Choose the correct answer

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 [, ,  Ã—  ] = Ï€/4, then the angle between and  is

(1) π/6

(2) π/4

(3) π/3

(4) π/2

8. If  and ( Ã—  )×  = 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  Ã—( Ã—  ) =  , 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  Ã— , Ã— ,  Ã—  as coterminous edges is 8 cubic units, then the volume of the parallelepiped with ( Ã—  )×( Ã—  ), ( Ã—  ) ×( Ã— ) and ( Ã— ) ×( Ã—  ) 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 (×  ) ×( Ã—  ) = 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  Ã—( Ã— ) = ( Ã—  ) × , where , ,  are any three vectors such that  .  â‰  0 and  .  â‰  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