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JEE Advanced

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.

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Matrices and Determinants

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Q 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

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Q 2. 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









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Q 3. 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







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Q 4. 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

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Q 5. 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

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Q 6. For the values of the value of
is

{tex}0{/tex}

B

C

D

Explanation



The determinants can be rewritten as 8 determinants and the value of each of these 8 determinants is zero.

Similarly, other determinants can be shown zero.

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Q 7. If are in AP, then the value of the det is, where

0

B

1

C

2

D

None of these

Explanation

We have,

[Applying ]
[Applying ]

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Q 8. is equal to

A

{tex}0{/tex}

C

D

None of the above

Explanation

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Q 9. If and then the value of is

A

B

D

Explanation



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Q 10. If and are the given determinants, then

A

C

D

Explanation

We have,

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Q 11. The value of is equal to

A

C

D

Explanation

We have,




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Q 12. If are non-zero real numbers, then vanishes, when

B

C

D

Explanation

Given,




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Q 13. =0, then

is one of the cube roots of unity

B

is one of the cube roots of unity

C

is one of the cube roots of unity

D

is one of the cube roots of unity

Explanation


is one of the cube roots of unity.

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Q 14. The repeated factor of the determinant
is

B

C

D

None of these

Explanation

We have,




Hence, the repeating factor is

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Q 15. The value of is equal to

0

B

1

C

D

Explanation






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Q 16. The value of the determinant, is

A

C

D

Explanation


 

 


 

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Q 17. The value of the determinant
is

Independent of

B

Independent of

C

Independent of and

D

None of these

Explanation

Applying


which is independent of

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Q 18. In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then, it can be decomposed into determinant, where has the value

A

1

B

9

C

16

24

Explanation

Since, the first column consists of sum of two terms, second column consists of sum of three terms and third column consists of sum four terms.

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Q 19. If then is equal to

A

1

B

3

4

D

2

Explanation



On comparing with , we get

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Q 20. If (where is a real valued function of then the value of

A

Depends upon the function

B

is 4

C

is

is zero

Explanation



So, is an odd function.
is an odd function

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Q 21. A root of the equation is

A

B

D

1

Explanation

On expanding the given determinant, we obtain

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Q 22. If then the value of

0

B

1

C

D

None of these

Explanation



[ two columns are identical]

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Q 23. The value of is equal to

A

C

D

Explanation

Applying we obtain




[Expanding along ]

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Q 24. The value of the determinant is

A

B

D

Explanation

Let

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Q 25. If to terms, to terms and to terms, then equals

A

B

0

D

Explanation

We have, upto terms

upto terms=
and upto terms=


two rows are identical]