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Applications of Vector Algebra One Mark Questions with Answer

12th Standard EM

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Maths

Time : 00:45:00 Hrs
Total Marks : 25
    25 x 1 = 25
  1. If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to

    (a)

    2

    (b)

    -1

    (c)

    1

    (d)

    0

  2. 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

  3. \(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , then the value of \([\vec { a } ,\vec { b } ,\vec { c } ]\) is

    (a)

    \(\left| \vec { a } \right| \left| \vec { b } \right| \left| \vec { c } \right| \)

    (b)

    \(\frac{1}{3}\)\(\left| \vec { a } \right| \left| \vec { b } \right| \left| \vec { c } \right| \)

    (c)

    1

    (d)

    -1

  4. If \(\vec { a } ,\vec { b } ,\vec { c } \) are three unit vectors such that \(\vec { a } \) is perpendicular to \(\vec { b } \) and is parallel to \(\vec { c } \) then \(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to

    (a)

    \(\vec { a } \)

    (b)

    \(\vec { b} \)

    (c)

    \(\vec { c } \)

    (d)

    \(\vec { 0 } \)

  5. If \([\vec { a } ,\vec { b } ,\vec { c } ]=1\)\(\frac { \vec { a } .(\vec { b } \times \vec { c } ) }{ (\vec { c } \times \vec { a } ).\vec { b } ) } +\frac { \vec { b } .(\vec { c } \times \vec { a } ) }{ (\vec { a } \times \vec { b } ).\vec { c } } +\frac { \vec { c } .(\vec { a } \times \vec { b } ) }{ (\vec { c } \times \vec { b } ).\vec { a } } \) is

    (a)

    1

    (b)

    -1

    (c)

    2

    (d)

    3

  6. The volume of the parallelepiped with its edges represented by the vectors \(\hat { i } +\hat { j } ,\hat { i } +2\hat { j } ,\hat { i } +\hat { j } +\pi \hat { k } \) is

    (a)

    \(\frac { \pi }{ 2 } \)

    (b)

    \(\frac { \pi }{ 3} \)

    (c)

    \( { \pi }\)

    (d)

    \(\frac { \pi }{ 4 } \)

  7. 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)

    \(\frac { \pi }{ 2 } \)

  8. If \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \)\(\vec { b } =\hat { i } +\hat { j } \)\(\vec { c } =\hat { i } \) and \((\vec { a } \times \vec { b } )\times\vec { c } \) = \(\lambda \vec { a } +\mu \vec { b } \) then the value of \(\lambda +\mu \) is

    (a)

    0

    (b)

    1

    (c)

    6

    (d)

    3

  9. If \(\vec { a } ,\vec { b } ,\vec { c } \) are non-coplanar, non-zero vectors such that \([\vec { a } ,\vec { b } ,\vec { c } ]\) = 3, then \({ \{ [\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } }]\} ^{ 2 }\) is equal to

    (a)

    81

    (b)

    9

    (c)

    27

    (d)

    18

  10. 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)

    \( { \pi }\)

  11. If the volume of the parallelepiped with \(\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } \)  as coterminous edges is 8 cubic units, then the volume of the parallelepiped with \((\vec { a } \times \vec { b } )\times (\vec { b } \times \vec { c } ),(\vec { b } \times \vec { c } )\times (\vec { c } \times \vec { a } )\) and \((\vec { c } \times \vec { a } )\times (\vec { a } \times \vec { b } )\)as coterminous edges is, 

    (a)

    8 cubic units

    (b)

    512 cubic units

    (c)

    64 cubic units

    (d)

    24 cubic units

  12. Consider the vectors \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { c } \) such that \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )\) = \(\vec { 0 } \) Let \({ P }_{ 1 }\) and \({ P }_{ 2 }\) be the planes determined by the pairs of vectors \(\vec { a } ,\vec { b } \) and \(\vec { c } ,\vec { d } \) respectively. Then the angle between \({ P }_{ 1 }\) and \({ P }_{ 2 }\) is

    (a)

    (b)

    45°

    (c)

    60°

    (d)

    90°

  13. If \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c } \) where \(\vec { a } ,\vec { b } ,\vec { c } \) are any three vectors such that \(\vec { a } ,\vec { b } \) \(\neq \) 0 and  \(\vec { a } .\vec { b } \) \(\neq \) 0 then \(\vec { a } \) and \(\vec { c } \) are

