { 1 } + z _ { 2 } + z _ { 3 } \right| {/tex} is

equal to 1

B

less than 1

C

greater than 3

D

equal to 3

Explanation