Vectors5

VECTOR LEVEL−I 

1.

2.







5.

6.

7.

      The scalar a . b  c  a  b  c



9.

 

 equals  (B) 2 a b c 

 (C) a b c





(D) None of these

The vectors 2ˆi  mˆj  3mkˆ and 1  m ˆi  2mˆj  kˆ include an acute angle for (A) all real m (B) m < –2 or m > –1/2 (C) m = –1/2 (D) m  [–2, –1/2]

   a  3, b  4, c  5 such that each is perpendicular to sum of the other two, then    abc = (B)

5 2

(C) 10 2

(D) 5 3

  1   If x and y are two vectors and  is the angle between them, then x  y is equal to 2    (A) 0 (B) (C) sin (D) cos 2 2 2     If u  iˆ  a  iˆ  ˆj  ( a  ˆj )  kˆ  ( a  kˆ ) , then





(A) u is unit vector (C) u = 2a 10.



If aˆ, bˆ, cˆ are three unit vectors, such that aˆ  bˆ  cˆ is also a unit vector, and 1, 2, 3 are angle between the vectors, aˆ, bˆ; bˆ, cˆ and cˆ, aˆ respectively then cos1 + cos2 + cos3 equals (A) 3 (B) -3 (C) 1 (D) -1      If angle between a and b is , then angle between 2a and  3b is 3 (A) /3 (B) -/3 (C) 2/3 (D) -2/3

(A) 5 2

8.





(A) 0 4.



     If b and c are two non-collinear vectors such that a. b  c  4 and      a  b  c  x 2  2x  6 b  sin y c , then the point ( x, y) lies on (A) x =1 (B) y =1 (C) y =  (D) x + y = 0

  

3.



OA and OB are two vectors such that | OA  OB | = | OA  2 OB | . Then (A) BOA = 90 (B) BOA > 90 (C) BOA < 90 (D) 60  BOA  90

(B) u = a + i + j + k (D) none of these

Let aˆ and bˆ be two unit vectors such that aˆ  bˆ is also a unit vector. Then the angle between aˆ and bˆ is (A) 30 (C) 90

(B) 60 (D) 120

ˆ    If a  ˆi  ˆj  kˆ , b  4ˆi  3ˆj  4kˆ and c  ˆi  ˆj  kˆ are linearly dependent vectors and c =

11.

3. (A)  =1,  = -1 (C)  = -1,   1

(B)  = 1,   1 (D)  =  1,  = 1

        Let a  2ˆi  ˆj  2kˆ and b  ˆi  ˆj . If c is a vector such that a  c = c , c  a  2 2 and the       angle between a  b and c is 30, then a  b  c =

12.







2 3 (C) 2



3 2 (D) 3

(A)

(B)





13.

Let a  i  k , b  x i  j  1  x  k and b  y i  x j  1  x  y  k . Then a b c depends on (A) only x (B) only y (C) NEITHER x NOR y (D) both x and y

14.

If | a  b || a | , then b. 2a  b equals (A) 0 (C) 2a.b





(B) 1 (D) none of these

15.

If | a | = 3, | b | = 5, | c | = 7 and a  b  c = 0, then angle between a and b is   (A) (B) 4 3  (C) (D) none of these 2

16.

Given that angle between the vectors a   ˆi  3ˆj  kˆ and b  2 ˆi  ˆj  kˆ is acute, whereas the vector b makes with the co-ordinate axes on obtuse angle then  belongs to (A) (-, 0) (B) (0, ) (C) R (D) none of these

   17. If a, b and c are unit coplanar vectors then the scalar triple product       2a  b, 2b  c, 2c  a = (A) 0 (B) 1





(C)  3 18. If

(D)

3

      a  b  a  b , then the angle between a and b is

(A) acute (C) /2

(B) obtuse (D) none of these

 b c  19. If the lines r  x and r  2b  y c  b intersect at a point with position vector  |b| | c |    b c  , then z  | b | | c |  

 

(A) z is the AM between | b | and | c |

(B) z is the GM between | b | & | c |

(C) z is the HM between | b | and | c |

(D) z = | b | + | c |

 20.







(B) Two (D) Infinite

If p and d are two unit vectors and  is the angle between them, then 2 1  p  d  sin 2 2 2 1 (C) p  d  1  cos 2





(B) p  d = sin (D)

1 pˆ  dˆ 2





2

 1  cos 2

The value of k for which the points A(1, 0, 3) , B(-1, 3,4) ,C(1, 2, 1) and D(k, 2, 5) are coplanar is (A) 1 (C) 0

24.





The number of unit vectors perpendicular to vectors a  1,1,0 and b =  0,1,1 is

(A)

23.





(B) a  b   (D) c  a

(A) One (C) Three 22.







(A) a  b  c   (C) b  c 21.



