Vectors5

VECTOR

LEVEL−I 1.   OB and

OA are two vectors such that |OA OB|

   =|OA 2OB|    . Then

(A) BOA = 90 (B) BOA > 90

(C) BOA < 90 (D) 60  BOA  90

2. If b and c are two non-collinear vectors such that a.



bc



4 and



b c





x 2x 6



b



siny



c

a   2   , then the point ( x, y) lies on

(A) x =1 (B) y =1

(C) y =  (D) x + y = 0

3. The scalar a.





bc

 

 abc





equals

(A) 0 (B) 2



abc



(C)



abc



(D) None of these

4. If aˆ,bˆ,cˆare 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 5. If angle between 3 is b and

a  , then angle between 2aand 3b is

(A) /3 (B) -/3 (C) 2/3 (D) -2/3

6. The vectors 2iˆmjˆ3mkˆ and



1m



iˆ2mjˆ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]

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

(A) 5 2 (B)

2 5

(C) 10 2 (D) 5 3

8. If x and y are two vectors and  is the angle between them, then 1 x y

2    is equal to (A) 0 (B) 2  (C) 2 sin (D) 2 cos 9. If u iˆ



aˆi



 ˆj( aˆj )kˆ( ak )ˆ , then

(A) u is unit vector (B) u = a + i + j + k

(C) u = 2a (D) none of these

10. Let aˆ andbˆbe two unit vectors such that aˆ  bˆis also a unit vector. Then the angle between aˆ and bˆis

(A) 30 (B) 60

11. If a iˆjˆkˆ, bˆ 4iˆ3jˆ4kˆ and c  iˆjˆkˆ are linearly dependent vectors and c = 3 .

(A)  =1,  = -1 (B)  = 1,   1

(C)  = -1,   1 (D)  =  1,  = 1

12. Let a 2iˆ jˆ2kˆ and b  iˆ jˆ. If c is a vector such that a = cc  , ca 2 2 and the angle between



ab



and c is 30, then



ab



c =

(A) 3 2 (B) 2 3 (C) 2 (D) 3

13. Let aik, bxij



1x



k and byixj



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 (B) 1

(C) 2a.b (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) 4  (B) 3  (C) 2  (D) none of these

16. Given that angle between the vectors aiˆ3jˆkˆ and b2iˆjˆ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 andc are unit coplanar vectors then the scalar triple product



2ab, 2bc, 2ca



=

(A) 0 (B) 1

(C)  3 (D) 3

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

(A) acute (B) obtuse

(C) /2 (D) none of these 19. If the lines           | c | c | b | b x

r and r2by



cb



intersect at a point with position vector          | c | c | b | b z , then

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

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

(A) abc (B) ab

(C) bc (D) ca

21. The number of unit vectors perpendicular to vectors



a  1 1 0



, , and b = 0,1,1 is









(A) One (B) Two

(C) Three (D) Infinite

22. If p and d are two unit vectors and  is the angle between them, then (A) 1 2 2 2   sin pd   (B) p   d= sin (C) 1





2 1 2   cos pd    (D)



ˆ ˆ



1 cos2



2 1 2    d p

23. 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 (2)2 (C) 0 (D) -1 24. If a a a b b b c c c 2 3 2 3 2 3 1 1 1 0   

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

non - coplanar, then the value of abc will be

(A) –1 (B) 1

(C) 0 (D) None of these

25. Let a, b, c be distinct non-negative numbers. If the vectors aiˆaˆ jckˆ,iˆkˆ ,ciˆcjˆbkˆlie 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

`

26. The unit vector perpendicular to the plane determined by P(1, -1, 2), Q(2, 0, -1), R(0, 2, 1) is (A)

6

2

j

k

i





(B)

6

2k

j

i





(C) 6 2i jk (D) None of these 27. If    C B

A, , are non-coplanar vectors then _{  }

              B A C C A B B A C C B A . . . . is equal to (A) 3 (B) 0 (B) 1 (D) None of there

28. If the vector aiˆ ˆjkˆ,

i

ˆ



b

ˆ

j



k

ˆ

and

i

ˆ



ˆ

j



c

k

ˆ

(a  b c1) are coplanar, then the value of c b a      1 1 1 1 1 1 is equal to (A) 1 (B) 0 (C) 2 (D) None of these

