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#### Model question paper 6

11th Standard

Reg.No. :
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Maths

Use blue pen Only

Time : 00:50:00 Hrs
Total Marks : 55

Part A

5 x 1 = 5
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. 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)

$1\over6$

3. If $\overrightarrow{r}={9\overrightarrow{a}+7\overrightarrow{b}\over16}$ ,then the point P whose position vector $\overrightarrow{r}$divides the line joining the points with position vectors $\overrightarrow{a}$and $\overrightarrow{b}$ in the ratio

(a)

7: 9 internally

(b)

9: 7 internally

(c)

9:7 externally

(d)

7:9 externally

4. 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)

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

5. 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

6. Part B

5 x 2 = 10
7. Represent graphically the displacement of 30 km 60° west of north

8. Prove that the relation R defined on the set V of all vectors by ‘ $\overrightarrow{a}R \overrightarrow{b} \ if \overrightarrow{a}=\overrightarrow{b}$ is an equivalence relation on V.

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

10. Find the direction cosines of a vector whose direction ratios are 3,-1,3

11. Find the direction cosines and direction ratios for the following vectors.5$\hat{i}$-3$\hat{j}$-48$\hat{k}$

12. Part C

5 x 3 = 15
13. Let A and B be two points with position vectors 2$\overrightarrow{a}$+ 4$\overrightarrow{b}$ and 2$\overrightarrow{a}$ −8$\overrightarrow{b}$. Find the position vectors of the points which divide the line segment joining A and B in the ratio 1:3 internally and externally.

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

15. Show that the following vectors are coplanar $\hat{i}$ −2$\hat{j}$ +3$\hat{k}$,-2 $\hat{i}$ +3$\hat{j}$ - 4 $\hat{k}$ ,-$\hat{j}$ +2 $\hat{k}$ .

16. Show that the points (4, - 3, 1), (2, - 4, 5) and (1, - 1, 0) form a right angled triangle.

17. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is ${1\over 2}|\overrightarrow{a}\times \overrightarrow{b}+ \overrightarrow{b}+\overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|$ .Also deduce the condition for collinearity of the points A, B, and C.

18. Part D

4 x 5 = 20
19. 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$\overrightarrow{EF}$.

20. The position vectors of the points P, Q, R, S are $\hat{i}$ + $\hat{j}$ + $\hat{k}$,2 $\hat{i}$+5$\hat{j}$,3$\hat{i}$+2$\hat{j}$-3$\hat{k}$, and $\hat{i}$-6$\hat{j}$-$\hat{k}$respectively. Prove that the line PQ and RS are parallel.

21. If $\overrightarrow{a},\overrightarrow{b}$are unit vectors and $\theta$ is the angle between them, show that $sin {\theta \over 2}={1\over2}|\overrightarrow{a}-\overrightarrow{b}|$

22. Find the projection of the vector $\hat{i}+3\hat{j}+7\hat{k}$ on the vector$2\hat{i}+6\hat{j}+3\hat{k}$.