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Question Bank No: 2

1. ( $\bar{a} \times \bar{b})^{2} is equal to$

a)		${\vec{a}}^{2} {\vec{b}}^{2} - (\bar{a} . \bar{b})^{2}$
b)		${\vec{a}}^{2} {\vec{b}}^{2} + (\bar{a} . \bar{b})^{2}$
c)		( $\bar{a} \bar{b})^{2}$
d)		$a^{2} b^{2}$

2. $\bar{i}$ (j $\times$ k)+j(k $\times$ i)+k(i $\times$ j) is equal to

a)		0
b)		-3
c)		-1
d)		3

3. Given two vectors $\bar{a} = 2 i - 3 j + 6 \bar{k}$ $\bar{b} = 2 i + 2 j - \bar{k} and λ = \frac{The projection of \bar{a} on \bar{b}}{The projection of \bar{b} on \bar{a}}$ then the value of $λ is$

a)		$\frac{3}{7}$
b)		7
c)		3
d)		$\frac{7}{3}$

4. The vector $\frac{1}{4} \bar{i} - \frac{3}{4} \bar{j} + \frac{1}{2} \bar{k} is$

a)		a unit matrix
b)		parallel to the vector 4 $\bar{i} - 12 \bar{j} + 8 \bar{k}$
c)		perpendicular to vector 2 $\bar{i} + \bar{j} + \bar{k}$
d)		none of these

5. If $\bar{a}$ $\times \bar{b} = \bar{b} \times \bar{c} \neq 0 then for any scalar k$

a)		$\bar{a}$ + $\bar{c} = K \bar{b}$
b)		$\bar{a}$ + $\bar{b} = K \bar{c}$
c)		5+ $\bar{c} = K \bar{a}$
d)		$\bar{a}$ - $\bar{b} = K (\bar{c} - \bar{b})$

6. If two vectors $\bar{a}$ and $\bar{b}$ be such that $| \bar{a} + \bar{b} | = | \bar{a} - \bar{b} | then the angle between them is$

a)		$0^{0}$
b)		$45^{0}$
c)		$90^{0}$
d)		$60^{0}$

7. $\bar{a} \times \bar{b} is a vector$

a)		parallel to $\bar{a}$
b)		perpendicular to $\bar{a}$ and $\bar{b}$ both
c)		parallel to $\bar{b}$
d)		perpendicular to $\bar{a}$

8. If Q is the angle between two unit vectors $\bar{a}, \bar{b}, then \cos θ is equal to$

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

9. If $\bar{a} and \bar{b}$ are position vectors of A and B respectively, then the position vectors of a point C in AB produced such that $\vec{AC}$ =3 $\overset{⇀}{AB}$ is

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

10. Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ , $\vec{b} = \hat{i} - \hat{j} + 2 \hat{k}$ and $\vec{c} = x \hat{i} + (x - 2) \hat{j} - \hat{k}$ . If the vector $\vec{c}$ lies in the plane of $\vec{a}$ and $\vec{b}$ , then x is

a)		1
b)		-4
c)		-2
d)		0

11. If $\vec{u}$ and $\vec{v}$ are unit vectors and $θ$ is the acute angle between them, then 2 $\vec{u} \times$ 3 $\vec{v}$ is a unit vector for

a)		more than two values of $θ$
b)		no value of $θ$
c)		exactly one value of $θ$
d)		exactly two values of $θ$

12. The resultant of two forces PN and 3N is a force 7N . If the direction of the 3N forces were reserved , the resultant would be $\sqrt{19}$ N. Then P =

a)		6
b)		3
c)		4
d)		5

13. Let $\vec{a}$ , $\vec{b}$ , $\vec{c}$ be unit vectors such that $\vec{a} + \vec{b} + \vec{c}$ = 0. Which one of the following is correct?

a)		$\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} = 0$
b)		$\vec{a} \times \vec{b} = \vec{b} \times$ $\vec{c} = \vec{c} \times \vec{a} \neq 0$
c)		$\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{a} \times \vec{c} \neq 0$
d)		$\vec{a} \times \vec{b}$ , $\vec{b} \times \vec{c}$ , $\vec{c} \times \vec{a}$ are mutually perpendicular.

