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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
a × b + ( )
a. b
2
(A)
a
b
2 (B)
a
b
2
(C)
a
b
2 (D) None of these
2. The area of a triangle whose vertices are, A (1, - 1, 2), B (2, 1, - 1) and C (3, - 1, 2) is :
(A) 13 (B) 13
(C) 6 (D) 6
3. If
a & b are two non - zero vectors,
then the component of
b along
a
is
(A)
^
a b a b b
(B)
^
a b b a a
(C)
^
a b b a b
(D)
^
a b a a a
4. If
^
a + b + c = 0, then which relation is correct.
(A)
^
a = b = c = 0
(B)
^ ^
a. b = b. c =c .a
(C) ^
a × b = b × c = c ×a (D) None of these
5. If ABCDEF is a regular hexagon and
AB AC AD AE AF
→ → → → →
→ , then λ = (A) 2 (B) 3 (C) 4 (D) 6
6. If O be the circumcentre and O′ be the orthocentre of a triangle ABC, then
OA OB OC
→ → →
(A) 2 OO′
→ (B) (^2) ′
→ O O
(C)
OO′
→
(D) ′
→ O O
7. If in the given fig.
OA
→
a (^) ,
OB
→
b
and AP : PB = m : n, then
OP
→
(A) mam^ n bn
+ (B)^
n a m b m n
(C) m
a - n
b
(D)
ma n b m n
8. If
^ a^ =^2 i^ +^2 j^ +^3 k^ , b = − i + 2 j^ +k and
c = 3 i +j^ , then
a
b is perpendicular to
c
if t = (A) 2 (B) 4 (C) 6 (D) 8
9. The area of the parallelogram whose diagonals are,
a = 3 i + j^ − 2 k
and
b = i − 3 j^ + 4 k
is :
(A) 10 3 (B) 5 3 (C) 8 (D) 4
10.
a. {(
b
c
) × (
a
b
c
(A) 0
(B) [
a
b
c
] + [
b
c
a
]
(C) [
a
b
c
] (D) None of these
O
A P B
Vectors
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
11. If the vectors
2 i^ − 3 j
, i + j^ − k and
3 i − k form three concurrent edges of a parallelopiped, then the volume of the parallelopiped is : (A) 8 (B) 10 (C) 4 (D) 14
12. (^ )^.
^
a × b c = a
b
c
, if :
(A)
a
b
b
c
(B)
b
c
c
a
(C)
c
a
a
b
(D)
a
b
b
c
c
a
13. If
a
b
c
are unit vectors such that
a
b
c
= 0, then
a
b
b
c
c
a
(A) 1 (B) 3
(C) - 3
(D) 3
14. If the position vectors of the points
A, B, C be
a
b
a
b
respectively then the points A, B, C are : (A) Collinear (B) Non − collinear (C) Form a right angled triangle (D) None of these
15. If
a
b
are the position vectors of A & B respectively, then the position vector of a point C on AB produced
such that
AC
→
= 3 AB
→ is : (A) 3
a -
b
(B) 3
b
a
(C) 3
a
b
(D) 3
b
a
16. The position vectors of the points A,
B & C are
i (^) +j
, j^ + k and k^ +i respectively. The vector area of the
∆ ABC = ± (^12)
α
, where
α
(A)
− i + j^ +k
(B) i − j^ +k
(C) i^ + j^ − k (D) i^ + j^ +k
17. If
a = (1, − 1, 1) &
c
then the vector
b
satisfying, a ×
b
c &
a
b
= 1, is : (A) (1, 0, 0) (B) (0, 0, 1) (C) (0, − 1, 0) (D) None of these
18. If
a ×
b
b ×^
c ≠^ 0, then for some scalar k : (A)
a
c
= k
b
(B)
a +
b
= k
c (C) b
c = k a
(D) None of these
19. P is the point of intersection of the diagonals of the parallelogram ABCD. If O is any point, then
OA OB OC OD
→ → → →
is :
(A) OP
→ (B) 2 OP
→
(C) 3 OP
→ (D) (^4) OP
→
20. A unit vector in the xy − plane which is perpendicular to 4 i − 3 j^ + k is :
(A)
i (^) +j 2
(B) 15 ( 3 i + 4 j)
(C) 15 ( 3 i^ − 4 j) (D) None of these
21. If the position vectors of the points
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(A) 169 ( i^ +^2 j^ −^8 k)
(B) 691 ( i^ + 2 j^ − 8 k)
(C) 691 ( −^ i^ −^2 j^ +^8 k)
(D) 691 ( − i − 2 j^ + 8 k)
30. Let
b = 3 j + 4 k^ ,
a = i +j
and let
b 1 &b 2
be component vectors of
b
parallel and perpendicular to
a
. If
b 1
= 32 i^ +^32 j, then
b 2 =
(A) 32 i^ +^32 j^ +^4 k
(B) − 32 i^ + 32 j^ + 4 k
(C) − 23 i + 32 j (D) None of these
31. If the points whose position vectors
are 3 i^ − 2 j^ − k, 2 i + 3 j^ − 4 k,
− i^ + j^ + 2 k and 4 i^ + 5 j^ + λ k lie on a plane, then λ =
(A) - 14617 (B)
(C) -
(D)
32. A and B are two points. The position vector of A is 6b − 2a. A point P divides the line AB in the ratio 1 : 2. If
a^ ^ −b
is the position vector of P, then the position vector of B is given by : (A) 7 a^ − 15 b (B) 7 a^ + 15 b
(C) 15 a^ −^7 b (D) 15 a^ +^7 b
33. If
a & b are unit vectors making an
angle θ with each other then
a − b is : (A) 1 (B) 0
(C) cos θ 2 (D) 2 sin
θ 2
34. If the vectors,
a i + j +k
, i + b j^ +k
and i^ + j^ + ck (a ≠ b ≠ c ≠ 1) are coplanar, then the value of, 1 1
− a +^ − b +^1 −c =
(A) - 1 (B) - (^12)
(C)
(D) 1
35. If C is the middle point of AB and P is any point outside AB, then :
(A)
PA PB PC
→ → →
(B) PA PB PC
→ → →
(C) PA PB PC
→ → →
(D) PA PB PC
→ → →
36.
