Untitled Document

Question Bank No: 1

1. If a and b are two unit vectors inclined at an angle $θ$ to each other, then $| \vec{a} + \vec{b} |$ $<$ 1 if

a)		$θ$ = $\frac{π}{6}$
b)		$θ$ = $\frac{π}{2}$
c)		$θ$ = $\frac{π}{3}$
d)		$\frac{2 π}{3} < θ$ $< π$

2. Volume of parallelopiped with sides given by $\vec{OA}$ $2 \hat{i} - 3 \hat{j} + \hat{k}$ , $\vec{OB}$ $= \hat{i} + \hat{j} - \hat{k}$ , $\vec{OC}$ $= 3 \hat{i} - \hat{k}$ is

a)		15
b)		17
c)		25
d)		None of these

3. The area of parallelogram constructed on the vectors $\vec{a} = \vec{m}$ $+ 2 \vec{n}$ and $\vec{b} = 2 \vec{m}$ $+ 3 \vec{n}$ where $\vec{m}$ and $\vec{n}$ are unit vectors forming an angle of $60^{0}$ , is

a)		3
b)		$3 \sqrt{3}$
c)		$\frac{3}{2} \sqrt{3}$
d)		None of these

4. Modulus of sum of three mutually perpendicular unit vectors is

a)		$\sqrt{3}$
b)		3
c)		0
d)		None of these

5. If $\vec{a} = 2 \hat{i} - \hat{j} + 6 \hat{k}$ and $\vec{b}$ $= \hat{i} + 5 \hat{j} + 11 \hat{k}$ , then $\vec{a}$ $\times \vec{b}$ is a vector

a)		Perpendicular to $\vec{a}$ only
b)		perpendicular to $\vec{b}$ only
c)		perpendicular to both $\vec{a}$ and $\vec{b}$
d)		None of these

6. The position vectors of the points A and B w.r.t an origin are $\hat{a} + \hat{i}, 3 \hat{j}$ $- 2 \hat{k}$ and $\hat{b}$ =3 $\hat{i} + \hat{j} - 2 \hat{k}$ respectively. If P is a point on AB, then $\vec{OP}$ which bisects $<$ AOB is

a)		2( $- \hat{i} + \hat{j} + \hat{k}$ )
b)		2( $\hat{i} - \hat{j} + \hat{k}$ )
c)		2( $\hat{i} + \hat{j} - \hat{k}$ )
d)		2( $\hat{i} + \hat{j} + \hat{k}$ )

7. The vectors 2 $\hat{i} + 3 \hat{j}, 5 \hat{i}$ +6 $\hat{j}$ and 8 $\hat{i}$ + $λ \hat{j}$ have their initial points at (1, 1). The value of $λ$ so that the vectors terminate on one straight line is

a)		0
b)		3
c)		6
d)		9

8. The points (2, -1, 1), (1, -3, -5), (3, -4, -4) are vertices of a triangle which is

a)		equilateral
b)		isosceles
c)		right angled
d)		None of these

9. If $\vec{a}, \vec{b}$ $, \vec{c}, \vec{d}$ are position vectors of vertices A, B, C, D of a quadrilateral ABCD and $\vec{a} + \vec{c}$ = $\vec{b}, \vec{d}$ , then ABCD is a

a)		parallelogram
b)		rhombus
c)		rectangle
d)		square

10. If $\vec{a} and \vec{b}$ are two vectors such that $\vec{a} . \vec{b}$ = 0 and $\vec{a} \vec{b}$ = $\vec{0}$ , then

a)		either $\vec{a} = \vec{0}$ or $\vec{b} = \vec{0}$
b)		$\vec{a} \| \| \vec{b}$
c)		$\vec{a} ⊥ \vec{0}$
d)		None of these

11. Vectors 2 $\hat{i} - \hat{j} + \hat{k}$ and 2 $\hat{i} - 4 \hat{j} + λ \hat{k}$ are perpendicular if $λ$ =

a)		16
b)		4
c)		-8
d)		8

12. The vector $\vec{b}$ which is collinear with vector $\vec{a}$ =(1,2,-1) and satisfies $\vec{a} . \vec{b}$ = 5 is

a)		$\frac{1}{3}$ (5,10,-5)
b)		$\frac{1}{6}$ (5,10,-5)
c)		(5,10,-5)
d)		6(5,10,-5)

