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#### Model question-Vector Algebra - I

11th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 60

Part A

10 x 1 = 10
1. The value of $\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}$ is

(a)

$\overrightarrow{AD}$

(b)

$\overrightarrow{CA}$

(c)

$\overrightarrow{0}$

(d)

$-\overrightarrow{AD}$

2. The unit vector parallel to the resultant of the vectors $\hat{i}+\hat{j}-\hat{k}$ and $\hat{i}-2\hat{j}+\hat{k}$ is

(a)

${\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}$

(b)

${2\hat{i}+\hat{j}\over\sqrt{5}}$

(c)

${2\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}$

(d)

${2\hat{i}-\hat{j}\over\sqrt{5}}$

3. The vectors $\overrightarrow{a}-\overrightarrow{b},\overrightarrow{b}-\overrightarrow{c},\overrightarrow{c}-\overrightarrow{a}$ are

(a)

parallel to each other

(b)

unit vectors

(c)

mutually perpendicular vectors

(d)

coplanar vectors.

4. One of the diagonals of parallelogram ABCD with $\overrightarrow{a}$ and $\overrightarrow{b}$ as adjacent sides is $\overrightarrow{a}+\overrightarrow{b}$The other diagonal $\overrightarrow{BD}$ is

(a)

$\overrightarrow{a}-\overrightarrow{b}$

(b)

$\overrightarrow{b}-\overrightarrow{a}$

(c)

$\overrightarrow{a}+\overrightarrow{b}$

(d)

$\overrightarrow{a}+\overrightarrow{b}\over 2$

5. If $\overrightarrow{a},\overrightarrow{b}$ are the position vectors A and B, then which one of the following points whose position vector lies on AB, is

(a)

$\overrightarrow{a}+\overrightarrow{b}$

(b)

${2\overrightarrow{a}-\overrightarrow{b}\over 2}$

(c)

${2\overrightarrow{a}+\overrightarrow{b}\over 3}$

(d)

${\overrightarrow{a}-\overrightarrow{b}\over 3}$

6. Two vertices of a triangle have position vectors $3\hat{i}+4\hat{j}-4\hat{k}$ and$2\hat{i}+3\hat{j}+4\hat{k}$If the position vector of the centroid is $\hat{i}+2\hat{j}+3\hat{k}$ ,then the position vector of the third vertex is

(a)

$-2\hat{i}-\hat{j}+9\hat{k}$

(b)

$-2\hat{i}-\hat{j}-6\hat{k}$

(c)

$2\hat{i}-\hat{j}+6\hat{k}$

(d)

$-2\hat{i}+\hat{j}+6\hat{k}$

7. If $\overrightarrow{a}$  and $\overrightarrow{b}$ having same magnitude and angle between them is 60° and their scalar product is ${1\over2}$ then $|\overrightarrow{a}|$ is

(a)

2

(b)

3

(c)

7

(d)

1

8. If   $\overrightarrow{a}$  and   $\overrightarrow{b}$ are two vectors of magnitude 2 and inclined at an angle 60° , then the angle between   $\overrightarrow{a}$  and $\overrightarrow{a}+\overrightarrow{b}$ is

(a)

30°

(b)

60°

(c)

45°

(d)

90°

9. If the projection of $5\hat{i}-\hat{j}-3\hat{k}$ on the vector $\hat{i}+3\hat{j}+\lambda\hat{k}$ is same as the projection of $\hat{i}+3\hat{j}+\lambda\hat{k}$ on $5\hat{i}-\hat{j}-3\hat{k}$then $\lambda$ is equal to

(a)

$\pm 4$

(b)

$\pm 3$

(c)

$\pm 5$

(d)

$\pm 1$

10. If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+x\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}$ and $\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=70,$ then x is equal to

(a)

5

(b)

7

(c)

26

(d)

10

11. Part B

10 x 2 = 20
12. Represent graphically the displacement of 60 km 50° south of east.

13. Represent graphically the displacement of 80km, 60° south of west

14. Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be the position vectors of the points A and B. Prove that the position vectors of the points which trisects the line segment AB are ${\overrightarrow{a}+2\overrightarrow{b}\over 3} and \ {\overrightarrow{b}+2\overrightarrow{a}\over3}.$

15. Find a direction ratio and direction cosines of the following vectors $3\hat{i}+4\hat{j}-6\hat{k}$

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

17. The direction ratios of a vector are 2, 3, 6 and it’s magnitude is 5. Find the vector

18. Find the direction cosines of a vector whose direction ratios are 1,2,3

19. Find the angle between the vectors $\hat{i}-\hat{j}$and$\hat{j}-\hat{k}$.

20. Find the vectors of magnitude $10\sqrt{3}$ that are perpendicular to the plane which contains $\hat{i}+2\hat{j}+\hat{k}$ and$\hat{i}+3\hat{j}+4\hat{k}$

21. Find the unit vectors perpendicular to each of the vectors $\overrightarrow{a}+\overrightarrow{b}$ and$\overrightarrow{a}-\overrightarrow{b}$ ,where $\overrightarrow{a}=\hat{i}+\hat{j} +\hat{k}$ and$\overrightarrow{b} =\hat{i}+2\hat{j} +3\hat{k}$.

22. Part C

5 x 3 = 15
23. If $\overrightarrow{PO}$ +$\overrightarrow{OQ}$ = $\overrightarrow{QO}$ +$\overrightarrow{OR}$, prove that the points P, Q, R are collinear.

24. Show that the points whose position vectors are 2$\hat{i}$+ 3$\hat{j}$− 5$\hat{k}$, 3$\hat{i}$+ $\hat{j}$− 2$\hat{k}$ and, 6$\hat{i}$− 5 $\hat{j}$+ 7$\hat{k}$ are collinear

25. If (a,a+b,a+b+c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c.

26. Show that the following vectors are coplanar 5$\hat{i}$ +6$\hat{j}$ +7$\hat{k}$ ,7 $\hat{i}$ -8$\hat{j}$ +9 $\hat{k}$,3$\hat{i}$+20$\hat{j}$ +5$\hat{k}$ .

27. If $\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k},$$\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}+\hat{j}$ be such that $\overrightarrow{a}+\lambda \overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$ then find $\lambda$.

28. Part D

3 x 5 = 15
29. If $\overrightarrow{a},\overrightarrow{b}$are unit vectors and $\theta$ is the angle between them, show that $cos {\theta \over 2}={1\over2}|\overrightarrow{a}+\overrightarrow{b}|$

30. If $\overrightarrow{a},\overrightarrow{b}$are unit vectors and $\theta$ is the angle between them, show that $tan {\theta \over 2}={|\overrightarrow{a}-\overrightarrow{b}|\over|\overrightarrow{a}+\overrightarrow{b}|}$

31. Three vectors $\overrightarrow{a},\overrightarrow{b}$and $\overrightarrow{c}$ are such that $|\overrightarrow{a}|=2,|\overrightarrow{b}|=3,|\overrightarrow{c}|=4,$ and $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$ .Find $4\overrightarrow{a}.\overrightarrow{b}+​​3\overrightarrow{b}.\overrightarrow{c}+3\overrightarrow{c}.\overrightarrow{a}.$