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Vector Algebra - I Model Question Paper

11th Standard

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Maths

Time : 01:30:00 Hrs
Total Marks : 50
    7 x 1 = 7
  1. The value of \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{DA}+\overrightarrow{CD}\) is

    (a)

    \(\overrightarrow{AD}\)

    (b)

    \(\overrightarrow{CA}\)

    (c)

    \(\overrightarrow{0}\)

    (d)

    \(-\overrightarrow{AD}\)

  2. A vector \(\overrightarrow{OP}\) makes 60° and 45° with the positive direction of the x and y axes respectively.  Then the angle between \(\overrightarrow{OP}\)and the z-axis is

    (a)

    45°

    (b)

    60°

    (c)

    90°

    (d)

    30°

  3. One of the diagonals of parallelogram ABCD with \(\overrightarrow{a}\) and \(\overrightarrow{b}\) as adjacent sides is \(\overrightarrow{a}+\overrightarrow{b}\)The other diagonal \(\overrightarrow{BD}\) is

    (a)

    \(\overrightarrow{a}-\overrightarrow{b}\)

    (b)

    \(\overrightarrow{b}-\overrightarrow{a}\)

    (c)

    \(\overrightarrow{a}+\overrightarrow{b}\)

    (d)

    \(\overrightarrow{a}+\overrightarrow{b}\over 2\)

  4. The value of  \(\theta \in (0,{\pi\over 2})\) for which the vectors \(\overrightarrow{a}=(sin \theta)\hat{i}+(cos\theta)\hat{j}\) and \(\overrightarrow{b}=\hat{i}-\sqrt{3}\hat{j}+2\hat{k}\) are perpendicular, is equal to

    (a)

    \({\pi\over 3}\)

    (b)

    \({\pi\over 6}\)

    (c)

    \({\pi\over 4}\)

    (d)

    \({\pi\over 2}\)

  5. If \(|\overrightarrow { a } |=|\overrightarrow { b } |\) then

    (a)

    \(\overrightarrow { a } =\overrightarrow { b } \)

    (b)

    \(\overrightarrow { a } =\overrightarrow { -b } \)

    (c)

    \(\overrightarrow { a } =\pm \overrightarrow { b } \)

    (d)

    both are null vectors

  6. If the direction cosines of a line are k, k and k, then

    (a)

    k>0

    (b)

    0<k<1

    (c)

    k=1

    (d)

    \(k=\frac { 1 }{ \sqrt { 3 } } or-\frac { 1 }{ \sqrt { 3 } } \)

  7. The value of \(\lambda\) when the vectors \(\vec{a}=2\vec{i}+\lambda\vec{j}+\vec{k}\) and \(​​\vec{b}=\vec{i}+2\vec{j}+3\vec{k}\) are orthogonal is

    (a)

    0

    (b)

    1

    (c)

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

    (d)

    -\(\frac{5}{2}\)

  8. 8 x 2 = 16
  9. Represent graphically the displacement of 30 km 60° west of north

  10. Represent graphically the displacement of 80km, 60° south of west

  11. Let \(\overrightarrow{a}\) and \(\overrightarrow{b}\) be the position vectors of the points A and B. Prove that the position vectors of the points which trisects the line segment AB are \({\overrightarrow{a}+2\overrightarrow{b}\over 3} and \ {\overrightarrow{b}+2\overrightarrow{a}\over3}.\)

  12. Can a vector have direction angles 30°,45°,60°?

  13. Verify whether the following ratios are direction cosines of some vector or not\({1\over\sqrt{2}},{1\over 2},{1\over 2}\)

  14. Find \(\overrightarrow{a}\).\(\overrightarrow{b}\)when \(\overrightarrow{a}=\hat{i}-2\hat{j}+\hat{k}\) and \(\overrightarrow{b}=3\hat{i}-4\hat{j}-2\hat{k}\)

  15. Show that \(\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{c})+\overrightarrow{b}\times (\overrightarrow{c}+\overrightarrow{a})+\overrightarrow{c}\times (\overrightarrow{a}+\overrightarrow{b})=\overrightarrow{0}\)

  16. If \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\) and \(\vec{b}=2\hat{i}+3\hat{j}-5\hat{k}\) then find \(\vec{a} \times \vec{b}\) . Verify that\(\vec{a}\) and \(\vec{b}\) are perpendicular to each other.

  17. 4 x 3 = 12
  18. If \(\overrightarrow{PO}\) +\(\overrightarrow{OQ}\) = \(\overrightarrow{QO}\) +\(\overrightarrow{OR}\), prove that the points P, Q, R are collinear.

  19. If \(\overrightarrow{a}=2\hat{i}+3\hat{j}-4\hat{k},\) \(\overrightarrow{b}=3\hat{i}-4\hat{j}-5\hat{k},\)  and \(\overrightarrow{c}=-3\hat{i}+2\hat{j}+3\hat{k},\)find the magnitude and direction cosines of 3\(\overrightarrow{a}\) − 2\(\overrightarrow{b}\) + 5\(\overrightarrow{c}\)

  20. Find the angle between the vectors \(5\hat{i}+3\hat{j}+4\hat{k}\)and\(6\hat{i}-8\hat{j}-\hat{k}\).

  21. If \((\overrightarrow { a } +\overrightarrow { b } ).(\overrightarrow { a } -\overrightarrow { b } )=0\) =0, then prove that \(\left| \overrightarrow { a } \right| =|\overrightarrow { b } |\)

  22. 3 x 5 = 15
  23. If D is the midpoint of the side BC of a triangle ABC, prove that \(\overrightarrow{AB}\) + \(\overrightarrow{AC}\) = 2\(\overrightarrow{AD}\) .

  24. If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then prove that \(\overrightarrow{AB}\) + \(\overrightarrow{AD}\) +\(\overrightarrow{CB}\)+\(\overrightarrow{CD}\) = 4\(\overrightarrow{EF}\).

  25. Show that the points (2, - 1, 3), (4, 3, 1) and (3, 1, 2) are collinear.

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