14. The number of distinct real values of $λ$ , for which the vectors $- λ^{2} \hat{i} +$ $\hat{j} + \hat{k}$ , $\hat{i} - λ^{2} \hat{j} + \hat{k}$ and $\hat{i} + \hat{j} - λ^{2} \hat{k}$ are coplanar, is

a)		0
b)		1
c)		2
d)		3

15. The values of a, for which the points A, B, C with position vectors 2 $\hat{i} - \hat{j} + \hat{k}$ , $\hat{i} - 3 \hat{j} - 5 \hat{k}$ , a $\hat{i} - 3 \hat{j} + \hat{k}$ respectively are the vertices of a right- angled triangle with C = $\frac{π}{2}$ are

a)		-2, 1
b)		2, -1
c)		1, 2
d)		-1, -2

16. ABC is a triangle, right angled at A. The resultant of forces acting along $\vec{AB}$ , $\vec{AC}$ with magnitudes $\frac{1}{AB}$ and $\frac{1}{AC}$ respectively is the force along $\vec{AD}$ , where D is the foot of the perpendicular from A onto BC. The square of the magnitude of the resultant is

a)		$\frac{1}{AB} + \frac{1}{AC}$
b)		$\frac{1}{AD}$
c)		$\frac{{AB}^{2} + {AC}^{2}}{{(AB)}^{2} {(AC)}^{2}}$
d)		$\frac{(AB) (AC)}{AB + AC}$

17. 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 $\vec{a} . \vec{b} \neq 0$ , $\vec{b} . \vec{c}$ $\neq 0$ , then $\vec{a}$ and $\vec{c}$ are

a)		perpendicular
b)		parallel
c)		inclined at an angle $\frac{π}{3}$
d)		inclined at an angle $\frac{π}{6}$

18. Point ( $α, β, γ$ ) lies on the plane x + y + z = 2. Let $\vec{a}$ = $α \hat{i}$ + $β \hat{j}$ + $γ \hat{k}$ , $\hat{k}$ $\times (\hat{k}$ $\times$ $\vec{a})$ = 0. Then $γ$ =

a)		0
b)		1
c)		2
d)		$\frac{1}{2}$

19. Let $\vec{A}$ be vector parallel to the line of intersection of planes $P_{1}$ and $P_{2}$ through the origin. $P_{1}$ is parallel to the vectors $\vec{a}$ = 2 $\hat{j}$ +3 $\hat{k}$ and $\vec{b}$ = 4 $\hat{j}$ -3 $\hat{k}$ and $P_{2}$ is parallel to the vectors $\vec{c}$ = $\hat{j} - \hat{k}$ and $\vec{d}$ = 3 $\hat{j}$ +3 $\hat{j}$ . The angle between $\vec{A}$ and 2 $\hat{i} + \hat{j} - 2 \hat{k}$ is

a)		$\frac{π}{2}$
b)		$\frac{π}{8}$
c)		$\frac{π}{6}$
d)		$\frac{3 π}{4}$

20. $\vec{a}$ and $\vec{b}$ are unit vectors imclined at an angle $α$ , $α \in [0, π]$ to each other and $| \vec{a} + \vec{b} |$ $<$ 1. Then $α \in$

a)		$(\frac{π}{3}, \frac{2 π}{3})$
b)		$(\frac{2 π}{3}, π)$
c)		$(0, \frac{π}{3})$
d)		$(\frac{π}{4}, \frac{3 π}{4})$

21. A line passes through the point 3 $\hat{i}$ and is parallel to the vector - $\hat{i} + \hat{j} + \hat{k}$ and another line passes through the point $\hat{i} + \hat{j}$ and is parallel to the vector $\hat{i} + \hat{k}$ , then they intersect at the point

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

22. The plane x+ 2y - z = 4 cuts the sphere $x^{2} + y^{2} + z^{2} - x + z - 2$ = 0 in a circle of radius

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

23. If the plane 2ax-3ay+4az+6 = 0 passes through the midpoint of the line joining the centres of the spheres $x^{2} + y^{2} + z^{2} + 6 x - 2 z$ =13 and $x^{2} + y^{2} + z^{2} - 10 x + 4 y - 2 z$ =8 , then a =

a)		1
b)		-1
c)		2
d)		-2

24. The resultant $\vec{R}$ of two forces acting on a particle is right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is