 

Let ABCDEF be a regular hexagon and AB  a , BC  b , CD  c then AE is

(2)2 (D) -1

a

a2

1  a3

If b

b2

1  b 3  0 and the vectors A = (1, a, a2), B = (1, b, b2), C = (1,c,c2) are

c

c2

1  c3

non - coplanar, then the value of abc will be (A) –1 (B) 1 (C) 0 (D) None of these 25.

` 26.

Let a, b, c be distinct non-negative numbers. If the vectors aˆi  aˆj  ckˆ, ˆi  kˆ, cˆi  cˆj  bkˆ lie in a plane, then c is (A) the arithmetic mean of a and b (B) the geometric mean of a and b (C) the harmonic mean of a and b (D) equal to zero The unit vector perpendicular to the plane determined by P(1, -1, 2), Q(2, 0, -1), R(0, 2, 1) is

i2jk 6 2i  j  k (C) 6 (A)

(B)

(D) None of these  

  

27.

If A, B, C are non-coplanar vectors then

i  j  2k 6



A . B C 

 

C  A. B (A) 3 (B) 1

 





B . A C  



C . A B

(B) 0 (D) None of there

is equal to

28. of

If the vector aiˆ  ˆj  kˆ, iˆ  bˆj  kˆ and iˆ  ˆj  ckˆ (a  b c1) are coplanar, then the value

1 1 1 is equal to   1 a 1 b 1 c (A) 1 (C) 2 29.

(B) 0 (D) None of these

  

 







If a , b , c are vectors such that a . b =0 and a  b  c . Then

2

2

(A) a  b

2

(C) b

2  c

2

2

2

(B) a  b  c

2 2  a c

(D) None of these

30.

The points with position vector 60i + 3j, 40i – 8j and ai –52j are collinear if (A) a = -40 (B) a = 40 (C) a = 20 (D) none of these .

31.

Let aˆ and bˆ be two unit vectors such that aˆ  bˆ is also a unit vector. Then the angle between aˆ and bˆ is (A) 30

(B) 60

(C) 90

(D) 120

32.

If vectors ax ˆi  3ˆj  5kˆ and x ˆi  2ˆj  2axkˆ make an acute angle with each other, for all x  R, then a belongs to the interval  1   6   3  (A)   ,0  (B) ( 0, 1) (C)  0, (D)   ,0    4   25   25 

33.

A vector of unit magnitude that is equally inclined to the vectors ˆi  ˆj , ˆj  kˆ and ˆi  kˆ is; 1 ˆ ˆ ˆ 1 ˆ ˆ ˆ i  jk i  jk (A) (B) 3 3 1 ˆ ˆ ˆ i  jk (C) (D) none of these 3













34.

Let a, b, c be three distinct positive real numbers. If p, q, r lie in plane, where p  a ˆi  aˆj  bkˆ , q  ˆi  kˆ and r  c ˆi  cˆj  b kˆ then b is (A) A.M of a, c (B) the G.M of a, c (C) the H.M of a, c (D) equal to c

85.

The scalar A . B  C  A  B  C is equal to ______________________

36.

If a, b, c are unit coplanar vectors, then the scalar triple product 2a  b, 2b  c, 2c  a is equal to _____________________

37.

The area of a parallelogram whose diagonals represent the vectors 3ˆi  ˆj  2kˆ and ˆi  3ˆj  4kˆ is









(A) 10 3 (C) 8

(B) 5 3 (D) 4



38.





The value of a  b b  c c  a is equal to

  (C) a b c 

 

(A) 2 a b c

(B) 3 a b c (D) 0

LEVEL−II 1.

 If a is any vector in the plane of unit vectors bˆ and cˆ , with bˆ  cˆ = 0, then the  magnitude of the vector a  bˆ  cˆ is  (A) | a | (B) 2 (C) 0 (D) none of these .   If a and b are two unit vectors and  is the angle between them, then the unit vector   along the angular bisector of a and b will be given by     ab ab (A) (B)   2 cos 2 cos 2 2   ab (C) (D) none of these.  2 sin 2 If a is a unit vector and projection of x along a is 2 units and a  x   b  x , then x is given by 1 1 a b  ab (A) (B) 2a  b  a  b 2 2 (C) a  a  b (D) none of these.



2.

3.



4.



 

 





6.

(B)

A scalar quantity

(D)

None of these

The shortest distance of the point (3, 2, 1) from the plane, which passes through a(1, 1, 1)  and which is perpendicular to vector   2ˆi  3kˆ , is 4 1 (A) (B) 2 (C) 3 (D) 3 13

  Let a  2ˆi  ˆj  kˆ , b  ˆi  2ˆj  kˆ and a unit vector   a then c = 1 1 (A) (B)  ˆj  kˆ 2 3 1 ˆ 1 (C) i ˆ 2 j (D) 5 2







7.



