#### Matrices And Determinants Important Question Paper

11th Standard

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Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
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. If $\triangle=\begin{vmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{vmatrix}$ then $\begin{vmatrix} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \end{vmatrix}$ is

(a)

$\triangle$

(b)

-$\triangle$

(c)

3$\triangle$

(d)

-3$\triangle$

3. 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

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. 6 x 2 = 12
7. The technology matrix of an economic system of two industries is$\begin{bmatrix} 0.50 & 0.30 \\ 0.41 & 0.33 \end{bmatrix}$. Test whether the system is viable as per Hawkins Simon conditions.

8. The technology matrix of an economic system of two industries is$\begin{bmatrix} 0.6 & 0.9 \\ 0.20 & 0.80 \end{bmatrix}$ .Test whether the system is viable as per Hawkins-Simon conditions.

9. The technology matrix of an economic system of two industries is $\begin{bmatrix} 0.50 & 0.25 \\ 0.40 & 0.67 \end{bmatrix}$. Test whether the system is viable as per Hawkins-Simon conditions.

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

11. Find the values of x if $\begin{vmatrix} 2 & 4 \\5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4\\6 & x \end{vmatrix}.$

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

13. 6 x 3 = 18
14. Solve: $\begin{vmatrix} x & 2 & -1 \\ 2 & 5 & x \\ -1 & 2 & x \end{vmatrix}=0.$

15. Without actual expansion show that the value of the determinant $\begin{vmatrix}5 &5^2 &5^3 \\5^2 & 5^3 & 5^4\\5^4&5^5&5^6 \end{vmatrix}$is zero.

16. Show that $\begin{vmatrix}0 &ab^2 &ac^2 \\a^2b & 0 & bc^2\\a^2c&b^2c&0\end{vmatrix}=2a^3b^3c^3.$

17. Using matrix method, solve x+2y+z=7, x+3z = 11 and 2x-3y =1.

18. if A=$\left[ \begin{matrix} cos\ \alpha & sin\ \alpha \\ -sin\ \alpha & \ cos\ \alpha \ \end{matrix} \right]$ is such that AT = A-1, find $\alpha$

19. Show that $\begin{vmatrix} a & a+b&a+b+c \\2a &3a+2b &4a+3b+2c\\3a&6a+3b&10a+6b+3c \end{vmatrix}=a^3.$

20. 3 x 5 = 15
21. Evaluate:$\begin{vmatrix} 1&a&a^2-bc\\1&b&b^2-ca\\1&c&c^2-ab \end{vmatrix}$

22. If A = $\begin{bmatrix}3 & -1 & 1 \\ -15 & 6 & -5\\5 & -2 & 2 \end{bmatrix}$ then, find the Inverse of A.

23. Let a, b and c denote the sides BC, CA and AB repectively of $\Delta$ ABC. If $\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| =0$, then find the value of sin2 A+sin2B+sin2C.