#### Important Question paper

11th Standard

Reg.No. :
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Maths

Use Blue Pen only
Time : 01:00:00 Hrs
Total Marks : 60

Part-A

10 x 1 = 10
1. If aij =${1\over2}(3i-2j)$ and A=[aij]2x2 is

(a)

$\begin{bmatrix} {1\over 2}& 2 \\ -{1\over2} & 1 \end{bmatrix}$

(b)

$\begin{bmatrix} {1\over 2}& -{1\over2} \\ 2& 1 \end{bmatrix}$

(c)

$\begin{bmatrix} 2& 2\\ {1\over 2}& -{1\over2} \end{bmatrix}$

(d)

$\begin{bmatrix} -{1\over 2}& {1\over2} \\ 1& 2 \end{bmatrix}$

2. What must be the matrix X, if 2x+$\begin{bmatrix} 1& 2 \\ 3 & 4 \end{bmatrix}=\begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}?$

(a)

$\begin{bmatrix} 1& 3 \\ 2 &-1 \end{bmatrix}$

(b)

$\begin{bmatrix} 1& -3 \\ 2 &-1 \end{bmatrix}$

(c)

$\begin{bmatrix} 2& 6 \\ 4 &-2 \end{bmatrix}$

(d)

$\begin{bmatrix} 2& -6 \\ 4 &-2 \end{bmatrix}$

3. The value of the determinant of A=$\begin{bmatrix} 0&a &-b \\ -a & 0 & c \\ b & -c & 0 \end{bmatrix}is$

(a)

-2abc

(b)

abc

(c)

0

(d)

a2+b2+c2

4. If x1,x2,x3 as well as y1,y2,y3 are in geometric progression with the same common ratio, then the points (x1, y1 ), (x2,y2), (x3,y3 ) are

(a)

vertices of an equilateral triangle

(b)

vertices of a right angled triangle

(c)

vertices of a right angled isosceles triangle

(d)

collinear

5. If a$\neq$b,b,c satisfy $\begin{vmatrix} a&2b &2c \\3 & b & c \\ 4 & a & b \end{vmatrix}=0,$ then abc=

(a)

a + b + c

(b)

0

(c)

b3

(d)

ab+bc

6. The value of $\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}$ is

(a)

$\overrightarrow{AD}$

(b)

$\overrightarrow{CA}$

(c)

$\overrightarrow{0}$

(d)

$-\overrightarrow{AD}$

7. If $\overrightarrow{a}+2\overrightarrow{b}$ and $3\overrightarrow{a}+m\overrightarrow{b}$ are parallel, then the value of m is

(a)

3

(b)

$1\over3$

(c)

6

(d)

$1\over6$

8. If $|\overrightarrow{a}|=13,|\overrightarrow{b}|=5$  and $\overrightarrow{a}.\overrightarrow{b}=60^o$ then $|\overrightarrow{a}\times\overrightarrow{b}|$ is

(a)

15

(b)

35

(c)

45

(d)

25

9. Vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are inclined at an angle $\theta =120^o$ .If $|\overrightarrow{a}|=1,|\overrightarrow{b}|=2,$ then $[(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})]^2$ is equal to

(a)

225

(b)

275

(c)

325

(d)

300

10. If $\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${\pi\over 6},$ then the area of the triangle formed by these two vectors as two sides, is

(a)

$7\over4$

(b)

$15\over4$

(c)

$3\over4$

(d)

$17\over4$

11. Part-B

10 x 2 = 20
12. Simplify : $sec \theta \begin{bmatrix} sec \theta & tan\theta \\ tan\theta & sec\theta \end{bmatrix}$-$tan \theta \begin{bmatrix} tan \theta & sec\theta \\ sec\theta & tan\theta \end{bmatrix}$

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

14. Consider the matrix Aa=$\begin{bmatrix} cos \alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{bmatrix}$
Find all possible real values of α satisfying the condition $A\alpha +A^T_{\alpha}=I$

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

16. Can a vector have direction angles 30°,45°,60°?

17. Find the direction cosines of $\overrightarrow{AB},$where A is (2, 3, 1) and B is (3, - 1, 2).

18. Find the direction cosines of the line joining (2, 3, 1) and (3, - 1, 2).

19. If $\overrightarrow{a}$ and $\overrightarrow{b}$are two vectors such that | $\overrightarrow{a}$ |=10,| $\overrightarrow{b}$ |=15 and $\overrightarrow{a}$.$\overrightarrow{b}$ = 75 $\sqrt{2}$, find the angle between $\overrightarrow{a}$and $\overrightarrow{b}$.

20. Find the angle between the vectors  $2\hat{i}+3\hat{j}-6\hat{k}$ and $6\hat{i}-3\hat{j}+2\hat{k}$

21. Five mangoes and 4 apples are in a box. If two fruits are chosen at random, find the probability that (i) one is a mango and the other is an apple (ii) both are of the same variety

22. Part-C

5 x 3 = 15
23. 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

24. 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

25. 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

26. If A1, B1, C1 are the cofactors of a1, b2,c3, respectively, i = 1 to 3 in
|A|=$\begin{vmatrix} a_1 &b_1 &c_1 \\ a_2 & b_2 &c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$, show that $\begin{vmatrix} A_1 &B_1 &C_1 \\ A_2 & B_2 &C_2 \\ A_3 & B_3 & C_3 \end{vmatrix}$ =|A|2

27. Identify the singular and non-singular matrices:$\begin{bmatrix} 1&2 &3 \\ 4 & 5 &6 \\ 7 & 8 & 9 \end{bmatrix}$

28. Part-D

29. If $\begin{bmatrix} 0 & p& 3 \\ 2 & q^2 & -1 \\ r & 1 & 0 \end{bmatrix}$ is skew-symmetric, find the values of p,q, and r.
30. Construct the matrix $A=[a_{ij}]_{3\times 3}$, where $a_{ij}=i-j.$ State whether A is symmetric or skew-symmetric.