# JEE Main

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

## Mathematics

Matrices and Determinants

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Q 1. Let {tex} \alpha , \beta , \gamma {/tex} are the real roots of the equation {tex} x ^ { 3 } + a x ^ { 2 } + b x + c = 0 {/tex} {tex} ( a , b , c \in R{/tex} and {tex} a \neq 0 ) {/tex}. If the system of equations (in {tex} u , v{/tex} and {tex} w {/tex}) given by
{tex} \alpha u + \beta v + \gamma w = 0 {/tex}
{tex} \beta u + \gamma v + \alpha w = 0 {/tex}
{tex} \gamma u + \alpha v + \beta w = 0 {/tex}
has non-trivial solutions, then the value of {tex} a ^ { 2 } / b {/tex} is.

3

B

2

C

5

D

1

##### Explanation

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Q 2. If {tex} a _ { 1 }\ , a _ { 2 } ,\ a _ { 3 } ,\ 5,\ 4 , a _ { 6 },\ a _ { 7 } , a _ { 8 } , a _ { 9 } {/tex} are in H.P., and {tex} D = \left| \begin{array} { c c c } { a _ { 1 } } & { a _ { 2 } } & { a _ { 3 } } \\ { 5 } & { 4 } & { a _ { 6 } } \\ { a _ { 7 } } & { a _ { 8 } } & { a _ { 9 } } \end{array} \right| {/tex} then the value of {tex} [ \mathrm { D } ] {/tex} is (where [ ] represents the greatest integer function)

2

B

3

C

1

D

4

##### Explanation

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Q 3.

{tex}x=0{/tex}

B

{tex}x=a{/tex}

C

{tex}x=b{/tex}

D

{tex}x=c{/tex}

##### Explanation

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Q 4. Let {tex} D _ { 1 } = \left| \begin{array} { c c c } { a } & { b } & { a + b } \\ { c } & { d } & { c + d } \\ { a } & { b } & { a - b } \end{array} \right| {/tex} and {tex} D _ { 2 } = \left| \begin{array} { c c c } { a } & { c } & { a + c } \\ { b } & { d } & { b + d } \\ { a } & { c } & { a + b + c } \end{array} \right| {/tex} then the value of {tex} \left| \frac { D _ { 1 } } { D _ { 2 } } \right| {/tex} is where {tex} b \neq 0 {/tex} and {tex} a d \neq b c {/tex}

2

B

3

C

1

D

0

##### Explanation

|{tex}\frac {D_{1}}{D_{2}}{/tex}|=2

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Q 5. If {tex} \left| \begin{array} { l l l } { x ^ { n } } & { x ^ { n + 2 } } & { x ^ { n + 4 } } \\ { y ^ { n } } & { y ^ { n + 2 } } & { y ^ { n + 4 } } \\ { z ^ { n } } & { z ^ { n + 2 } } & { z ^ { n + 4 } } \end{array} \right| = \left( \frac { 1 } { y ^ { 2 } } - \frac { 1 } { x ^ { 2 } } \right) \left( \frac { 1 } { z ^ { 2 } } - \frac { 1 } { y ^ { 2 } } \right) \left( \frac { 1 } { x ^ { 2 } } - \frac { 1 } { z ^ { 2 } } \right) {/tex} then {tex}- n{/tex} is.

A

2

B

3

4

D

-4

##### Explanation

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Q 6. If {tex}p+q+r+0=a+b+c{/tex}, then the value of the determinant is {tex}\left| \begin{array} { l l l } { p a } & { q b } & { r c } \\ { q c } & { r a } & { p b } \\ { r b } & { p c } & { q a } \end{array} \right| {/tex}

0

B

{tex} p a + q b + r c {/tex}

C

1

D

none of these

##### Explanation

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Q 7. If a determinant of order {tex}3\times3{/tex} is formed by using the numbers {tex}1{/tex} or {tex}-1{/tex},then the minimum value of the determinant is

A

{tex} - 2 {/tex}

{tex} - 4{/tex}

C

{tex} 0 {/tex}

D

{tex} - 8{/tex}

##### Explanation

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Q 8. If {tex}\mathrm {a =cos \theta+ i\ sin \theta ,b=cos \ 2\theta - i \ sin \ 2\theta;c=cos\ 3\theta+ i \ sin \ 3\theta}{/tex} and if =0,then