13. If < $\vec{a}, \vec{b} = θ$ and $| \vec{a} \times \vec{b} | = | \vec{a} . \vec{b} |$ then $θ$ is

a)		0
b)		$45^{0}$
c)		$110^{0}$
d)		$135^{0}$

14. If $θ$ is the angle between two vectors $\vec{a}$ and $\vec{b}$ then $\vec{a} . \vec{b} > 0$ only if

a)		0 $\leq θ \leq π$
b)		$\frac{π}{2}$ $\leq θ \leq π$
c)		0 $\leq θ \leq \frac{π}{2}$
d)		0 $\leq θ < \frac{π}{2}$

15. The vector $\frac{1}{8} \hat{i} - \frac{3}{8} \hat{j} + \frac{1}{4} \hat{k}$ is

a)		Unit vector
b)		parallel to the vector 2 $\hat{i} - 6 \hat{j} + 4 \hat{k}$
c)		perpendicular to the vector 2 $\hat{i} + \hat{j} + \hat{k}$
d)		makes an angle $\frac{π}{3}$ with 2 $\hat{i} - 4 \hat{j} + 3 \hat{k}$

16. If the cross product of two non-zero vectors is zero,then the vectors are

a)		Collinear
b)		coinitial
c)		co-terminus
d)		parallel

17. If $| \vec{a} | =$ $| \vec{b} |$ then $(\vec{a} + \vec{b}) . (\vec{a} - \vec{b}) =$

a)		zero
b)		negative
c)		positive
d)		none of these

18. If $| \vec{a} \times \vec{b} | = \vec{| a |}$ $| \vec{b} |$ then $\vec{a}$ and $\vec{b}$ are

a)		Like parallel
b)		Unlike parallel
c)		coincident
d)		Perpendicular

19. If $\vec{b}$ is a unit vector, then $(\vec{a} . \vec{b}) \vec{b} + \vec{b} \times (\vec{a} \times \vec{b}) =$

a)		$\vec{a^{2}} \vec{b}$
b)		$(\vec{a} . \vec{b}) \vec{a}$
c)		$\vec{a}$
d)		None of these

20. The vectors $\vec{a}$ = 5 $\hat{i} + 4 \hat{j}$ and $\vec{b}$ = 20 $\hat{i} - 16 \hat{j}$ are

a)		coincident
b)		parallel
c)		perpendicular
d)		neither parallel nor perpendicular

21. Let $\bar{OA}$ = $\bar{a}, \bar{OB}$ =10 $\bar{a} + 2 \bar{b} and \bar{OC}$ $= \bar{b} where O, A, C are non collinear . Let P denote the area of the quadrilateral OABC and Q denote the area of the parallelogram with OA and OC as adjacent sides, then \frac{p}{q}$ is equal to

a)		4
b)		6
c)		$\frac{1}{2} \frac{\| \bar{a} - \bar{b} \|}{\| \bar{a} \|}$
d)		none of these

22. If $\bar{a} + \bar{b} + \bar{c} = 0 and | \bar{a} | = 3, | \bar{b} | = 5, | \bar{c} | = 7 then the angle between \bar{a} and \bar{b}$

a)		$\frac{Π}{3}$
b)		$\frac{Π}{2}$
c)		$\frac{Π}{4}$
d)		$\frac{Π}{6}$

23. If $\bar{a}, \bar{b}, \bar{c}, \bar{d} are the position vectors of points A, B, C and D such that no three of them collinear and a + c = b + d then ABCD is$

a)		a parallelogram
b)		a rhombus
c)		a rectangle
d)		a square

24. The sum of two unit vectors is a unit vector, The magnitude of their difference is

a)		2
b)		$\sqrt{3}$
c)		$\sqrt{2}$
d)		1

25. $\bar{a}, \bar{b}, \bar{c} are three vectors such that \bar{c} = \bar{a} + \bar{b} and angle between \bar{a} and \bar{b} is \frac{Π}{2} then$

a)		$a^{2} + b^{2} + c^{2} = 0$
b)		$a^{2} - b^{2} = c^{2}$
c)		$a^{2} + b^{2} = c^{2}$
d)		$\bar{c} = \bar{a} \times \bar{b}$

26. Let $\bar{a}, \bar{b}, \bar{c} be three non zero vectors, then \bar{a} . \bar{b} = \bar{a} . \bar{c} holds$

a)		only if $\bar{b} = \bar{c}$
b)		only if $\bar{a} is orthogonal to both \bar{b} and \bar{c}$
c)		only if $\bar{a} is orthogonal to \bar{b} - \bar{c}$
d)		only if either $\bar{b} = \bar{c} or \bar{a} is orthogonal to \bar{b} - \bar{c}$

