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12th Standard Maths English Medium Applications of Vector Algebra Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

      Multiple Choice Questions


    15 x 1 = 15
  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. 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)

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

  3. 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)

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

  4. 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°

  5. 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  \(\frac{\pi}{6}\)

  6. 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

  7. 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)

  8. If \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \), then

    (a)

    \(\left| \overset { \rightarrow }{ d } \right| \)

    (b)

    \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \)

    (c)

    \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ 0 } \)

    (d)

    a, b, c are coplanar

  9. If the vector \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ 2k } \)\(\overset { \wedge }{ -i } +\overset { \wedge }{ 2k } \) and \(2\overset { \wedge }{ i } +x\overset { \wedge }{ j } -y\overset { \wedge }{ k } \)  are mutually orthogonal, then the values of x, y, z are

    (a)

    (10, 4, 1)

    (b)

    (-10, 4, 1)

    (c)

    (-10, -4, \(\frac { 1 }{ 2 } \))

    (d)

    (-10, 4, \(\frac { 1 }{ 2 } \))

  10. If the work done by a force \(\overset { \rightarrow }{ F } =\overset { \wedge }{ i } +m\overset { \wedge }{ j } -\overset { \wedge }{ k } \) in moving the point of application from(1, 1, 1) to (3, 3, 3) along a straight line is 12 units, then m is

    (a)

    5

    (b)

    2

    (c)

    3

    (d)

    6

  11. If \(\lambda \overset { \wedge }{ i } +2\lambda \overset { \wedge }{ j } +2\lambda \overset { \wedge }{ k } \) is a unit vector, then the value of λ is 

    (a)

    土 \(\frac { 1 }{ 3 } \)

    (b)

    土 \(\frac { 1 }{ 4 } \)

    (c)

    土 \(\frac { 1 }{ 9 } \)

    (d)

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

  12. For any three vectors \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \)\(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) \times \left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) \) is

    (a)

    0

    (b)

    \(\left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right] \)

    (c)

    2\(\left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right] \)

    (d)

    \({ \left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right] }^{ 2 }\)

  13. Let \(\overset { \rightarrow }{ u } ,\overset { \rightarrow }{ v } ,\overset { \rightarrow }{ w } \) be vectors such that \(\overset { \rightarrow }{ u } +\overset { \rightarrow }{ v } +\overset { \rightarrow }{ w } =\overset { \rightarrow }{ 0 } \). If \(\left| \overset { \rightarrow }{ u } \right| \)= 3\(\left| \overset { \rightarrow }{ v } \right| \)= 4\(\left| \overset { \rightarrow }{ w } \right| \)=5  then \(\overset { \rightarrow }{ u } .\overset { \rightarrow }{ v } +\overset { \rightarrow }{ v } .\overset { \rightarrow }{ w } +\overset { \rightarrow }{ w } .\overset { \rightarrow }{ u } \) is ______________

    (a)

    25

    (b)

    -25

    (c)

    5

    (d)

    \(\sqrt { 5 } \)

  14. The length of the 丄r from the origin to plane \(\overset { \rightarrow }{ r } .\left( \overset { \wedge }{ 3i } +4\overset { \wedge }{ j } +12\overset { \wedge }{ k } \right) \)26 is _____________

    (a)

    2

    (b)

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

    (c)

    26

    (d)

    \(\frac { 26 }{ 169 } \)

  15. 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)

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

    1. 2 Marks

    10 x 2 = 20
  16. If \(\vec{ a } =\hat { -3i } -\hat { j } +\hat { 5k } \)\(\vec{b}=\hat{i}-\hat{2j}+\hat{k} \),  \(\vec{c}=\hat{4i}-\hat{4k} \ and \) \(find \ {\vec a } .(\vec { b } \times \vec { c } )\)

  17. 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.