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

25. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are non- coplanar vectors and $λ$ is a real number, then $[λ (\vec{a} + \vec{b}), λ^{2} \vec{b}, λ \vec{c}]$ = $[\vec{a}, \vec{b} + \vec{c}, \vec{b}]$ for

a)		no value of $λ$
b)		exactly one value of $λ$
c)		exactly two values of $λ$
d)		exactly three values of $λ$

26. For any vector $\vec{a}$ , the value of ${(\vec{a} \times \hat{i})}^{2} + {(\vec{a} \times \hat{j)}}^{2} + {(\vec{a} \times \hat{k)}}^{2}$ =

a)		${\| \vec{a} \|}^{2}$
b)		${3 \| \vec{a} \|}^{2}$
c)		${4 \| \vec{a} \|}^{2}$
d)		${2 \| \vec{a} \|}^{2}$

27. The distance between the line $\vec{r} = 2 \hat{i} - 2 \hat{j} + 3 \hat{k} + λ$ ( $\hat{i} - \hat{j} + 4 \hat{k}$ ) and the plane $\vec{r} . (\hat{i} + 5 \hat{j} + \hat{k})$ = 5 is

a)		$\frac{10}{3 \sqrt{3}}$
b)		$\frac{10}{9}$
c)		$\frac{10}{3}$
d)		$\frac{3}{10}$

28. If C is the midpoint of AB and P is any point outside AB, then

a)		$\vec{PA} + \vec{PB} = \vec{PC}$
b)		$\vec{PA} + \vec{PB} = 2 \vec{PC}$
c)		$\vec{PA} + \vec{PB} + \vec{PC} = 0$
d)		$\vec{PA} + \vec{PB} + 2 \vec{PC} = 0$

29. Three forces $\vec{P}$ , $\vec{Q}$ , $\vec{R}$ acting along IA, IB, IC, where I is the incentre of a triangle ABC, are in equilibrium. Then $| \vec{P} |$ : $| \vec{Q} |$ : $| \vec{R} |$ =

a)		cos $\frac{A}{2}$ : cos $\frac{B}{2}$ : cos $\frac{C}{2}$
b)		sin $\frac{A}{2}$ : sin $\frac{B}{2}$ : sin $\frac{C}{2}$
c)		sec $\frac{A}{2}$ : sec $\frac{B}{2}$ : sec $\frac{C}{2}$
d)		cosec $\frac{A}{2}$ : cosec $\frac{B}{2}$ : cosec $\frac{C}{2}$

30. Let $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are nonzero vectors such that ( $\vec{a} \times \vec{b}$ ) $\times \vec{c}$ = $\frac{1}{3}$ $| \vec{b} | | \vec{c} |$ $\vec{a}$ . If $θ$ is the acute angle between the vectors $\vec{b}$ and $\vec{c}$ , then sin $θ$ =

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

31. Let $\vec{u}$ , $\vec{v}$ , $\vec{w}$ be such that $| \vec{u} |$ =1, $| \vec{v} |$ = 2, $| \vec{w} |$ =3. If the projection of $\vec{v}$ along $\vec{u}$ is equal to that of $\vec{w}$ along $\vec{u}$ and $\vec{v}$ , $\vec{w}$ are perpendicular to each other, then $| \vec{u} - \vec{v} + \vec{w} |$ =

a)		2
b)		$\sqrt{7}$
c)		$\sqrt{14}$
d)		14

32. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are non coplanar vectors and $λ$ is a scalar, then the vectors $\vec{a} + 2 \vec{b} + 3 \vec{c}$ , $λ \vec{b} + 4 \vec{c}$ , ( $2 λ$ -1) $\vec{c}$ are noncoplanar for

a)		all values of $λ$
b)		all except one value
c)		all except two values
d)		none value of $λ$