27. If the vector 2 $\bar{i}$ - $\bar{j}$ + $\bar{k}$ , $\bar{i}$ +2 $\bar{j}$ -3 $\bar{k}$ and 3 $\bar{i}$ - $λ \bar{j} + 5 \bar{k} are coplanar then λ$

a)		-4
b)		2
c)		4
d)		3

28. The projection of the vector $\bar{i}$ -2 $\bar{j}$ + $\bar{k}$ on the vector 4 $\bar{i}$ -4 $\bar{j}$ +7 $\bar{k}$ is

a)		$\frac{5}{19} \sqrt{5}$
b)		2 $\frac{1}{9}$
c)		$\frac{9}{19}$
d)		$\frac{1}{19} \sqrt{6}$

29. The vector 2 $\bar{i}$ - $\bar{j}$ + $\bar{k}$ , $\bar{i}$ -3 $\bar{j}$ -5 $\bar{k}$ and $\sqrt{3} \bar{i} - 4 \bar{j} - 4 \bar{k} are the sides of a triangle which is$

a)		equilateral
b)		isosceles only
c)		right angled only
d)		right angled and isisceles

30. Area of a parrallelogram whose diagonals are given by the vectors 3i+j-2k and i-3j+4k is

a)		10 $\sqrt{3}$
b)		5 $\sqrt{3}$
c)		8
d)		4

31. The angle between the vectors 2 $\bar{i}$ +3 $\bar{j}$ + $\bar{k}$ and 2 $\bar{k}$ - $\bar{j}$ - $\bar{k}$ is

a)		$\frac{Π}{2}$
b)		$\frac{Π}{4}$
c)		$\frac{Π}{3}$
d)		0

32. If $\bar{a} and \bar{b} are two vectors such that \bar{a} - \bar{b} = 0 and \bar{a} \times \bar{b} = \vec{0} then$

a)		$\bar{a} is parallel to \bar{b}$
b)		$\bar{a} is ⊥ r to \bar{b}$
c)		either $\bar{a} or \bar{b} is a null vector$
d)		none of these

33. If $| \bar{\propto} + \bar{β} | = | \bar{\propto} - \bar{β} | then$

a)		$\bar{\propto} is parallel to \bar{β}$
b)		$\bar{α}$ is $⊥ r to \bar{β}$
c)		$\bar{α} = \frac{1}{2} \bar{β}$
d)		none of these

34. Let $\bar{a} and \bar{b} be two unit vectors . Let \propto be the angle between them, then \bar{a} + \bar{b} is a unit vector if$

a)		$\propto = \frac{Π}{4}$
b)		$\propto = \frac{Π}{3}$
c)		$\propto = 2 \frac{Π}{3}$
d)		$\propto = \frac{Π}{2}$

35. If Q is the angle between two vectors $\vec{a} and \vec{b} then | \frac{\bar{a} \times \bar{b}}{\bar{a} \times \bar{b}} | equal$

a)		cot $θ$
b)		-cot $θ$
c)		tan $θ$
d)		-tan $θ$

36. If $\bar{a}, \bar{b}, \bar{c} are unit vectors such that \bar{a} + \bar{b} + \bar{c} = 0 then the value of \bar{a}$ . $\bar{b} + \bar{b} . \bar{c} + \bar{c} . \bar{a =}$

a)		$\frac{2}{3}$
b)		$\frac{- 2}{3}$
c)		$\frac{- 3}{2}$
d)		$\frac{3}{2}$

37. If $| \bar{a} | = 6, | \bar{b} | = 8, | \bar{a} - \bar{b} | = 10 then | \bar{a} + \bar{b} | is equal to$

a)		10
b)		24
c)		36
d)		40

38. Volume of the parallel piped whose coterminus edges are 2 $\bar{i}$ -3 $\bar{j}$ +4 $\bar{k}$ , $\bar{i}$ +2 $\bar{j}$ -2 $\bar{k}$ , 3 $\bar{i}$ - $\bar{j}$ + $\bar{k}$ is

