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JEE Advanced > Matrices and Determinants

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

Q 1.

Correct4

Incorrect-1

Consider the set {tex}A{/tex} of all determinants of order 3 with entries {tex}\mathrm 0 \quad or \quad1{/tex} only. Let {tex}B{/tex} be the subset of {tex}A{/tex} consisting of all determinants With value {tex}1{/tex}.Let {tex} C{/tex} the subset Of {tex}A{/tex} consisting of all determinants with value {tex}-1{/tex}.Then

A

{tex}C{/tex} is empty

{tex}B{/tex} has as many elements as {tex}C{/tex}

C

{tex} A = B \cup C {/tex}

D

{tex} B {/tex} has twice as many elements as elements as {tex} C {/tex}

Explanation

Q 2.

Correct4

Incorrect-1

If {tex} \omega ( \neq 1 ) {/tex} is a cube root of unity, then
{tex} \left| \begin{array} { c c c } { 1 } & { 1 + i + \omega ^ { 2 } } & { \omega ^ { 2 } } \\ { 1 - i } & { - 1 } & { \omega ^ { 2 } - 1 } \\ { - i } & { - i + \omega - 1 } & { - 1 } \end{array} \right| = {/tex}

{tex} 0 {/tex}

B

1

C

{tex}\mathrm {i}{/tex}

D

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

Explanation



Q 3.

Correct4

Incorrect-1

Let {tex} a , b , c {/tex} be the real numbers. Then following system of equations in {tex} x , y {/tex} and {tex} z {/tex}
{tex} \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } - \frac { z ^ { 2 } } { c ^ { 2 } } = 1 , \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } + \frac { z ^ { 2 } } { c ^ { 2 } } = 1 {/tex},
{tex} - \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } + \frac { z ^ { 2 } } { c ^ { 2 } } = 1 {/tex} has

A

no solution

B

unique solution

C

infinitely many solutions

finitely many solutions

Explanation









Q 4.

Correct4

Incorrect-1

If {tex} A {/tex} and {tex} B {/tex} are square matrices of equal degree, then which one is correct among the followings?

{tex} A + B = B + A {/tex}

B

{tex} A + B = A - B {/tex}

C

{tex} A - B = B - A {/tex}

D

{tex} A B = B A {/tex}

Explanation

Q 5.

Correct4

Incorrect-1

The parameter, on which the value of the determinant
{tex} \left| \begin{array} { c c c } { 1 } & { a } & { a ^ { 2 } } \\ { \cos ( p - d ) x } & { \cos p x } & { \cos ( p + d ) x } \\ { \sin ( p - d ) x } & { \sin p x } & { \sin ( p + d ) x } \end{array} \right| {/tex} does not depend
upon is

A

{tex} a {/tex}

{tex} p {/tex}

C

{tex} d {/tex}

D

{tex} x {/tex}

Explanation







Q 6.

Correct4

Incorrect-1

If {tex} f ( x ) = \left| \begin{array} { c c c } { 1 } & { x } & { x + 1 } \\ { 2 x } & { x ( x - 1 ) } & { ( x + 1 ) x } \\ { 3 x ( x - 1 ) } & { x ( x - 1 ) ( x - 2 ) } & { ( x + 1 ) x ( x - 1 ) } \end{array} \right| {/tex} then
{tex} f ( 100 ) {/tex} is equal to

{tex}\mathrm {0}{/tex}

B

1

C

{tex}100{/tex}

D

{tex}-100{/tex}

Explanation





Q 7.

Correct4

Incorrect-1

If the system of equations {tex} x - k y - z = 0 , k x - y - z = 0 , x + y - z = 0 {/tex} has a non-zero solution, then the possible values of {tex} k {/tex} are

A

{tex} - 1,2 {/tex}

B

{tex} 1,2 {/tex}

C

{tex} 0,1 {/tex}

{tex} - 1,1 {/tex}

Explanation

Q 8.

Correct4

Incorrect-1

Let {tex} \omega = - \frac { 1 } { 2 } + i \frac { \sqrt { 3 } } { 2 } . {/tex} Then the value of the determinant
{tex} \left| \begin{array} { c c c } { 1 } & { 1 } & { 1 } \\ { 1 } & { - 1 - \omega ^ { 2 } } & { \omega ^ { 2 } } \\ { 1 } & { \omega ^ { 2 } } & { \omega ^ { 4 } } \end{array} \right| {/tex} is

A

{tex}\mathrm {3\omega}{/tex}

{tex} 3 \omega ( \omega - 1 ) {/tex}

C

{tex} 3 \omega ^ { 2 } {/tex}

D

{tex} 3 \omega ( 1 - \omega ) {/tex}

Explanation



Q 9.

Correct4

Incorrect-1

The number of values of {tex} k {/tex} for which the system of equations {tex} ( k + 1 ) x + 8 y = 4 k ; k x + ( k + 3 ) y = 3 k - 1 {/tex} has infinitely many solutions is

A

0

1

C

2

D

infinite

Explanation

Q 10.

