Model question-Vector Algebra - I

11th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 60

    Part A

    Answer all the questions

    10 x 1 = 10
  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. The unit vector parallel to the resultant of the vectors \(\hat{i}+\hat{j}-\hat{k}\) and\(\hat{i}-2\hat{j}+\hat{k}\) is

    (a)

    \({\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}\)

    (b)

    \({2\hat{i}+\hat{j}\over\sqrt{5}}\)

    (c)

    \({2\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}\)

    (d)

    \({2\hat{i}-\hat{j}\over\sqrt{5}}\)

  3. The vectors \(\overrightarrow{a}-\overrightarrow{b},\overrightarrow{b}-\overrightarrow{c},\overrightarrow{c}-\overrightarrow{a}\) are

    (a)

    parallel to each other

    (b)

    unit vectors

    (c)

    mutually perpendicular vectors

    (d)

    coplanar vectors.

  4. 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)

    \(\overrightarrow{a}+\overrightarrow{b}\over 3\)

  5. If \(\overrightarrow{a},\overrightarrow{b}\) are the position vectors A and B, then which one of the following points whose position vector lies on AB, is

    (a)

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

    (b)

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

    (c)

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

    (d)

    \({\overrightarrow{a}-\overrightarrow{b}\over 3}\)

  6. Two vertices of a triangle have position vectors \(3\hat{i}+4\hat{j}-4\hat{k}\) and\(2\hat{i}+3\hat{j}+4\hat{k}\)If the position vector of the centroid is \(\hat{i}+2\hat{j}+3\hat{k}\) ,then the position vector of the third vertex is

    (a)

    \(-2\hat{i}-\hat{j}+9\hat{k}\)

    (b)

    \(-2\hat{i}-\hat{j}-6\hat{k}\)

    (c)

    \(2\hat{i}-\hat{j}+6\hat{k}\)

    (d)

    \(-2\hat{i}+\hat{j}+6\hat{k}\)

  7. If \(\overrightarrow{a}\)  and \(\overrightarrow{b}\) having same magnitude and angle between them is 60° and their scalar product is \({1\over2}\) then \(|\overrightarrow{a}|\) is

    (a)

    2

    (b)

    3

    (c)

    7

    (d)

    1

  8. If   \(\overrightarrow{a}\)  and   \(\overrightarrow{b}\) are two vectors of magnitude 2 and inclined at an angle 60° , then the angle between   \(\overrightarrow{a}\)  and \(\overrightarrow{a}+\overrightarrow{b}\) is 

    (a)

    30°

    (b)

    60°

    (c)

    45°

    (d)

    90°

  9. If the projection of \(5\hat{i}-\hat{j}-3\hat{k}\) on the vector \(\hat{i}+3\hat{j}-\lambda\hat{k}\) is same as the projection of \(\hat{i}+3\hat{j}+\lambda\hat{k}\) on \(5\hat{i}-\hat{j}-3\hat{k}\),then \(\lambda\) is equal to

    (a)

    \(\pm 4\)

    (b)

    \(\pm 3\)

    (c)

    \(\pm 5\)

    (d)

    \(\pm 1\)

  10. If \(\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+x\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}\) and \(\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=70,\) then x is equal to

    (a)

    5

    (b)

    7

    (c)

    26

    (d)

    10

  11. Part B

    Answer all the questions

    10 x 2 = 20
  12. Represent graphically the displacement of 60 km 50° south of east.

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

  14. 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 \({\overrightarrow{a}+2\overrightarrow{b}\over 3} and \ {\overrightarrow{b}+2\overrightarrow{a}\over3}.\)

  15. Find a direction ratio and direction cosines of the following vectors \(3\hat{i}+4\hat{j}-6\hat{k}\)
     

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

  17. The direction ratios of a vector are 2, 3, 6 and it’s magnitude is 5. Find the vector

  18. Find the direction cosines of a vector whose direction ratios are 1,2,3

  19. Find the angle between the vectors \(\hat{i}-\hat{j}\)and\(\hat{j}-\hat{k}\).

  20. Find the vectors of magnitude \(10\sqrt{3}\) that are perpendicular to the plane which contains \(\hat{i}+2\hat{j}+\hat{k}\) and\(\hat{i}+3\hat{j}+4\hat{k}\)

  21. Find the unit vectors perpendicular to each of the vectors \(\overrightarrow{a}+\overrightarrow{b}\) and\(\overrightarrow{a}-\overrightarrow{b}\) ,where \(\overrightarrow{a}=\hat{i}+\hat{j} +\hat{k} \) and\(\overrightarrow{b} =\hat{i}+2\hat{j} +3\hat{k} \).

  22. Part C

    Answer all the questions

    5 x 3 = 15
  23. If \(\overrightarrow{PO}\) +\(\overrightarrow{OQ}\) = \(\overrightarrow{QO}\) +\(\overrightarrow{OR}\), prove that the points P, Q, R are collinear.

  24. Show that the points whose position vectors are 2\(\hat{i}\)+ 3\(\hat{j}\)− 5\(\hat{k}\), 3\(\hat{i}\)+ \(\hat{j}\)− 2\(\hat{k}\) and, 6\(\hat{i}\)− 5 \(\hat{j}\)+ 7\(\hat{k}\) are collinear

  25. If (a,a+b,a+b+c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c.

  26. Show that the following vectors are coplanar 5\(\hat{i}\) +6\(\hat{j}\) +7\(\hat{k}\) ,7 \(\hat{i}\) -8\(\hat{j}\) +9 \(\hat{k}\),3\(\hat{i}\)+20\(\hat{j}\) +5\(\hat{k}\) .

  27. If \(\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k},\)\(\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}\) and \(\overrightarrow{c}=3\hat{i}+\hat{j}\) be such that \(\overrightarrow{a}+\lambda \overrightarrow{b}\) is perpendicular to \(\overrightarrow{c}\) then find \(\lambda\).

  28. Part D

    Answer all the questions

    3 x 5 = 15
  29. If \(\overrightarrow{a},\overrightarrow{b}\)are unit vectors and \(\theta\) is the angle between them, show that \(cos {\theta \over 2}={1\over2}|\overrightarrow{a}+\overrightarrow{b}|\)

  30. If \(\overrightarrow{a},\overrightarrow{b}\)are unit vectors and \(\theta\) is the angle between them, show that \(tan {\theta \over 2}={|\overrightarrow{a}-\overrightarrow{b}|\over|\overrightarrow{a}+\overrightarrow{b}|}\)

  31. Three vectors \(\overrightarrow{a},\overrightarrow{b}\)and \(\overrightarrow{c}\) are such that \(|\overrightarrow{a}|=2,|\overrightarrow{b}|=3,|\overrightarrow{c}|=4,\) and \(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}\) .Find \(4\overrightarrow{a}.\overrightarrow{b}+​​3\overrightarrow{b}.\overrightarrow{c}+3\overrightarrow{c}.\overrightarrow{a}.\)

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