a)		5 units
b)		6 units
c)		7 units
d)		8 units

39. If ${(\bar{a} \times \bar{b)}}^{2} + {(\vec{a} \vec{b)}}^{2} = 144 and | \vec{a} | = 4 then | \vec{b} | =$

a)		16
b)		8
c)		3
d)		12

40. Let $\bar{a} = 2 \bar{i} + \bar{j} + \bar{k}, \bar{b}$ = $\bar{i}$ +2 $\bar{j}$ - $\bar{k} and a unit vector \bar{c} be coplanar if \bar{c} is perpendicular to \bar{a}$ then $\vec{c} =$

a)		$\frac{1}{\sqrt{2}} (- j + \bar{k})$
b)		$\frac{1}{\sqrt{3}} (- i - j - \bar{k})$
c)		$\frac{1}{\sqrt{5}} (\bar{i} - 2 \bar{j})$
d)		$\frac{1}{\sqrt{3}} (\bar{i} - \bar{j} - \bar{k})$

41. The area of the parallelogram whose adjacent sides are 2 $\bar{i}$ -3 $\bar{k}$ and 4 $\bar{j}$ +2 $\bar{k}$ is

a)		2 $\sqrt{14}$
b)		3 $\sqrt{14}$
c)		4 $\sqrt{14}$
d)		5 $\sqrt{14}$

42. The angle between the vectors $\bar{a} \times \bar{b} and \bar{b} \times \bar{a} is$

a)		$180^{0}$
b)		$90^{0}$
c)		$0^{0}$
d)		$45^{0}$

43. Area of the parallelogram whose diagonals are $\bar{i}$ +2 $\bar{j}$ +3 $\bar{k}$ and - $\bar{i} - 2 \bar{j} + \bar{k} is$

a)		$\frac{1}{2} \sqrt{20}$
b)		$\frac{1}{2} \sqrt{5}$
c)		$\frac{1}{2} \sqrt{80}$
d)		$\frac{1}{2} \sqrt{70}$

44. If $| \vec{a} | = 3 | \vec{b} | = 4 and | \bar{a} + \bar{b} | = 5 then | \bar{a} - \bar{b} | =$

a)		8
b)		5
c)		4
d)		3

45. For any two vectors $\vec{u} and \vec{v} (\bar{u} . \bar{v})^{2} + (\bar{u} . \bar{v})^{2} =$

a)		$\| \vec{u} \|^{2} \| \vec{v} \|^{2}$
b)		2 $\| \vec{u} \|^{2} \| \vec{v} \|^{2}$
c)		4 $\| \vec{u} \|^{2} \| \vec{v} \|^{2}$
d)		none of these

46. If $\bar{a} is a unit vector ⊥ r to \bar{b} and \bar{c} then the second unit vector ⊥ r to \bar{b} and \bar{c}$ is

a)		$\bar{b} \times \bar{c}$
b)		$\bar{a} \times \bar{b}$
c)		- $\vec{a}$
d)		none of these

47. If $\bar{a}, \bar{b}, \bar{c} are non co planar vectors then the vectors 5 \bar{a}$ +6 $\bar{b} + 7 \bar{c}, 7 \bar{a} - 8 \bar{b} + 9 \bar{c} and 3 \bar{a} + 20 \bar{b} + 5 \bar{c} are$

a)		collinear
b)		coplanar
c)		non coplanar
d)		none of these

48. The unit vector parallel to the resultant of the vectors 2 $\bar{i}$ +3 $\bar{j}$ - $\bar{k}$ and 4 $\bar{i}$ -3 $\bar{j}$ +2 $\bar{k}$ is

a)		$\frac{1}{\sqrt{37}} (6 \bar{i} + \bar{k})$
b)		$\frac{1}{\sqrt{37}} (6 \bar{i} + \bar{j})$
c)		$\frac{1}{\sqrt{37}} (6 \bar{j} + \bar{k})$
d)		none of these

49. If G is the centroid of a triangle ABC then $\vec{GA}$ + $\vec{GB}$ + $\vec{GC}$

a)		$\vec{o}$
b)		3 $\vec{GA}$
c)		3 $\vec{GB}$
d)		3 $\vec{GC}$

50. The unit vector $⊥ r to each of the vectors \vec{i}$ + $\vec{2 j}$ +3 $\vec{k} and - 3 \vec{i} - 2 \vec{j} + \vec{k} is$

a)		$\frac{1}{6 \sqrt{5}} (8 \vec{i} - 10 \vec{j} + 4 \vec{k})$
b)		8 $\vec{i}$ -10 $\vec{j}$ +4 $\vec{k}$
c)		8 $\vec{i}$ +10 $\vec{j}$ +4 $\vec{k}$
d)		none of these