^
a , b , c are three non − zero, non − coplanar vectors and
p , q , r are three other vectors such that,
p (^) ^ b c a b c
= ×
. ×
^
q (^) ^ c a a b c
= ×
. ×
and
^
r (^) ^
a b a b c
×
. ×
, then [
p , q , r] equals
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(A) ^
a. b × c (B)^ ^1 a. b ×c (C) 0 (D) None of these
37. Let
a = 2 i − j +k^ ,
b = i + 2 j −k
c = i + j^ − 2 k be three vectors. A
vector in the plane of
b & c whose
projection on
a is of magnitude
is : (A) 2 i − 3 j^ + 3 k (B) 2 i + 3 j^ + 3 k
(C) − 2 i^ − j^ + 5 k (D) 2 i^ + j^ + 5 k
38. The magnitudes of mutually
perpendicular forces
^
a , b & c are 2, 10 and 11 respectively. Then the magnitude of its resultant is : (A) 12 (B) 15 (C) 9 (D) None of these
39. A vector (^) a has components 2p & 1 with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If (^) a has components p + 1 & 1 w.r.t. the new system, then :
(A) p = 0 (B) p = 1 or − (^13)
(C) p = − 1 or
(D) p = 1 or − 1
40. Let the value of ,
p
= (x + 4y) a + (2x + y + 1)
b and
q
= (y − 2x + 2) a + (2x - 3y - 1)
b , where
a
b are^ non-collinear vectors. If 3
p
q , then the value of x & y, will be : (A) - 1, 2 (B) 2, - 1 (C) 1, 2 (D) 2, 1
41. If (x, y, z) ≠ (0, 0, 0) and
( i + j^ + 3 k)x +^ ( 3 i − 3 j^ + k)y +
( − 4 i^ + 5 j) z =^ λ^ (^ x i^ +^ y j^ +^ z k), then
the value of λ will be : (A) - 2, 0 (B) 0 , - 2 (C) - 1, 0 (D) 0, - 1
42. If three non - zero are, a = a i 1 + a j 2 +a k 3 ^ ,
b = b i 1 + b j 2 ^ +b k 3
and
c = c i 1 + c j 2 +c k 3
. If
c
is the unit vector perpendicular to the vectors a & b and the angle between^
a &b
is π 6 , then
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
2
is equal to :
(A) 0 (B)
1
2 1
2 1 ( Σ a ) ( Σ b ) ( Σc^2 )
(C) 1 (D)
( Σ a 12 ) ( Σb 12 ) 4
43. The position vector of coplanar points A, B, C, D are
^
a , b , c &d
respectively in such a way that,
^ ^
a − d b − c =^ ( ) .( )
b − d c − a = 0, then the point D of the ∆ ABC is : (A) Incentre (B) Circumcentre
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
53. If
^
a , b , c are coplanar vectors, then
(A)
a b c
b c a
c a b
(B)
a a a b a
b a b b b
c a c b c
(C)
a c a b a
b c b b c
c c c b b
(D)
a a b c a
b a a c c
c a c c b
54. A unit vector which is coplanar to
vector, i + j^ + 2 k and i^ + 2 j^ +k and
perpendicular to i^ + j^ + k, is :
(A)
i (^) −j 2 (B)^ ±^
j^ −k
(C)
k (^) −j 2
(D)
i + j (^) +k 3
55. If
x. a = 0 ,
x. b = 0 &^
x. c= 0 for
some non - zero vector
x , then the true statement is :
(A) [ ^ ]
a b c = 0^ (B)^ [ ]
^
a b c ≠^0
(C) [ ]
^
a b c = 1^ (D) None of these
56. If
a has magnitude 5 and points north-east & vector
b
has magnitude 5 & points north-west, then
a
b
is equal to : (A) 25 (B) 5
(C) 7
(D) 5
57. In a regular hexagon ABCDEF, AE
→
is equal to :
(A) AC AF AB
→ → →
→ → →
(C) AC AB AF
→ → →
58. OD DA DB DC
→ → → →
(A) OA OB OC
→ → →
→ → →
(C) OA OB OC
→ → →
59. In a ∆ ABC, if (^2) AC
→ = 3 (^) CB
→ , then
2 OA
→
→ equals :
(A) 5 OC
→ (B) - OC
→
(C) OC
→ (D) None of these
60. If AO^ OB
→ →
→ →
- (^) , then A, B, C form : (A) Equilateral triangle (B) Eight angled triangle (C) Isosceles triangle (D) Line
61. If the position vectors of A and B are i (^) + 3 j (^) − 7 k and 5 i − 2 j (^) + 4 k, then
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
the direction cosine of (^) AB
→ along y - axis is :