  18. 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 \(2\hat { i } +4\hat { j } -2\hat { k } \)

  19. Determine whether the three vectors \(2\hat { i } +3\hat { j } +\hat { k } \)\(\hat { i } -2\hat { j } +2\hat { k } \) and \(\hat { 3i } +\hat { j } +2\hat { k } \) are coplanar.

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

  21. Find the parametric form of vector equation and Cartesian equations of the straight line passing through the point (−2,3, 4) and parallel to the straight line \(\frac { x-1 }{ -4 } =\frac { y-3 }{ 5 } =\frac { z-8 }{ 6 } \)

  22. Show that the points (2, 3, 4),(−1, 4, 5) and (8,1, 2) are collinear.

  23. A variable plane moves in such a way that the sum of the reciprocals of its intercepts on the coordinate axes is a constant. Show that the plane passes through a fixed point

  24. Find the direction cosines of the normal to the plane 12x + 3y − 4z = 65 . Also, find the non-parametric form of vector equation of a plane and the length of the perpendicular to the plane from the origin.

  25. Find the Cartesian equation of a.line passing through the pointsA(2, -1, 3) and B(4, 2, 1)

    ()

    -1

    1. 3 Marks


    10 x 3 = 30
  26. With usual notations, in any triangle ABC, prove the following by vector method.
    (i) a2=b2+c2−2bc cos A
    (ii) b2=c2+a2−2ca cos B
    (iii) c2= a2+b2−2ab cos C

  27. If D is the midpoint of the side BC of a triangle ABC, then show by vector method that \({ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })\)

  28. 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 \(\frac{1}{4}\)(area of ΔABC )

  29. Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord.

  30. Forces of magnit \(5\sqrt { 2 } \) and \(5\sqrt { 2 } \) units acting in the directions \(\hat { 3i } +\hat { 4j } +\hat { 5k } \) and \(\hat { 10i } +\hat { 6j } -\hat { 8k } \) respectively, act on a particle which is displaced from the point with position vector \(\hat { 4i } +\hat { 3j } -\hat { 2k } \) to the point with position vector \(\hat { 6i } +\hat { j } -\hat { 3k } \). Find the work done by the forces.

  31. Find the torque of the resultant of the three forces represented by \(-\hat { 3i } +\hat { 6j } +\hat { 3k } \)\(\hat { 4i } -\hat { 10j } +\hat { 12k } \) and \(\hat { 4i } +\hat { 7j } \) acting at the point with position vector \(\hat { 8i } -\hat { 6j } -\hat { 4k } \), about the point with position vector \(\hat { 18i } +\hat { 3j } -\hat { 9k } \)

  32. Find the vector equation in parametric form and Cartesian equations of the line passing through (-4, 2, -3) and is parallel to the line  \(\frac { -x-2 }{ 4 } =\frac { y+3 }{ -2 } =\frac { 2z-6 }{ 3 } \)

  33. Show that the straight lines x + 1=  2y = −12z and x = y + 2 = 6z − 6 are skew and hence find the shortest distance between them.

  34. Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)

    ()

    0

  35. 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

    1. 5 Marks


    6 x 5 = 30
  36. By vector method, prove that cos(α + β) = cos α cos β -  sin α sin β

  37. With usual notations, in any triangle ABC, prove by vector method that \(\frac { a }{ sinA } =\frac { b }{ sinB }=\frac { c }{ sinc }\)

  38. Prove by vector method that the perpendiculars (attitudes) from the vertices to the opposite sides of a triangle are concurrent.

  39. Using vector method, prove that cos(α − β )=cos α cos β +sin α sin β

  40. Determine whether the pair of straight lines \(\vec { r } (2\hat { i } +\hat { 3j } -\hat { k } )+t(2\hat { i } +3\hat { j } +2\hat { k } )\)\(\vec { r } =(2\hat { j } -3\hat { k } )+s(\hat { i } +2\hat { j } +3\hat { k } )\) are parallel. Find the shortest distance between them.

  41. Show that the points A, B, C with position vector \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k } \) and \(3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.

    ()

    points on the plane

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