{tex} \theta = 2 k \pi , k \in Z {/tex}

B

{tex} \theta = ( 2 k + 1 ) \pi , k \in Z {/tex}

C

{tex} \theta = ( 4 k + 1 ) \pi , k \in Z {/tex}

D

none of these

##### Explanation

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Q 9. The determinant {tex} \left| \begin{array} { c c c } { y ^ { 2 } } & { - x y } & { x ^ { 2 } } \\ { a } & { b } & { c } \\ { a ^ { \prime } } & { b ^ { \prime } } & { c ^ { \prime } } \end{array} \right| {/tex} is equal to

A

{tex} \left| \begin{array} { l l } { b x + a y } & { c x + b y } \\ { b ^ { \prime } x + a ^ { \prime } y } & { c ^ { \prime } x + b ^ { \prime } y } \end{array} \right| {/tex}

B

{tex} \left| \begin{array} { l l } { a x + b y } & { b x + c y } \\ { a ^ { \prime } x + b ^ { \prime } y } & { b ^ { \prime } x + c ^ { \prime } y } \end{array} \right| {/tex}

C

{tex} \left| \begin{array} { c c } { b x + c y } & { a x + b y } \\ { b ^ { \prime } x + c ^ { \prime } y } & { a ^ { \prime } x + b ^ { \prime } y } \end{array} \right| {/tex}

{tex} \left| \begin{array} { c c } { a x + b y } & { b x + c y } \\ { a ^ { \prime } x + b ^ { \prime } y } & { b ^ { \prime } x + c ^ { \prime } y } \end{array} \right| {/tex}

##### Explanation

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Q 10. If {tex} a ^ { 2 } + b ^ { 2 } + c ^ { 2 } = - 2 \operatorname { and } f ( x ) = \left| \begin{array} { c c c } { 1 + a ^ { 2 } x } & { \left( 1 + b ^ { 2 } \right) x } & { \left( 1 + c ^ { 2 } \right) x } \\ { \left( 1 + a ^ { 2 } \right) x } & { 1 + b ^ { 2 } x } & { \left( 1 + c ^ { 2 } \right) x } \\ { \left( 1 + a ^ { 2 } \right) x } & { \left( 1 + b ^ { 2 } \right) x } & { 1 + c ^ { 2 } x } \end{array} \right| {/tex} then {tex} f ( x ) {/tex} is a polynomial of degree

A

0

B

1

2

D

3

##### Explanation

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Q 11. The value of the determinant {tex}\begin{bmatrix}{1} & 1 & 1\\{^{m}C_1} & {^{m+1}C_1}&{^{m+2}C_1}\\{^{m}C_2} & {^{m+1}C_2}&{^{m+2}C_2} \end{bmatrix}{/tex} is equal to

1

B

-1

C

0

D

none of these

##### Explanation

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Q 12. If {tex} x \neq y \neq z {/tex} and {tex} \left| \begin{array} { c c c } { x } & { x ^ { 2 } } & { 1 + x ^ { 3 } } \\ { y } & { y ^ { 2 } } & { 1 + y ^ { 3 } } \\ { z } & { z ^ { 2 } } & { 1 + z ^ { 3 } } \end{array} \right| = 0 , {/tex} then the value of {tex} x y z {/tex} is

A

1

B

2

-1

D

-2

##### Explanation

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Q 13. If {tex} x \neq 0 , y \neq 0 , z \neq 0 {/tex} and {tex} \left| \begin{array} { c c c } { 1 + x } & { 1 } & { 1 } \\ { 1 + y } & { 1 + 2 y } & { 1 } \\ { 1 + z } & { 1 + z } & { 1 + 3 z } \end{array} \right| = 0 , {/tex} then {tex} x ^{-1} + y^{-1}+z^{-1}{/tex} is equal to

A

{tex} 1 {/tex}

B

{tex} 2 {/tex}

{tex} - 3 {/tex}

D

none of these

##### Explanation

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Q 14. The value of the determinant {tex} \Delta = \left| \begin{array} { c c c c } { 1 ^ { 2 } } & { 2 ^ { 2 } } & { 3 ^ { 2 } } & { 4 ^ { 2 } } \\ { 2 ^ { 2 } } & { 3 ^ { 2 } } & { 4 ^ { 2 } } & { 5 ^ { 2 } } \\ { 3 ^ { 2 } } & { 4 ^ { 2 } } & { 5 ^ { 2 } } & { 6 ^ { 2 } } \\ { 4 ^ { 2 } } & { 5 ^ { 2 } } & { 6 ^ { 2 } } & { 7 ^ { 2 } } \end{array} \right| {/tex} is equal to

