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Applications of Vector Algebra Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 35
    5 x 1 = 5
  1. 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

  2. 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  \(\frac{\pi}{6}\)

  3. 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)

  4. The angle between the vector \(3\overset { \wedge }{ i } +4\overset { \wedge }{ j } +\overset { \wedge }{ 5k } \) and the z-axis is

    (a)

    30o

    (b)

    60o

    (c)

    45o

    (d)

    90o

  5. The distance from the origin to the plane \(\overset { \rightarrow }{ r } .\left( \overset { \wedge }{ 2i } -\overset { \wedge }{ j } +5\overset { \wedge }{ k } \right) =7\) is ______________ 

    (a)

    \(\frac { 7 }{ \sqrt { 30 } } \)

    (b)

    \(\frac { \sqrt { 30 } }{ 7 } \)

    (c)

    \(\frac { 30 }{ 7 } \)

    (d)

    \(\frac { 7 }{ 30 } \)

  6. 1 x 2 = 2
  7. \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) are said to be coplanar if
    (1) \(\left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right] \)=0
    (2) \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) lie on the same plane
    (3) They are either parallel or intersecting
    (4) Skew lines

  8. 4 x 2 = 8
  9. If \(\hat { 2i } -\hat { j } +\hat { 3k } ,\hat { 3i } +\hat { 2j } +\hat { k } ,\hat { i } +\hat { mj } +\hat { 4k } \) are coplanar, find the value of m.

  10. If \(\vec { a } =\hat { i } -2\hat { j } +3\hat { k }, \vec { 6 } =2\hat { i } +\hat { j } -2\hat { k }, \vec { c } =3\hat { i } +2\hat { j } +\hat { k } \) find \(\vec { a } .(\vec { b } \times \vec { c } )\).

  11. The volume of the parallelepiped whose coterminus edges are \(7\hat { i } +\lambda \hat { j } -3\hat { k } ,\hat { i } +2\hat { j } -\hat { k } \)\(-3\hat { i } +7\hat { j } +5\hat { k } \) is 90 cubic units. Find the value of λ.

  12. Flnd the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and
    2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

    ()

    x = -1 is one root

  13. 5 x 3 = 15
  14. With usual notations, in any triangle ABC, prove the following by vector method.
    (i) a=bcosC+ccos B
    (ii) b=ccosA+acosC
    (iii) c=acosB+bcos A

  15. In triangle, ABC the points, D, E, F are the midpoints of the sides, BC, CA and AB respectively. Using vector method, show that the area of ΔDEF is equal to \(\frac{1}{2}\)(area of ΔABC )

  16. 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

  17. If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } -\overset { \wedge }{ j } ,\overset { \rightarrow }{ b } =\overset { \wedge }{ j } -\overset { \wedge }{ k } ,\overset { \rightarrow }{ c } =\overset { \wedge }{ k } -\overset { \wedge }{ i } \) then find \(\left[ \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } -\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } -\overset { \rightarrow }{ a } \right] \)

    ()

    parametric form od vector equation

  18. If \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } =0\) then show that \(\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } =\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } =\overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \)

    ()

    lies in the plane containing \(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \)

  19. 1 x 5 = 5
  20. Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

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