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12th Standard Maths Applications of Vector Algebra English Medium Free Online Test One Mark Questions with Answer Key 2020 - 2021

12th Standard

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Maths

Time : 00:10:00 Hrs
Total Marks : 10

    Answer all the questions

    9 x 1 = 9
  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 \(\vec { a } ,\vec { b } ,\vec { c } \) 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)

    \( { \pi }\)

  3. 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)

  4. 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)

  5. Let \(\overset { \rightarrow }{ a } \),\(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) be three non- coplanar vectors and let \(\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r } \) be the vectors defined by the relations \(\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \) Then the value of  \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r } \)=

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    3

  6. The volume of the parallelepiped whose sides are given by \(\overset { \rightarrow }{ OA } =2\overset { \wedge }{ i } -3\overset { \wedge }{ j } \)\(\overset { \rightarrow }{ OB } =\overset { \wedge }{ i } +\overset { \wedge }{ j } -\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ OC } =3\overset { \wedge }{ i } -\overset { \wedge }{ k } \) is

    (a)

    \(\frac { 4 }{ 13 } \)

    (b)

    4

    (c)

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

    (d)

    \(\frac { 4 }{ 9 } \)

  7. If \(\left| \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right| =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \)then the angle between the vector \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) is _____________

    (a)

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

    (b)

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

    (c)

    \(\frac { \pi }{ 6 } \)

    (d)

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

  8. If \(\overset { \rightarrow }{ d } \) = \(\lambda \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right) +\mu \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\omega \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) \) and \({ \left| \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right| }=\frac { 1 }{ 8 } \) then λ + μ + ω is _____________

    (a)

    0

    (b)

    1

    (c)

    8

    (d)

    8\(\overset { \rightarrow }{ d } .\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) \)

  9. Let \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\) and \(\overset { \rightarrow }{ c } \) be three vectors having magnitudes 1, 1, 2 respectively.
    If \(\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } =0\) then the acute angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ c } \) is ____________________

    (a)

    0

    (b)

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

    (c)

    \(\frac { \pi }{ 6 } \)

    (d)

    \(\frac { 2\pi }{ 3 } \)

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