33. The resultant of forces $\vec{P}$ and $\vec{Q}$ is $\vec{R}$ . If $\vec{Q}$ is doubled, then $\vec{R}$ is doubled. If the direction of $\vec{Q}$ is reserved , then $\vec{R}$ is again doubled. Then $P^{2}$ : $Q^{2}$ : $R^{2}$ =

a)		3 : 1 : 1
b)		2: 3 : 2
c)		1 : 2 : 3
d)		2 : 3 : 1

34. Let $\vec{u}$ = $\hat{i}$ + $\bar{j}$ , $\vec{v}$ = $\hat{i}$ - $\hat{j}$ , $\vec{w}$ = $\hat{i}$ + 2 $\hat{j}$ +3 $\hat{k}$ . If $\vec{n}$ is a unit vector such that $\vec{u} . \vec{n}$ = 0 and $\vec{v} . \vec{n}$ = 0, then $| \vec{w} . \vec{n} |$ =

a)		0
b)		1
c)		2
d)		3

35. A particle acted on by constant forces 4 $\hat{i}$ + $\hat{j}$ _3 $\hat{k}$ and 3 $\hat{i}$ + $\hat{j}$ - $\hat{k}$ is displaced from the point (1, 2, 3) to the point (5, 4, 1). The work done =

a)		20
b)		30
c)		40
d)		50

36. The vectors $\vec{AB}$ = 3 $\hat{i}$ +4 $\hat{k}$ and $\vec{AC}$ = 5 $\hat{i}$ -2 $\hat{j}$ +4 $\hat{k}$ are two sides of a triangle ABC. The length of the median through A is

a)		$\sqrt{18}$
b)		$\sqrt{72}$
c)		$\sqrt{33}$
d)		$\sqrt{288}$

37. If $\vec{u}$ , $\vec{v}$ , $\vec{w}$ are three vectors, then ( $\vec{u} + \vec{v} - \vec{w}$ ) $.$ ( $\vec{u} - \vec{v}$ ) $\times$ ( $\vec{v} - \vec{w}$ ) =

a)		0
b)		$\vec{u} . \vec{v} \times$ $\vec{w}$
c)		$\vec{u} . \vec{w} \times \vec{v}$
d)		3 $\vec{u} . \vec{v} \times \vec{w}$

38. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are three vectors such that $\vec{a} +$ $\vec{b} + \vec{c} = 0$ , $| \vec{a} |$ = 1, $| \vec{b} |$ = 2, $| \vec{c} |$ = 3, then $\vec{a} .$ $\vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}$ =

a)		0
b)		-7
c)		7
d)		1

39. A tetrahedron has vertices O (0, 0, 0), A (1, 2, 1), B (2, 1, 3), C(-1, 1, 2). The angle between the faces OAB and ABC is

a)		$\cos^{- 1} (\frac{19}{35})$
b)		$\cos^{- 1}$ $(\frac{17}{31})$
c)		$30^{\circ}$
d)		$90^{\circ}$

40. If $\vec{a} \times$ $\vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} \neq \vec{0}$ , then $\vec{a} + \vec{b} + \vec{c}$ =

a)		- $\vec{b}$
b)		$\vec{0}$
c)		2 $\vec{a}$
d)		none of these

41. If $\vec{a}$ = 3 $\hat{i} -$ 5 $\hat{j}$ , $\vec{b}$
= 6 $\hat{i} +$ 3 $\hat{j}$ and $\vec{c}$ = $\vec{a} \times \vec{b}$ , then $| \vec{a} |$ : $| \vec{b} |$ : $| \vec{c} |$ =

a)		$\sqrt{34} : \sqrt{45} : \sqrt{39}$
b)		$\sqrt{34} : \sqrt{45} : 39$
c)		34 : 39 : 45
d)		39 : 35 : 34