    (a)

    perpendicular

    (b)

    parallel

    (c)

    inclined at an angle \(\frac{\pi}{3}\)

    (d)

    inclined at an angle  \(\frac{\pi}{6}\)

  14. If \(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =\hat { i } +2\hat { j } +5\hat { k } ,\vec { c } =3\hat { i } +5\hat { j } -\hat { k } \) then a vector perpendicular to \(\vec { a } \) and lies in the plane containing \(\vec { b } \) and \(\vec { c } \) is

    (a)

    \(-17\hat { i } +21\hat { j } -\hat { 97k } \)

    (b)

    \(17\hat { i } +21\hat { j } -\hat { 123k } \)

    (c)

    \(-17\hat { i } -21\hat { j } +\hat { 197k } \)

    (d)

    \(-17\hat { i } -21\hat { j } -\hat { 197k } \)

  15. 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)

    \(\frac{\pi}{2}\)

  16. If the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -5 }= \frac { x+2 }{ 2 } \) lies in the plane x + 3y + - αz + β = 0, then (α, β) is

    (a)

    (-5, 5)

    (b)

    (-6, 7)

    (c)

    (5, 5)

    (d)

    (6, -7)

  17. The angle between the line \(\vec { r } =(\hat { i } +2\hat { j } -3\hat { k } )+t(2\hat { i } +\hat { j } -2\hat { k } )\) and the plane \(\vec { r } .(\hat { i } +\hat { j } )+4=0\) is

    (a)

    (b)

    30°

    (c)

    45°

    (d)

    90°

  18. The coordinates of the point where the line \(\vec { r } =(6\hat { i } -\hat { j } -3\hat { k } )+t(\hat { i } +4\hat { j } )\) meets the plane \(\vec { r } =(\hat { i } +\hat { j } -\hat { k } )\) = 3 are

    (a)

    (2,1,0)

    (b)

    (7,1,7)

    (c)

    (1,2,6)

    (d)

    (5,1,1)

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

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    3

  20. The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0

    (a)

    \(\frac { \sqrt { 7 } }{ 2\sqrt { 2 } } \)

    (b)

    \(\frac{7}{2}\)

    (c)

    \(\frac { \sqrt { 7 } }{ 2 } \)

    (d)

    \(\frac { 7 }{ 2\sqrt { 2 } } \)

  21. If the direction cosines of a line are \(\frac { 1 }{ c } ,\frac { 1 }{ c } ,\frac { 1 }{ c } \), then

    (a)

    \(c=\pm 3\)

    (b)

    \(c=\pm \sqrt { 3 } \)

    (c)

    c > 0

    (d)

    0 < c < 1

  22. The vector equation \(\vec { r } =(\hat { i } -2\hat { j } -\hat { k } )+t(6\hat { i } -\hat { k) } \) represents a straight line passing through the points

    (a)

    (0,6,1)− and (1,2,1)

    (b)

    (0,6,-1) and (1,4,2)

    (c)

    (1,-2,-1) and (1,4,-2)

    (d)

    (1,-2,-1) and (0,-6,1)

  23. If the distance of the point (1,1,1) from the origin is half of its distance from the plane x + y + z + k =0, then the values of k are

    (a)

    \(\pm 3\)

    (b)

    \(\pm 6\)

    (c)

    -3, 9

    (d)

    3, 9

  24. If the planes \(\vec { r } =(2\hat { i } -\lambda \hat { j } +\hat { k } )=3\) and \(\vec { r } =(4+\hat { j } -\mu \hat { k } )=5\) are parallel, then the value of λ and μ are

    (a)

    \(\frac { 1 }{ 2 } ,-2\)

    (b)

    \(-\frac { 1 }{ 2 } ,2\)

    (c)

    \(-\frac { 1 }{ 2 } ,-2\)

    (d)

    \(\frac { 1 }{ 2 } ,2\)

  25. If the length of the perpendicular from the origin to the plane 2x + 3y + λz =1, λ > 0 is \(\frac{1}{5}\) then the value of λ is

    (a)

    \(2\sqrt { 3 } \)

    (b)

    \(3\sqrt { 2 } \)

    (c)

    0

    (d)

    1

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