A

1

0

C

2

D

3

##### Explanation

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Q 15. If {tex} D _ { k } = \left| \begin{array} { c c c } { 1 } & { n } & { n } \\ { 2 k } & { n ^ { 2 } + n + 1 } & { n ^ { 2 } + n } \\ { 2 k - 1 } & { n ^ { 2 } } & { n ^ { 2 } + n + 1 } \end{array} \right| {/tex} and {tex} \sum _ { k = 1 } ^ { n } D _ { k } = 56, {/tex} then {tex} n{/tex} equals

A

4

B

6

C

8

none of these

##### Explanation

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Q 16. The value of {tex} \sum _ { r = 2 } ^ { n } ( - 2 ) ^ { r } \left| \begin{array} { c c c } { ^ { n - 2 } C _ { r - 2 } } & { ^{n-2}C _ { r - 1 } } & { ^ { n - 2 } C _ { r } } \\ { - 3 } & { 1 } & { 1 } \\ { 2 } & { - 1 } & { 0 } \end{array} \right| ( n > 2 ) {/tex} is

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

B

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

C

{tex} 2 n - 3 + ( - 1 ) ^ { n } {/tex}

D

none of these

##### Explanation

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Q 17. The value of the determinant {tex} \left| \begin{array} { c c c } { ^ { n } C _ { r - 1 } } & { ^ { n } C _ { r } } & { ( r + 1 ) ^ { n + 2 } C _ { r + 1 } } \\ { ^ { n } C _ { r } } & { ^ { n } C _ { r + 1 } } & { ( r + 2 ) ^ { n + 2 } C _ { r + 2 } } \\ { ^ { n } C _ { r + 1 } } & { ^ { n } C _ { r + 2 } } & { ( r + 3 ) ^ { n + 2 } C _ { r + 3 } } \end{array} \right| {/tex} is

A

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

{tex} 0 {/tex}

C

{tex} ^{n + 3} C _ { r + 3 } {/tex}

D

{tex} ^ { n } C _ { r - 1 } + ^ { n } C _ { r } + ^ { n } C _ { r + 1 } {/tex}

##### Explanation

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Q 18. {tex} \triangle_1{/tex} = {tex} \left| \begin{array} { c c c } { y^5 } z^6(z^3-y^3) & { x^4z^6(x^3-z^3) } & {x^4 y^5(y^3-x^3) } \\ { y^2 z^3(y^6-z^6) } & { xz^3(z^6-x^6) } & {xy^2(x^6-y^6) } \\ { y^2z^3(z^3-y^3) } & { xz^3(x^3-z^3) } & { xy^2(y^3-x^3)} \end{array} \right| {/tex} and
{tex} \triangle_2{/tex}={tex} \left| \begin{array} { c c c } { x} & { y^2 } & {z^3 } \\ { x^4 } & { y^5 } & { z^6 } \\ { x^7 } & { y^8 } & { z^9 } \end{array} \right|{/tex}. Then {tex} \triangle_1{/tex}{tex} \triangle_2{/tex} is equal to

{tex} \triangle{^3}_2{/tex}

B

{tex} \triangle{^2}_2{/tex}

C

{tex} \triangle{^4}_2{/tex}

D

None of these

##### Explanation

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Q 19. The value of the determinant
{tex} \left| \begin{array} { c c c } {( a_1-b_1)^2 } & { ( a_1-b_2)^2 } & { ( a_1-b_3)^2 } & ( a_1-b_4)^2\\ { ( a_2-b_1)^2 } & { ( a_2-b_2)^2 } & { ( a_2-b_3)^2 } & ( a_2-b_4)^2\\ { ( a_3-b_1)^2 } & { ( a_3-b_2)^2 } & { ( a_3-b_3)^2 } & ( a_3-b_4)^2\\ {( a_4-b_1)^2 } &{( a_4-b_2)^2} & {( a_4-b_3)^2} & {( a_4-b_4)^2} \end{array} \right| {/tex} is

