Choose the correct or the most suitable answer from the given four alternatives :
1. If and
are parallel vectors, then [
,
,
] is equal to
(1) 2
(2) -1
(3) 1
(4) 0
2. If a vector lies in the plane of
and
, then
Ans: (3)
3. If = 0, then the value of [
,
,
] is
(3) 1
(4) -1
Ans: (1)
4. If ,
,
are three unit vectors such that
is perpendicular to
, and is parallel to
then
x (
x
) is equal to
(1)
(2)
(3)
(4)
Ans: (2)
5. If [,
,
] = 1, then the value of
is
(1) 1
(2) -1
(3) 2
(4) 3
6. The volume of the parallelepiped with its edges represented by the vectors iˆ + ˆj, iˆ + 2 ˆj, iˆ + ˆj + π kˆ is
(1) π/2
(2) π/3
(3) π
(4) π/4
7. If and
are unit vectors such that [
,
,
×
] = π/4, then the angle between
and
is
(1) π/6
(2) π/4
(3) π/3
(4) π/2
8. If and (
×
)×
= λ
+ μ
, then the value of λ + μ is
(1) 0
(2) 1
(3) 6
(4) 3
9. If ,
,
are non-coplanar, non-zero vectors such that [
,
,
] = 3, then
is equal to
(1) 81
(2) 9
(3) 27
(4)18
10. If ,
,
are three non-coplanar vectors such that
×(
×
) =
, then the angle between
and
is
(1) π/2
(2) 3Ï€/4
(3) π/4
(4) π
11. If the volume of the parallelepiped with ×
,
×
,
×
as coterminous edges is 8 cubic units, then the volume of the parallelepiped with (
×
)×(
×
), (
×
) ×(
×
) and (
×
) ×(
×
) as coterminous edges is,
(1) 8 cubic units
(2) 512 cubic units
(3) 64 cubic units
(4) 24 cubic units
12. Consider the vectors ,
,
,
such that (
×
) ×(
×
) = 0 . Let P1 and P2 be the planes determined by the pairs of vectors
,
and
,
respectively. Then the angle between P1 and P2 is
(1) 0Ëš
(2) 45Ëš
(3) 60Ëš
(4) 90Ëš
13. If ×(
×
) = (
×
) ×
, where
,
,
are any three vectors such that
.
≠0 and
.
≠0 ,then
and
are
(1) perpendicular
(2) parallel
(3) inclined at an angle π/3
(4) inclined at an angle π/3
14. If = 2ˆi + 3ˆ j - ˆk,
= iˆ + 2 ˆj - 5kˆ,
= 3ˆi + 5 ˆj - ˆk, then a vector perpendicular to
and lies in the plane containing
and
is
(1) -17iˆ + 21 ˆj - 97kˆ
(2) 17iˆ + 21 ˆj -123kˆ
(3) -17iˆ - 21 ˆj + 97kˆ
(4) -17iˆ - 21 ˆj - 97kˆ
15. The angle between the lines is
(1) π/6
(2) π/4
(3) π/3
(4) π/2
16. If the line lies in the plane x + 3y - α z + β = 0, then (α , β ) is
(1) (-5, 5)
(2) (-6, 7)
(3) (5, -5)
(4) (6, -7)
17. The angle between the line = (ˆ i + 2 ˆ j - 3 ˆ k ) + t(2 ˆ i + ˆ j - 2 ˆ k ) and the plane
= (ˆ i + ˆ j) + 4 = 0 is
(1) 0Ëš
(2) 30Ëš
(3) 45Ëš
(4) 90Ëš
18. The coordinates of the point where the line = (6 ˆ i - ˆ j - 3 ˆ k ) + t(-ˆ i + 4 ˆ k ) meets the plane
.( ˆ i + ˆ j - ˆ k ) = 3 are
(1) (2,1, 0)
(2) (7, -1, -7)
(3) (1, 2, -6)
(4) (5, -1,1)
19. Distance from the origin to the plane 3x - 6 y + 2z + 7 = 0 is
(1) 0
(2) 1
(3) 2
(4) 3
20. The distance between the planes x + 2 y + 3z + 7 = 0 and 2x + 4 y + 6z + 7 = 0 is
(1) √7 / 2√2
(2) 7/2
(3) √7 / 2
(4) 7 / 2√2
21. If the direction cosines of a line are then
(1) c = ±3
(2) c = ±√3
(3) c > 0
(4) 0 < c < 1
22. The vector equation points = (ˆ i - 2 ˆ j - ˆ k ) + t(6 ˆ i - ˆ k ) represents a straight line passing through the
(1) (0, 6, -1) and (1, -2, -1)
(2) (0, 6, -1) and (-1, -4, -2)
(3) (1, -2, -1) and (1, 4, -2)
(4) (1, -2, -1) and (0, -6,1)
23. If the distance of the point (1,1,1) from the origin is half of its distance from the plane x + y + z + k = 0 , then the values of k are
(1) ±3
(2) ±6
(3) -3, 9
(4) 3, -9
24. If the planes .(2 ˆi - λ ˆj + ˆk ) = 3 and
.(4 ˆi + ˆj - μ ˆk ) = 5 are parallel, then the value of λ and μ are
(1) 1/2 , -2
(2) –1/2 ,2
(3) – 1/2 , -2
(4) 1/2 ,2
25. If the length of the perpendicular from the origin to the plane 2x + 3y + λ z = 1 , λ > 0 is 1/5 , then the value of λ is
(1) 2√3
(2) 3√2
(3) 0
(4) 1
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