If 4 a  5 b  9c  0 , then ( a  b ) [ ( b  c )  ( c  a ) ]is equal to (A) A vector perpendicular to plane of a, b and c  (C) 0

5.



 Let a and  (A) u   (C) u  u 



  c be coplanar. If c is perpendicular to

ˆi  ˆj  kˆ  ˆi  ˆj  kˆ 

         b be the two non–collinear unit vector. If u  a  a  b b and v  a  b , then v is    (B) u  u  a   ab (D) none of these

 





2

8.

(A) 4 (C) 8 9.

2

2

If a, b and c are unit vectors, then a  b  b  c  c  a does NOT exceed (B) 9 (D) 6

If a  r  b  ta and a.r  3, where a  2ˆi  ˆj  kˆ and b  ˆi  2ˆj  kˆ then r equals 7 2 7 1 (A) ˆi  ˆj (B) ˆi  ˆj 6 5 6 3 7ˆ 2ˆ 1ˆ (C) i  j  k (D) none of these 6 3 3

      10. If  a  b   b  c  c  a = 0 and at least one of the numbers ,  and  is non-zero,    then the vectors a, b and c are (A) perpendicular (B) parallel (C) co-planar (D) none of these



 



  11. The vectors a and b are non-zero and non-collinear. The value of x for which vector       c = (x –2) a + b and d = (2x +1) a – b are collinear. (A) 1 (B) 1/2 (C) 1/3 (D) 2

12

13.

14.

15.

   a b  c ,  (A) a = 1,  (C) b = 2,

   b  c  a , then   b  c   b  2a









(D) b = 1, b  a

      If a , b , c are three non - coplanar vectors and p, q, r are vectors defined by the          b c ca ab relations p     , q     , r     then the value of expression a b c  a b c  a b c                 (a + b).p + (b + c).q + (c + a).r is equal to (A) 0 (B) 1 (C) 2 (D) 3    ˆ2 The value of |a  ˆi |2 + |a  ˆj|2 + |a  k| (A) a2 (C) 3a2

is (B) 2a2 (D) None of these

If a  ˆi  ˆj, b  2ˆj  kˆ and r  a  b  a, r  b  a  b , then a unit vector in the direction of r is; 1 ˆ 1 ˆ i  3ˆj  kˆ i  3ˆj  kˆ (A) (B) 11 11 1 ˆ ˆ ˆ i  jk (C) (D) none of these 3





16.



(B) c = 1, a = 1







a.ˆi a  ˆi   a.ˆja  ˆj a.kˆ a  kˆ  is equal to; (A) 3 a

(B) r

(C) 2 r

(D) none of these



17.

If the vertices of a tetrahedron have the position vectors 0, ˆi  ˆj, 2ˆj  kˆ and ˆi  kˆ then the volume of the tetrahedron is (A) 1/6 (B) 1 (C) 2 (D) none of these

18.

A = (1, -1, 1), C = (-1, -1, 0) are given vectors; then the vector B which satisfies A  B  C

and A.B  1 is ___________________________________ 19.

If a, b, c are given non-coplanar unit vectors such that a  (b  c ) 

bc , then the angle 2

between a and c is ________________________________ 20.

Vertices of a triangle are (1, 2, 4) (3, 1, -2) and (4, 3, 1) then its area is_______________

21.

A unit vector coplanar with i  j  2k and i  2 j  k and perpendicular to i  j  k _______________________

is

LEVEL−III





 





1. If a, b, c are coplanar vectors and a is not parallel to b then c  b  a  b a  a  c  a  b b is equal to (A) a  b  a  b c (B) a  b  a  b c

    (C) a  b   a  b c







(D) none of these

2.

The projection of ˆi  ˆj  kˆ on the line whose equation is r = (3 + ) ˆi + (2 -1) ˆj + 3 kˆ ,  being the scalar parameter is; 1 (A) (B) 6 14 6 (C) (D) none of these 14

3.

If p, q are two non-collinear and non-zero vectors such that (b –c) p  q +(c –a) p + (a –b) q = 0 where a, b, c are the lengths of the sides of a triangle, then the triangle is (A) right angled (B) obtuse angled (C) equilateral (D) isosceles

L−I 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37.

B A C A C B C B A C B D B B A D C O B

2. 4. 6. 8. 10. 12. 14. 16. 18. 20. 22. 24. 26. 28. 30. 32. 34. 36. 38.

A D B

A B A A D C D A A   /3

2. 4. 6. 8. 10. 12. 14. 16. 18. 20.

B C A B C D B D K 5 5/2

D B A A A C C A C A A C C O A

L−II 1. 3. 5. 7. 9. 11. 13. 15. 17. 19.

21.

−

 



J K

J K

2

ON



2

L−III

1. 3.

2. C

C