(A) 4 162
(B) - 5
(C) - 5 (D) 11
62. The point B divides the arc AC of a quadrant of a circle in the ratio 1 : 2.
If O is the centre and (^) OA
→
a and
OB
→
b , then the vector
OC
→
is : (A)
b − 2 a (B) 2
a −b
(C) (^3 )
b − a (D)^ None of these
63. The points D, E, F divide BC, CA & AB of the triangle ABC in the ratio 1 : 4, 3 : 2 & 3 : 7 respectively & the point K divides AB in the ratio 1 : 3,
then (^ AD^ BE^ CF)
→ → →
→
(A) 1 : 1 (B) 2 : 5 (C) 5 : 2 (D) None of these
64. If vector
a = 2 i − 3 j + 6 k^ and vector
b = − 2 i + 2 j −k
, then
Pr Pr
ojection of vector a on vector b ojection of vector b on vector a
(^)
(A) 37 (B)
(C) 3 (D) 7
65. If
r
be position vector of any point on a sphere &
a &b
are respectively position vectors of the extremities of a diameter, then : (A)
r.^ (
a −b
) = 0
(B)
r. (
r −a
) = 0
(C) (
r +a
). (
r +b
) = 0
(D) (
r − a). (
r −b
) = 0
66. If
^
a × (b ×c) = 0, then :
(A)
| | | | .| |
^
a = b c
(B)
b parallel to
c
(C)
a
parallel to
b
(D)
b
perpendicular to
c
67. If
^
a , b &c
are three non-coplanar vectors, then
( ). [ ( ) ( )]
^ ^
a + b + c a + b × a + c =
(A) [ ]
^
a b c (B) 2 [ ]
^
a b c
(C) - (^) [ ^ ]
a b c (D)^0
68. If
^
a , b , c are non-coplanar unit
vectors such that,
^
a × (b ×c) =
b +c 2
then the angle between
a & b is :
(A) π 4 (B)
π 2
(C)
π
(D) π
69. Given
a = i + j^ −k
b = − i + 2 j +k
c = − i + 2 j −k. A unit vector perpen-
dicular to both
a + b &^
b + c is :
(A) i (B)
j
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
b
a
≠ λ
b
a
is not perpen. to
b
, then
r
(A)
a
b
(B)
a
b
(C)
a
×
b
a
(D)
a
×
b
b
79. A non-zero vector
a
is parallel to the line of intersection of the plane determined by the vectors,
i (^) , i (^) +j
the plane determined by the vectors, i − j (^) , i (^) + k. The angle between (^) a &
the vector
i (^) − 2 j (^) +k
is :
(A) π^
π 4
or (^4) (B)
π π or
(C) π^
π 2
or 2 (D) None of these
80. If
b & c are any two non-collinear unit vectors and
a is any vector, then
( )
( )
..
( )
^ ^ ^
a b b a c c a^ ^ b^ c b c
×
= ×
(A) a (B)
b
(C)
c
(D) 0
81. The value of x for which the angle
between the vectors,
a = − 2 i + x j^ +k
and
b = x i + 2 x j +k
is acute & the
angle between
b and x^ -^ axis lies between π/2 and π satisfy : (A) x > 0 (B) x < 0 (C) x > 1 only (D) x < - 1 only
82. If the sum of two unit vectors is a unit vector, then the magnitude of their difference is :
(A)
(B) 3
(C) 2 (D) 5
ANSWERS
1. C 2. B 3. D 4. C 5. B 6. C
7. B 8. D 9. B 10. B 11. C 12.D
13. C 14. A 15. D 16. D 17. B 18.A
19. D 20. B 21. B 22. C 23. A 24.A
25. D 26. B 27. A 28. A 29. C 30.B
31. A 32. A 33. D 34. D 35. B 36.B
37. C 38. B 39. B 40. B 41. D 42.D
43. C 44. C 45. C 46. C 47. A 48.D
49. B 50. D 51. B 52. A 53. B 54.B
55. A 56. D 57. B 58. C 59. A 60.C
61. B 62. C 63. B 64. B 65. D 66.B
67. C 68. C 69. C 70. C 71. C 72.B
73. D 74. B 75. A 76. C 77. C 78.B
79. A 80. A 81. B 82. B
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