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Applications of Vector Algebra 2 Mark Book Back Question Paper With Answer Key

12th Standard

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Maths

Time : 00:30:00 Hrs
Total Marks : 52

    2 Marks

    26 x 2 = 52
  1. With usual notations, in any triangle ABC, prove the following by vector method.
    (i) a= b+ c− 2bc cos A
    (ii) b= c+ a− 2ca cos B
    (iii) c= a+ b− 2ab cos C

  2. With usual notations, in any triangle ABC, prove the following by vector method.
    (i) a = b cos C + c cos B
    (ii) b = c cos A + a cos C
    (iii) c = a cos B + b cos A

  3. A particle is acted upon by the forces \((\hat { 3i } -\hat { 2j } +\hat { 2k } )\) and \((\hat { 2i } +\hat { j } -\hat { k } )\) is displaced from the point (1, 3, -1 ) to the point (4, -1, λ). If the work done by the forces is 16 units, find the value of λ.

  4. 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.

  5. Prove by vector method that the median to the base of an isosceles triangle is perpendicular to the base.

  6. Prove by vector method that an angle in a semi-circle is a right angle.

  7. Prove by vector method that the diagonals of a rhombus bisect each other at right angles.

  8. Using vector method, prove that if the diagonals of a parallelogram are equal, then it is a rectangle

  9. Prove by vector method that the parallelograms on the same base and between the same parallels are equal in area.

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

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

  12. 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 \(2\hat { i } +4\hat { j } -2\hat { k } \)

  13. If \(\vec { a } =\hat { i } -\hat { k } ,\vec { b } =x\hat { i } +\hat { j } +(1-x)\hat { k } ,\vec { c } =y\hat { i } +x\hat { j } +(1+x+y)\hat { k } \) show that \([\vec { a } ,\vec { b } ,\vec { c } ]\) depends on neither x nor y.

  14. If the vectors \(a\hat { i } +a\hat { j } +c\hat { k } ,\hat { i } +\hat { k } \) and \(c\hat { i } +c\hat { j } +b\hat { k } \) are coplanar, prove that c is the geometric mean of a and b.

  15. Let \(\vec { a } ,\vec { b } ,\vec { c } \)  be three non-zero vectors such that \(\vec { c } \) is a unit vector perpendicular to both \(\vec { a } \) and \(\vec { b } \). If the angle between  \(\vec { a } \) and \(\vec { b } \) is \(​​\frac { \pi }{ 6 } \), show that \({ [\vec { a } ,\vec { b } ,\vec { c } ] }^{ 2 }\) = \(\frac { 1 }{ 4 } { \left| \vec { a } \right| }^{ 2 }{ \left| \vec { b } \right| }^{ 2 }\)

  16. For any four vectors \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { d } \) we have \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { b } ,\vec { d } ]\vec { c } -[\vec { a } ,\vec { b } ,\vec { c } ]\vec { d } =[\vec { a } ,\vec { c } ,\vec { d } ]\vec { b } -[\vec { b } ,\vec { c } ,\vec { d } ]\vec { a } \)

  17. For any vector \(\vec { a } \), prove that \(\hat { i } \times (\vec { a } \times \hat { i } )+\hat { j } \times (\vec { a } \times \hat { j } )+\hat { k } \times \vec { a } \times \hat { k } =2\vec { a } \).

  18. Prove that \([\vec { a } -\vec { b } ,\vec { b } -\vec { c } ,\vec { c } -\vec { a } ]\) = 0

  19. If \(\hat { a } ,\hat { b } ,\hat { c } \) are three unit vectors such that \(\hat { b } \) and \(\hat { c } \) are non-parallel and \(\hat { a } \times \hat { b } \times \hat { c } =\frac { 1 }{ 2 } \hat { b } \) the angle between \(\hat { a } \) and \(\vec{ c } \).

  20. Find the vector and Cartesian form of the equations of a plane which is at a distance of 12 units from the origin and perpendicular to \(6\hat { i } +2\hat { j } -3\hat { k } \)

  21. If the Cartesian equation of a plane is 3x - 4y + 3z = -8, find the vector equation of the plane in the standard form.

  22. Find the vector and Cartesian equations of the plane passing through the point with position vector \(4\hat { i } +2\hat { j } -3\hat { k } \) and normal to vector \(2\hat { i } -\hat { j } +\hat { k } \)

  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. Verify whether the line \(\frac { x-3 }{ -4 } =\frac { y-4 }{ -7 } =\frac { z+3 }{ 12 } \) lies in the plane 5x-y+z = 8.

  25. Find the angle between the line \(\vec { r } =(2\hat { i } -\hat { j } +\hat { k } )+t(\hat { i } +2\hat { j } -2\hat { k } )\) and the plane \(\vec { r } =(6\hat { i } +3\hat { j } +2\hat { k } )=8\)

  26. Find the angle between the planes \(\vec { r } .(\hat { i } +\hat { j } -2\hat { k } )\) = 3 and 2x - 2y + z =2

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