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#### Applications of Vector Algebra Model Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
10 x 1 = 10
1. If a vector $\vec { \alpha }$ lies in the plane of $\vec { \beta }$ and $\vec { \gamma }$ , then

(a)

$[\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]$ = 1

(b)

$[\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]$= -1

(c)

$[\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]$= 0

(d)

$[\vec { \alpha } ,\vec { \beta } ,\vec { \gamma } ]$= 2

2. If $\vec { a }$ and $\vec { b }$ are unit vectors such that $[\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { \pi }{ 4 }$, then the angle between $\vec { a }$ and $\vec { b }$ is

(a)

$\frac { \pi }{ 6 }$

(b)

$\frac { \pi }{ 4 }$

(c)

$\frac { \pi }{ 3 }$

(d)

$\frac { \pi }{ 2 }$

3. If $\vec { a } ,\vec { b } ,\vec { c }$ are three non-coplanar vectors such that $\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } }$, then the angle between

(a)

$\frac { \pi }{ 2 }$

(b)

$\frac { 3\pi }{ 6 }$

(c)

$\frac { \pi }{ 4 }$

(d)

${ \pi }$

4. The angle between the lines $\frac { x-2 }{ 3 } =\frac { y+1 }{ -2 }$, z=2 and $\frac { x-1 }{ 1 } =\frac { 2y+3 }{ 3 } =\frac { z+5 }{ 2 }$

(a)

$\frac{\pi}{6}$

(b)

$\frac{\pi}{4}$

(c)

$\frac{\pi}{3}$

(d)

$\frac{\pi}{2}$

5. Distance from the origin to the plane 3x - 6y + 2z 7 = 0 is

(a)

0

(b)

1

(c)

2

(d)

3

6. The number of vectors of unit length perpendicular to the vectors $\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right)$ and $\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right)$is

(a)

1

(b)

2

(c)

3

(d)

7. If $\overset { \rightarrow }{ a }$,$\overset { \rightarrow }{ b }$ and $\overset { \rightarrow }{ c }$ are any three vectors, then $\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right)$ if and only if

(a)

$\overset { \rightarrow }{ b }$$\overset { \rightarrow }{ c }$ are collinear

(b)

$\overset { \rightarrow }{ a }$ and $\overset { \rightarrow }{ c }$ are collinear

(c)

$\overset { \rightarrow }{ a }$ and $\overset { \rightarrow }{ b }$ are collinear

(d)

none

8. If  $\left| \overset { \rightarrow }{ a } \right| =\left| \overset { \rightarrow }{ b } \right| =1$such that $\overset { \rightarrow }{ a } +2\overset { \rightarrow }{ b }$ and $5\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b }$ are perpendicular to each other, then the angle between $\overset { \rightarrow }{ a }$ and $\overset { \rightarrow }{ b }$ is

(a)

45o

(b)

60o

(c)

cos-1 $\left( \frac { 1 }{ 3 } \right)$

(d)

cos-1 $\left( \frac { 2 }{ 7 } \right)$

9. If $\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ 3k }$$\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +\overset { \wedge }{ 2j } +\overset { \wedge }{ k }$$\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j }$ then $\overset { \rightarrow }{ a } +\left( -\overset { \rightarrow }{ b } \right)$will be perpendiculur to $\overset { \rightarrow }{ c }$ only when t =

(a)

5

(b)

4

(c)

3

(d)

$\frac { 7 }{ 3 }$

10. The straight lines $\frac { x-3 }{ 2 } =\frac { y+5 }{ 4 } =\frac { z-1 }{ -13 }$ and $\frac { x+1 }{ 3 } =\frac { y-4 }{ 5 } =\frac { z+2 }{ 2 }$ are

(a)

parallel

(b)

perpendicular

(c)

inclined at 45o

(d)

none

11. 5 x 2 = 10
12. If $\hat { a } =\hat { -3i } -\hat { j } +\hat { 5k }$$\hat{b}=\hat{i}-\hat{2j}+\hat{k}$$\hat{c}=\hat{4i}-\hat{4k}$and $\hat { a } .(\hat { b } \times \hat { c } )$

13. Find the volume of the parallelepiped whose coterminous edges are represented by the vectors $-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k }$ and $2\hat { i } +4\hat { j } -2\hat { k }$

14. Prove that $(\vec { a } .(\vec { b } \times \vec { c } ))\vec { a } =(\vec { a } \times \vec { b } )\times (\vec { a } \times \vec { c } )$

15. Find a parametric form of vector equation of a plane which is at a distance of 7 units from the origin having 3, −4,5 as direction ratios of a normal to it.

16. Find the distance between the parallel planes x+2y-2z=0 and 2x+4y-4z+5=0

17. 5 x 3 = 15
18. Find the distance of the point (5,-5,-10) from the point of intersection of a straight line passing through the points A(4,1,2) and B(7,5,4) with the plane x-y+z=5

19. Find the coordinates of the point where the straight line $\vec { r } =(2\hat { i } -\hat { j } +2\hat { k } )+t(3\hat { i } +4\hat { j } +2\hat { k } )$ intersects the plane x−y+z−5=0.

20. Dot product of a vector with vector $\overset { \wedge }{ 3i } -5\overset { \wedge }{ k }$$2\overset { \wedge }{ i } +7\overset { \wedge }{ j }$ and $\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k }$ are respectively -1, 6 and 5. Find the vector.

()

$\overset { \rightarrow }{ b }$$\overset { \rightarrow }{ d }$

21. Find the equation of the plane through the intersection of the planes lx-3y+ z-4 -0 and x - y + Z + 1 - 0 and perpendicular to the plane x + 2y - 3z + 6 = 0

()

$\overset { \rightarrow }{ a }$$\overset { \rightarrow }{ c }$

22. Show that the lines $\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 }$ and $\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 }$ do not intersect

()

a, b, c

23. 3 x 5 = 15
24. Prove by vector method that sin(α −β )=sinα cosβ −cosα sinβ

25. Find the vector parametric, vector non-parametric and Cartesian form of the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1)and parallel to the straight line  $\frac { x-1 }{ 1 } =\frac { 2y+1 }{ 2 } =\frac { z+1 }{ -1 }$

26. Find the shortest distance between the following pairs of lines $\frac { x-3 }{ 3 } =\frac { y-8 }{ -1 } =\frac { z-3 }{ 1 }$and $\frac { x+3 }{ -3 } =\frac { y+7 }{ 2 } =\frac { z-6 }{ 4 }$