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 \(\frac { x+3 }{ -3 } =\frac { y+7 }{ 2 } =\frac { z-6 }{ 4 } \) 

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