#### Vector Algebra - I Model Question Paper

11th Standard

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Maths

Time : 01:30:00 Hrs
Total Marks : 50
7 x 1 = 7
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. A vector $\overrightarrow{OP}$ makes 60° and 45° with the positive direction of the x and y axes respectively.  Then the angle between $\overrightarrow{OP}$and the z-axis is

(a)

45°

(b)

60°

(c)

90°

(d)

30°

3. 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)

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

4. The value of  $\theta \in (0,{\pi\over 2})$ for which the vectors $\overrightarrow{a}=(sin \theta)\hat{i}+(cos\theta)\hat{j}$ and $\overrightarrow{b}=\hat{i}-\sqrt{3}\hat{j}+2\hat{k}$ are perpendicular, is equal to

(a)

${\pi\over 3}$

(b)

${\pi\over 6}$

(c)

${\pi\over 4}$

(d)

${\pi\over 2}$

5. If $|\overrightarrow { a } |=|\overrightarrow { b } |$ then

(a)

$\overrightarrow { a } =\overrightarrow { b }$

(b)

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

(c)

$\overrightarrow { a } =\pm \overrightarrow { b }$

(d)

both are null vectors

6. If the direction cosines of a line are k, k and k, then

(a)

k>0

(b)

0<k<1

(c)

k=1

(d)

$k=\frac { 1 }{ \sqrt { 3 } } or-\frac { 1 }{ \sqrt { 3 } }$

7. The value of $\lambda$ when the vectors $\vec{a}=2\vec{i}+\lambda\vec{j}+\vec{k}$ and $​​\vec{b}=\vec{i}+2\vec{j}+3\vec{k}$ are orthogonal is

(a)

0

(b)

1

(c)

$\frac{3}{2}$

(d)

-$\frac{5}{2}$

8. 8 x 2 = 16
9. Represent graphically the displacement of 30 km 60° west of north

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

11. 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 ${\overrightarrow{a}+2\overrightarrow{b}\over 3} and \ {\overrightarrow{b}+2\overrightarrow{a}\over3}.$

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

13. Verify whether the following ratios are direction cosines of some vector or not${1\over\sqrt{2}},{1\over 2},{1\over 2}$

14. Find $\overrightarrow{a}$.$\overrightarrow{b}$when $\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\overrightarrow{b}=3\hat{i}-4\hat{j}-2\hat{k}$

15. Show that $\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{c})+\overrightarrow{b}\times (\overrightarrow{c}+\overrightarrow{a})+\overrightarrow{c}\times (\overrightarrow{a}+\overrightarrow{b})=\overrightarrow{0}$

16. If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\vec{b}=2\hat{i}+3\hat{j}-5\hat{k}$ then find $\vec{a} \times \vec{b}$ . Verify that$\vec{a}$ and $\vec{b}$ are perpendicular to each other.

17. 4 x 3 = 12
18. If $\overrightarrow{PO}$ +$\overrightarrow{OQ}$ = $\overrightarrow{QO}$ +$\overrightarrow{OR}$, prove that the points P, Q, R are collinear.

19. If $\overrightarrow{a}=2\hat{i}+3\hat{j}-4\hat{k},$ $\overrightarrow{b}=3\hat{i}-4\hat{j}-5\hat{k},$  and $\overrightarrow{c}=-3\hat{i}+2\hat{j}+3\hat{k},$find the magnitude and direction cosines of 3$\overrightarrow{a}$ − 2$\overrightarrow{b}$ + 5$\overrightarrow{c}$

20. Find the angle between the vectors $5\hat{i}+3\hat{j}+4\hat{k}$and$6\hat{i}-8\hat{j}-\hat{k}$.

21. If $(\overrightarrow { a } +\overrightarrow { b } ).(\overrightarrow { a } -\overrightarrow { b } )=0$ =0, then prove that $\left| \overrightarrow { a } \right| =|\overrightarrow { b } |$

22. 3 x 5 = 15
23. If D is the midpoint of the side BC of a triangle ABC, prove that $\overrightarrow{AB}$ + $\overrightarrow{AC}$ = 2$\overrightarrow{AD}$ .

24. If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then prove that $\overrightarrow{AB}$ + $\overrightarrow{AD}$ +$\overrightarrow{CB}$+$\overrightarrow{CD}$ = 4$\overrightarrow{EF}$.

25. Show that the points (2, - 1, 3), (4, 3, 1) and (3, 1, 2) are collinear.