Correct4

Incorrect-1

If {tex} A = \left[ \begin{array} { l l } { \alpha } & { 0 } \\ { 1 } & { 1 } \end{array} \right] {/tex} and {tex} B = \left[ \begin{array} { l l } { 1 } & { 0 } \\ { 5 } & { 1 } \end{array} \right] , {/tex} then value of {tex} \alpha {/tex} for which
{tex} A ^ { 2 } = B , {/tex} is

A

1

B

- 1

C

4

no real values

Explanation

Q 11.

Correct4

Incorrect-1

If the system of equations {tex} x + a y = 0 , a z + y = 0 {/tex} and {tex} a x + z = 0 {/tex} has infinite solutions, then the value of {tex} a {/tex} is

-1

B

1

C

{tex}\mathrm {0}{/tex}

D

no real values

Explanation



Q 12.

Correct4

Incorrect-1

If {tex} A = \left[ \begin{array} { l l } { \alpha } & { 2 } \\ { 2 } & { \alpha } \end{array} \right] {/tex} and {tex} \left| A ^ { 3 } \right| = 125 {/tex} then the value of {tex} \alpha {/tex} is

A

{tex} \pm 1 {/tex}

B

{tex} \pm 2 {/tex}

{tex} \pm 3 {/tex}

D

{tex} \pm 5 {/tex}

Explanation

Q 13.

Correct4

Incorrect-1

If {tex} P = \left[ \begin{array} { c c } { \frac { \sqrt { 3 } } { 2 } } & { \frac { 1 } { 2 } } \\ { - \frac { 1 } { 2 } } & { \frac { \sqrt { 3 } } { 2 } } \end{array} \right] {/tex} and {tex} A = \left[ \begin{array} { c c } { 1 } & { 1 } \\ { 0 } & { 1 } \end{array} \right] {/tex} and {tex} Q = P A P ^ { T } {/tex} and

{tex} x = P ^ { T } Q ^ { 2005 } P {/tex} then {tex} x {/tex} is equal to

{tex} \left[ \begin{array} { c c } { 1 } & { 2005 } \\ { 0 } & { 1 } \end{array} \right] {/tex}

B

{tex} \left[ \begin{array} { c c } { 4 + 2005 \sqrt { 3 } } & { 6015 } \\ { 2005 } & { 4 - 2005 \sqrt { 3 } } \end{array} \right] {/tex}

C

{tex} \frac { 1 } { 4 } \left[ \begin{array} { c c } { 2 + \sqrt { 3 } } & { 1 } \\ { - 1 } & { 2 - \sqrt { 3 } } \end{array} \right] {/tex}

D

{tex} \frac { 1 } { 4 } \left[ \begin{array} { c c } { 2005 } & { 2 - \sqrt { 3 } } \\ { 2 + \sqrt { 3 } } & { 2005 } \end{array} \right] {/tex}

Explanation





Q 14.

Correct4

Incorrect-1

The number of {tex} 3 \times 3 {/tex} matrices {tex} A {/tex} whose entries are either 0 or 1 and for which the system {tex} A \left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] = \left[ \begin{array} { l } { 1 } \\ { 0 } \\ { 0 } \end{array} \right] {/tex} has exactly two distinct solutions, is

{tex}\mathrm {0}{/tex}

B

{tex} 2 ^ { 9 } - 1 {/tex}

C

{tex}168{/tex}

D

{tex}2{/tex}

Explanation

Q 15.

Correct4

Incorrect-1

Let {tex} \omega \neq 1 {/tex} be a cube root of unity and {tex} S {/tex} be the set of all non-singular matrices of the form {tex} \left[ \begin{array} { c c c } { 1 } & { a } & { b } \\ { \omega } & { 1 } & { c } \\ { \omega ^ { 2 } } & { \omega } & { 1 } \end{array} \right] {/tex}
where each of {tex} a , b {/tex} and {tex} c {/tex} is either {tex} \omega {/tex} or {tex} \omega ^ { 2 } . {/tex} Then the number of distinct matrices in the set {tex} \mathrm { S } {/tex} is

2

B

6

C

4

D

8

Explanation



Q 16.

Correct4

Incorrect-1

If {tex} P {/tex} is a {tex} 3 \times 3 {/tex} matrix such that {tex} P^T = 2 P + I , {/tex} where {tex} P ^ { T } {/tex} is the transpose of {tex} P {/tex} and {tex} I {/tex} is the {tex} 3 \times 3 {/tex} identity matrix, then there
exists a column matrix {tex} X = \left[ \begin{array} { l } { x } \\ { y } \\ { z } \end{array} \right] \neq \left[ \begin{array} { l } { 0 } \\ { 0 } \\ { 0 } \end{array} \right] {/tex} such that

A

{tex} P X = \left[ \begin{array} { l } { 0 } \\ { 0 } \\ { 0 } \end{array} \right] {/tex}

B

{tex} P X = X {/tex}

C

{tex} P X = 2 X {/tex}

{tex} P X = - X {/tex}

Explanation