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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

{tex}C{/tex} is empty

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

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

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

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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

no solution

unique solution

infinitely many solutions

finitely many solutions

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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

{tex} a {/tex}

{tex} p {/tex}

{tex} d {/tex}

{tex} x {/tex}

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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

{tex} - 1,2 {/tex}

{tex} 1,2 {/tex}

{tex} 0,1 {/tex}

{tex} - 1,1 {/tex}

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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

1

- 1

4

no real values

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Q 6. For the values of the value of

is

{tex}0{/tex}

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

0

1

2

None of these

We have,

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Q 8.

{tex}0{/tex}

None of the above

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Q 9. If

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Q 10. If

We have,

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Q 11. The value of

We have,

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Q 12. If

Given,

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Q 13.

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Q 14. The repeated factor of the determinant

None of these

We have,

Hence, the repeating factor is

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Q 15. The value of

0

1

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Q 16. The value of the determinant,

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Q 17. The value of the determinant

Independent of

Independent of

Independent of

None of these

Applying

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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

1

9

16

24

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

1

3

4

2

On comparing with

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Q 20. If

Depends upon the function

is 4

is

is zero

So,

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Q 21. A root of the equation

1

On expanding the given determinant, we obtain

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Q 22. If

0

1

None of these

[

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Q 23. The value of

Applying

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Q 24. The value of the determinant

Let

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Q 25. If

0

We have,

and

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