A

dependant on {tex} a _ { i } , i = 1,2,3,4 {/tex}

B

dependant on {tex} b _ { i ^ { ,} } i = 1,2,3,4 {/tex}

C

dependant on {tex} a _ { ij } , b_{i} , i = 1,2,3,4 {/tex}

0

##### Explanation

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Q 20. The inverse of a skew-symmetric matrix of odd order is

A

a symmetric matrix

B

a skew symmetric

C

diagonal matrix

does not exist

##### Explanation

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Q 21. Let {tex} A {/tex} and {tex} B {/tex} be two {tex} 2 \times 2 {/tex} matrices. Consider the statements
(i) {tex} A B = O \Rightarrow A = O {/tex} or {tex} B = O {/tex}
(ii) {tex} A B = I_2 , \Rightarrow A = B ^ { - 1 } {/tex}
(iii) {tex} ( A + B ) ^ { 2 } = A ^ { 2 } + 2 A B + B ^ { 2 } {/tex}. Then

A

(i) and (ii) are false, (iii) is true

B

(ii) and (iii) are false, (i) is true

C

(i) is false, (ii) and (iii) are true

(i) and (iii) are false, (ii) is true

##### Explanation

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Q 22. The equation {tex} [1 \ x\ y ]{/tex}{tex} \left[ \begin{array} { l l l l } { 1 } & { 3 } & { 1 } \\ { 0 } & { 2 } & { - 1 } \\ { 0 } & { 0 } & { 1 } \end{array} \right] \left[ \begin{array} { l } { 1 } \\ { x } \\ { y } \end{array} \right] = [ 0 ] {/tex} has
{tex} \begin{array} { l l } { \text { (i) for } y = 0 } & { \text { (p) rational roots } } \\ { \text { (ii) for } y = - 1 } & { \text { (q) irrational roots } } \\ { } & { \text { (r) integral roots } } \end{array} {/tex}
Then

A

B

{tex} (i) -( p ) \quad (ii)-(q){/tex}

D

{tex} (i) -( { r } ) \quad (ii)-(p){/tex}

##### Explanation

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Q 23. If {tex} A = \left[ \begin{array} { l l } { a } & { b } \\ { 0 } & { a } \end{array} \right] {/tex} is {tex} n ^ { \text {th } } {/tex} root of {tex} I _ { 2 } {/tex}, then choose the correct statements:
(i) if {tex} n {/tex} is odd, {tex} a = 1 , b = 0 {/tex}
(ii) if {tex} n {/tex} is odd, {tex} a = - 1 , b = 0 {/tex}
(iii) if {tex} n {/tex} is even, {tex} a = 1 , b = 0 {/tex}
(iv) if {tex} n {/tex} is even, {tex} a = - 1 , b = 0 {/tex}

A

i, ii, iii

B

ii, iii, iv

C

i, ii, iii, iv

i, iii, iv

##### Explanation

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Q 24. The product of matrices {tex} A = \left[ \begin{array} { c c } { \cos ^ { 2 } \theta } & { \cos \theta \sin \theta } \\ { \cos \theta \sin \theta } & { \sin ^ { 2 } \theta } \end{array} \right] {/tex} and {tex} B = \left[ \begin{array} { c c } { \cos ^ { 2 } \phi } & { \cos \phi \sin \phi } \\ { \cos \phi \sin \phi } & { \sin ^ { 2 } \phi } \end{array} \right] {/tex} is a null matrix if {tex} \theta - \phi = {/tex}

A

{tex} 2 n \pi , n \in Z {/tex}

B

{tex} n \frac { \pi } { 2 } , n \in Z {/tex}

{tex} ( 2 n + 1 ) \frac { \pi } { 2 } , n \in Z {/tex}

D

{tex} n \pi , n \in Z {/tex}

##### Explanation

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Q 25. If {tex} A = \left[ a _ { ij } \right] _ { 4 \times 4 } {/tex}, such that {tex}a_{ij}{/tex} ={tex}\begin{cases}2 & i = j\\0 & i \neq j\end{cases}{/tex} then {tex}\left\{{\det\frac{(adj{(adj A))}}{7}}\right\}{/tex} is (where {.} represents fractional part function)

{tex} 1 / 7 {/tex}

B

{tex} 2 /7{/tex}

C

{tex} 3 / 7 {/tex}

D

none of these