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#### Applications of Matrices and Determinants Important Question Paper

12th Standard EM

Reg.No. :
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Time : 01:00:00 Hrs
Total Marks : 50
10 x 1 = 10
1. If A=(1 2 3), then the rank of AAT is

(a)

0

(b)

2

(c)

3

(d)

1

2. The rank of m×n matrix whose elements are unity is

(a)

0

(b)

1

(c)

m

(d)

n

3. if T=$_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.4 } & \overset { B }{ 0.6 } \\ 0.2 & 0.8 \end{matrix} \right)$ is a transition probability matrix, then at equilibrium A is equal to

(a)

$\frac { 1 }{ 4 }$

(b)

$\frac { 1 }{ 5 }$

(c)

$\frac { 1 }{ 6 }$

(d)

$\frac { 1 }{ 8 }$

4. if T= $_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.7 } & \overset { B }{ 0.3 } \\ 0.6 & x \end{matrix} \right)$ is a transition probability matrix, then the value of x is

(a)

0.2

(b)

0.3

(c)

0.4

(d)

0.7

5. Which of the following is not an elementary transformation?

(a)

${ R }_{ i }\leftrightarrow { R }_{ j }$

(b)

${ R }_{ i }\rightarrow { 2R }_{ i }+{ 2c }_{ j }$

(c)

${ R }_{ i }\rightarrow { 2R }_{ i }-{ 4R }_{ i }$

(d)

${ C }_{ i }\rightarrow { C }_{ i }+{ 5C }_{ j }$

6. if $\left| A \right| \neq 0,$ then A is

(a)

non- singular matrix

(b)

singular matrix

(c)

zero matrix

(d)

none of these

7. For what value of k, the matrix $A=\left( \begin{matrix} 2 & k \\ 3 & 5 \end{matrix} \right)$ has no inverse?

(a)

$\cfrac { 3 }{ 10 }$

(b)

$\cfrac { 10 }{ 3 }$

(c)

3

(d)

10

8. The rank of an n x n matrix each of whose elements is 2 is

(a)

1

(b)

2

(c)

n

(d)

n2

9. The value of $\left| \begin{matrix} { 5 }^{ 2 } & { 5 }^{ 3 } & { 5 }^{ 4 } \\ { 5 }^{ 3 } & { 5 }^{ 4 } & { 5^{ 5 } } \\ { 5 }^{ 4 } & { 5 }^{ 5 } & { 5 }^{ 6 } \end{matrix} \right|$

(a)

52

(b)

0

(c)

513

(d)

59

10. If $\left| \begin{matrix} 2x & 5 \\ 8 & x \end{matrix} \right| =\left| \begin{matrix} 6 & -2 \\ 7 & 3 \end{matrix} \right|$ then x =

(a)

3

(b)

± 3

(c)

± 6

(d)

6

11. 5 x 2 = 10
12. Find the rank of the following matrices.
$\left( \begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix} \right)$

13. If A=$\left( \begin{matrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{matrix} \right)$ and B=$\left( \begin{matrix} 1 & -2 & 3 \\ -2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right)$, then find the rank of AB and the rank
of BA.

14. Solve the following system of equations by rank method
x+y+z=9,2x+5y+7z=52,2x−y−z =0

15. Find the rank of the matrix $\left[ \begin{matrix} 7 & -1 \\ 2 & 1 \end{matrix} \right]$

16. Find the rank of the matrix $\left( \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right)$

17. 5 x 3 = 15
18. Find the rank of the matrix $\begin{pmatrix} 1 & 5 \\ 3 & 9 \end{pmatrix}$

19. Find the rank of the matrix $\begin{pmatrix} -5 & -7 \\ 5 & 7 \end{pmatrix}$

20. Find the rank of the matrix $\left( \begin{matrix} 0 & -1 & 5 \\ 2 & 4 & -6 \\ 1 & 1 & 5 \end{matrix} \right)$

21. Find the rank of the matrix
$A=\left( \begin{matrix} 2 & 4 & 5 \\ 4 & 8 & 10 \\ -6 & -12 & -15 \end{matrix} \right)$

22. Findtherankofthematrix $A=\left( \begin{matrix} 1 & 2 & -4 \\ 2 & -1 & 3 \\ 8 & 1 & 9 \end{matrix}\begin{matrix} 5 \\ 6 \\ 7 \end{matrix} \right)$

23. 3 x 5 = 15
24. Solve the equations 2x + 3y = 7, 3x + 5y = 9 by Cramer’s rule.

25. The following table represents the number of shares of two companies A and B during the month of January and February and it also gives the amount in rupees invested by Ravi during these two months for the purchase of shares of two companies. Find the the price per share of A and B purchased during both the months

 Months Number of Shares of the company Amount invested by Ravi (in Rs) A B January 10 5 125 February 9 12 150
26. The sum of three numbers is 6. If we multiplythe third number by 2 and add the first number to the result we get 7. By adding second and third numbers to three times the first number we get 12. Find the numbers using rank method