Matrices And Determinants - Important Question Paper

11th Standard

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Time : 01:00:00 Hrs
Total Marks : 50

Part A

10 x 1 = 10
1. The value of x, if $\begin{vmatrix} 0 & 1 & 0 \\ x & 2 & x \\ 1 & 3 & x \end{vmatrix}=0$ is

(a)

0, - 1

(b)

0, 1

(c)

- 1, 1

(d)

- 1, - 1

2. The value of the determinant ${\begin{vmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & c \end{vmatrix}}^{2}$is

(a)

abc

(b)

0

(c)

a2b2c2

(d)

-abc

3. The number of Hawkins-Simon conditions for the viability of an input - output analysis is

(a)

1

(b)

3

(c)

4

(d)

2

4. Which of the following matrix has no inverse

(a)

$\begin{pmatrix} -1 & 1 \\ 1 &-4 \end{pmatrix}$

(b)

$\begin{pmatrix} 2 & -1 \\ -4 &2 \end{pmatrix}$

(c)

$\begin{pmatrix} cos\ a & sin\ a \\ -sin\ a & cos\ a \end{pmatrix}$

(d)

$\begin{pmatrix} sin\ a & cos\ a \\ -cos\ a & sin\ a \end{pmatrix}$

5. The Inverse of matrix of$\begin{pmatrix} 3 & 1 \\ 5 & 2\end{pmatrix}$ is

(a)

$\begin{pmatrix} 2 & -1 \\-5 & 3 \end{pmatrix}$

(b)

$\begin{pmatrix} -2 & 5 \\1 & -3 \end{pmatrix}$

(c)

$\begin{pmatrix} 3 & -1 \\-5 & -3 \end{pmatrix}$

(d)

$\begin{pmatrix} -3 & 5 \\1 & -2 \end{pmatrix}$

6. If A $=\begin{pmatrix} -1 & 2 \\ 1 & -4 \end{pmatrix}$ then A (adj A) is

(a)

$\begin{pmatrix} -4 & -2 \\ -1 & -1 \end{pmatrix}$

(b)

$\begin{pmatrix} 4 & -2 \\ -1 & 1 \end{pmatrix}$

(c)

$\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$

(d)

$\begin{pmatrix} 0 & 2 \\ 2 & 0 \end{pmatrix}$

7. If A is an invertible matrix of order 2, then det (A-1) be equal to

(a)

det (A)

(b)

${{1}\over{det(A)}}$

(c)

1

(d)

0

8. If A is a square matrix of order 3 and IAI = 3 then | adj A| is equal to

(a)

81

(b)

27

(c)

3

(d)

9

9. If $\begin{vmatrix} x & 2 \\ 8 &5 \end{vmatrix}=0$ then the value of x is

(a)

${{-5}\over{6}}$

(b)

${{5}\over{6}}$

(c)

${{-16}\over{5}}$

(d)

${{16}\over{5}}$

10. If $\begin{vmatrix} 4 & 3 \\ 3 & 1 \end{vmatrix}=-5$ then value of $\begin{vmatrix} 20 & 15 \\ 15 & 5 \end{vmatrix}$ is

(a)

-5

(b)

-125

(c)

-25

(d)

0

11. Part B

6 x 2 = 12
12. Suppose the inter-industry flow of the product of two sectors X and Yare given as under.

Production Sector Consumption Sector Domestic demand Gross output
X Y
X 15 10 10 35
Y 20 30 15 65

Find the gross output when the domestic demand changes to 12 for X and 18 for Y.

13. Solve: $\begin{vmatrix}2& x&3\\4&1&6\\1&2&7 \end{vmatrix}=0$

14. Two commodities A and B are produced such that 0.4 tonne of A and 0.7 tonne of B are required to produce a tonne of A. Similarly 0.1 tonne of A and 0.7 tonne of B are needed to produce a tonne of B. Write down the technology matrix. If 6.8 tonnes of A and 10.2 tonnes of B are required, find the gross production of both of them.

15. If $A=\begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$ then show that |2A| = 4 |A|.

16. Using the property of determinant, evaluate $\begin{vmatrix} 6 &5 &12 \\ 2 & 4 &4 \\2 & 1 & 4 \end{vmatrix}.$

17. Using the property of determinants show that $\begin{vmatrix} x &a &x+a \\ y & b &y+b \\z & c & z+c \end{vmatrix}=0.$

18. Part C

6 x 3 = 18
19. If$A=\begin{bmatrix}1&3&3\\1&4&3\\1&3&4 \end{bmatrix}$then verify that A (adj A) = |A| I and also find A-1.

20. Find the inverse of $\begin{bmatrix}-1 & 5 \\-3 & 2 \end{bmatrix}.$

21. Solve: 2x+ 5y = 1 and 3x + 2y = 7 using matrix method.

22. Show that $\begin{vmatrix}x+a &b&c \\a &x+b&c\\a&b&x+c \end{vmatrix}=x^2(x+a+b+c)$

23. Using the properties of determinants, show that $\left| \begin{matrix} 2 & 7 & 65 \\ 3 & 8 & 75 \\ 5 & 9 & 86 \end{matrix} \right|$=0

24. Find the adjoint of the matrix $\left[ \begin{matrix} 2 & -1 & 3 \\ 0 & 5 & 1 \\ 3 & 6 & 8 \end{matrix} \right]$

25. Part D

2 x 5 = 10
26. If $A=\left[ \begin{matrix} 1 & tan\quad x \\ -tan\quad x & \quad \quad \quad 1 \end{matrix} \right]$, then show that ATA-1=$\left[ \begin{matrix} cos\quad 2x & -sin2x \\ sin\quad 2x & cos2x \end{matrix} \right] .$

27. Solve by matrix inversion method: 3x - y + 2z = 13 ; 2x + Y - z = 3 ; x + 3y - 5z = - 8.