#### Matrices and Determinants Model Question Paper

11th Standard

Reg.No. :
•
•
•
•
•
•

Maths

Time : 01:30:00 Hrs
Total Marks : 50
5 x 1 = 5
1. If A=$\begin{bmatrix}\lambda & 1 \\ -1 & -\lambda \end{bmatrix}$ ,then for what value of $\lambda$, A2 = O?

(a)

0

(b)

$\pm 1$

(c)

-1

(d)

1

2. If A is a square matrix, then which of the following is not symmetric?

(a)

A+ AT

(b)

AAT

(c)

AT A

(d)

A− AT

3. If A and B are symmetric matrices of order n, where (A $\neq$ B), then

(a)

A + B is skew-symmetric

(b)

A + B is symmetric

(c)

A + B is a diagonal matrix

(d)

A + B is a zero matrix

4. If the points (x,−2), (5, 2), (8,8) are collinear, then x is equal to

(a)

-3

(b)

${1\over 3}$

(c)

1

(d)

3

5. If A is skew-symmetric of order n and C is a column matrix of order n × 1, then CT AC is

(a)

an identity matrix of order n

(b)

an identity matrix of order 1

(c)

a zero matrix of order 1

(d)

an identity matrix of order 2

6. 7 x 2 = 14
7. Suppose that a matrix has 12 elements. What are the possible orders it can have? What if it has 7 elements?

8. If A=$\begin{bmatrix} 0 &c &b \\ c & 0 &a \\ b & a & 0 \end{bmatrix}$,compute A2

9. Construct an m × n matrix A= [aij], where a ij is given by
$a_{ij}={|3i-4j|\over 4}with \ m=3,n=4$

10. If A=$\begin{bmatrix} 4 & 2 \\ -1 & x \end{bmatrix}$ and such that (A- 2I)(A-3I)=O, find the value of x.

11. Show that the points (a, b + c), (b, c + a), and (c, a + b) are collinear

12. If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k.

13. In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section of Express the given information as a column matrix. Using sclar multiplication find the total number ofp of each kind in all the colleges.

14. 7 x 3 = 21
15. Solve for x if [x 2 -1] $\begin{bmatrix} 1&1 &2 \\ -1 & -4 &1 \\ -1 &-1 &-2 \end{bmatrix}$$\begin{bmatrix} x \\ 2 \\ 1 \end{bmatrix}$=0

16. A fruit shop keeper prepares 3 different varieties of gift packages. Pack-I contains 6 apples, 3 oranges, and 3 pomegranates. Pack-II contains 5 apples, 4 oranges, and 4 pomegranates and Pack –III contains 6 apples, 6 oranges, and 6 pomegranates. The cost of an apple, an orange, and a pomegranate respectively are Rs. 30, Rs. 15 and Rs. 45. What is the cost of preparing each package of fruits?

17. If A =$\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}$ and B= $\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}$
verify(AB)T=BTAT

18. If A =$\begin{bmatrix} 4 & 6 & 2 \\ 0 & 1 & 5 \\ 0 & 3 & 2 \end{bmatrix}$ and B= $\begin{bmatrix} 0 & 1 & -1 \\ 3 & -1 & 4 \\ -1 & 2 & 1 \end{bmatrix}$
verify(A-B)T=AT-BT

19. If AT=$\begin{bmatrix} 4 & 5 \\ -1 & 0 \\ 2 & 3 \end{bmatrix}$ and B= $\begin{bmatrix} 2 & -1&1 \\7 & 5&-2 \end{bmatrix}$,verify (A+B)T=AT+BT=BT+AT

20. Identify the singular and non-singular matrices:$\begin{bmatrix} 0&a-b &k \\ b-a & 0 &5 \\ -k & -5 & 0 \end{bmatrix}$

21. If A=$\left[ \begin{matrix} \alpha & 0 \\ 1 & 1 \end{matrix} \right]$ and B=$\left[ \begin{matrix} 1 & 0 \\ 5 & 1 \end{matrix} \right]$ find the values of $\alpha$ for which A2=B.

22. 2 x 5 = 10
23. If A=$\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y \end{bmatrix}$ is a matrix such that AAT =9I, find the values of x and y.

24. If $\lambda =-2$ , determine the value of $\begin{vmatrix} 0& 2\lambda &1 \\ \lambda^2 &0 &3\lambda^3+1 \\ -1 &6\lambda-1 &0 \end{vmatrix}$ .