42. If $| \vec{a} |$ = 5 , $| \vec{b} |$ = 4 , $| \vec{c} |$ = 3 and $\vec{a} + \vec{b} + \vec{c}$ = 0 , then $| \vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} |$ =

a)		25
b)		50
c)		-25
d)		-50

43. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are vectors such that $\vec{a} + \vec{b} + \vec{c}$ = 0 and $| \vec{a} |$ = 7, $| \vec{b} |$ =5, $| \vec{c} |$ =3, then angle ( $\vec{b}$ , $\vec{c}$ ) =

a)		$60^{\circ}$
b)		$30^{\circ}$
c)		$45^{\circ}$
d)		$90^{\circ}$

44. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are vectors such that $[\vec{a} \vec{b} \vec{c}]$ = 4 , then $[\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}]$ =

a)		16
b)		64
c)		4
d)		8

45. If $| \vec{a} |$ = 4, $| \vec{b} |$ = 2 and ( $\vec{a}$ , $\vec{b}$ ) = $\frac{π}{6}$ , then ${(\vec{a} \times \vec{b})}^{2}$ =

a)		48
b)		16
c)		$\vec{a}$
d)		none of these

46. If $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are non zero , non coplanar vectors and ${\vec{b}}_{1}$ = $\vec{b} - \frac{\vec{b} . \vec{a}}{{| \vec{a} |}^{2}} \vec{a}$ , ${\vec{b}}_{2}$ = $\vec{b} + \frac{\vec{b} . \vec{a}}{{\vec{a}}^{2}} \vec{a}$ , ${\vec{c}}_{1}$ = $\vec{c} - \frac{\vec{c} . \vec{a}}{{| \vec{a} |}^{2}} \vec{a} + \frac{\vec{b} . \vec{c}}{{| \vec{c} |}^{2}} {\vec{b}}_{1}$ , ${\vec{c}}_{2}$ = $\vec{c} - \frac{\vec{c} . \vec{a}}{{| \vec{a} |}^{2}} \vec{a} - \frac{{\vec{b}}_{1} . \vec{c}}{{| \vec{b_{1}} |}^{2}} {\vec{b}}_{1}$ , ${\vec{c}}_{3}$ = $\vec{c} - \frac{\vec{c} . \vec{a}}{{| \vec{c} |}^{2}}$ $\vec{a} + \frac{\vec{b} . \vec{c}}{{| \vec{c} |}^{2}}$ ${\vec{b}}_{1}$ , ${\vec{c}}_{4}$ = $\vec{c} - \frac{\vec{c} . \vec{a}}{{| \vec{c} |}^{2}} \vec{a} - \frac{\vec{b} . \vec{c}}{{| \vec{b} |}^{2}} {\vec{b}}_{1}$ , then the set of orthogonal vectors is

a)		( $\vec{a}$ , ${\vec{b}}_{1}$ , ${\vec{c}}_{3}$ )
b)		( $\vec{a}$ , ${\vec{b}}_{1}$ , ${\vec{c}}_{2}$ )
c)		( $\vec{a}$ , ${\vec{b}}_{1}$ , ${\vec{c}}_{1}$ )
d)		( $\vec{a}$ , ${\vec{b}}_{2}$ , ${\vec{c}}_{2}$ )

47. The unit vector which is orthognal to the vector which is orthogonal to the vector $5 \hat{i} + \hat{j} + \hat{k}$ and $\hat{i} -$ $\hat{j} +$ $\hat{k}$ is

a)		$\frac{2 \hat{i} - 6 \hat{j} + \hat{k}}{\sqrt{41}}$
b)		$\frac{2 \hat{i} - 5 \hat{j}}{\sqrt{29}}$
c)		$\frac{3 \hat{j} - \hat{k}}{\sqrt{10}}$
d)		$\frac{2 \hat{i} - 8 \hat{j} + \hat{k}}{\sqrt{69}}$

48. If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ , $\vec{a} .$ $\vec{b} = 1$ and $\vec{a}$ $\times \vec{b}$ = $\hat{j} - \hat{k}$ , then $\vec{b}$ is

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

49. If the volume of the parallelopiped with coterminal edges $\hat{i} + a \hat{j} + \hat{k}$ , $\hat{j} + a \hat{k}$ , $a \hat{i} + \hat{k}$ is minimum, then a =

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

50. Let $\vec{V}$ = 2 $\hat{i}$ + $\hat{j}$ - $\hat{k}$ and $\vec{W}$ = $\hat{i}$ +3 $\hat{k}$ . If $\vec{U}$ is a unit vector, then the maximum value of $[\vec{U} \vec{V} \vec{W}]$ is

a)		-1
b)		$\sqrt{10} + \sqrt{6}$
c)		$\sqrt{59}$
d)		$\sqrt{60}$