29. If a,b,c are vectors such that a.b=0 and

a





b





c



. Then (A) a2 b2  c2 (B) a2  b2 c2

(C) b2  a2 c2 (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 aˆandbˆbe two unit vectors such that aˆ bˆis also a unit vector. Then the angle

between aˆandbˆis

(A) 30 (B) 60 (C) 90 (D) 120

32. If vectors axiˆ3jˆ5kˆ and xiˆ2jˆ2axkˆ make an acute angle with each other, for all x  R, then a belongs to the interval

(A)        ,0 4 1 (B) ( 0, 1) (C)       25 6 , 0 (D)        ,0 25 3

33. A vector of unit magnitude that is equally inclined to the vectors iˆ  , jˆ jˆ  and kˆ iˆ  is; kˆ (A)



iˆ jˆ kˆ



3 1   (B)



iˆ jˆ kˆ



3 1   (C)



iˆ jˆ kˆ



3 1   (D) none of these

34. Let a, b, c be three distinct positive real numbers. If p,q,r lie in plane, where kˆ b jˆ a iˆ a

p   , q iˆkˆ and rciˆcjˆbkˆ 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 3iˆjˆ2kˆ and kˆ 4 jˆ 3 iˆ  is (A) 10 3 (B) 5 3 (C) 8 (D) 4

38. The value of



ab bc ca



is equal to

(A) 2

 

abc (B) 3

 

abc

(C)

 

abc (D) 0

LEVEL−II

1. If a is any vector in the plane of unit vectors bˆandcˆ, with bˆ  = 0, then the cˆ magnitude of the vector a



bˆcˆ



is

(A) |a| (B) 2

(C) 0 (D) none of these .

2. If aandb are two unit vectors and  is the angle between them, then the unit vector along the angular bisector of aandb will be given by

(A) 2 cos 2 b a    (B) 2 cos 2 b a    (C) 2 sin 2 b a    (D) none of these.

3. 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 (A)



a b



a b





2 1    (B)



2a b



a b





2 1    (C)



a



ab





(D) none of these. 4. If 4a5b9c0, then ( a b)[( b c)( c a)]is equal to

(A) A vector perpendicular to plane of a, b and c (B) A scalar quantity

(C) 0 (D) None of these

5. 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  2 iˆ 3kˆ, is

(A) 3 4 (B) 2 (C) 3 (D) 13 1

6. Let a 2iˆjˆkˆ, b iˆ2jˆkˆ and a unit vector c be coplanar. If c is perpendicular to a then c = (A)



jˆ kˆ



2 1   (B)



iˆ jˆ kˆ



3 1   (C)



iˆ ˆ2j



5 1  (D)



iˆ jˆ kˆ



2 1  

7. Let a and b be the two non–collinear unit vector. If u a

 

abb and v ab, then v is

(A) u (B) u ua

8. If a, b and c are unit vectors, then 2 2 2 a c c b b

a     does NOT exceed

(A) 4 (B) 9

(C) 8 (D) 6

9. If ar btaanda.r3,where a2iˆjˆkˆ and biˆ2jˆkˆ then r equals

(A) jˆ 5 2 iˆ 6 7  (B) jˆ 3 1 iˆ 6 7  (C) kˆ 3 1 jˆ 3 2 iˆ 6 7   (D) none of these

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

a





b





c



,

b





c





a



, then

(A)

a



= 1, b  c (B)

c



= 1,

a



= 1 (C) b = 2, b 2a (D) b = 1, b  a

13. If a, b, c are three non - coplanar vectors and p, q, r   are vectors defined by the relationsp b c a b c        _     , q c a a b c             , r a b a b c          

  then the value of expression

(a + b).p + (b + c).q + (c + a).r         is equal to (A) 0 (B) 1 (C) 2 (D) 3 14. The value of ˆ 2 ˆ2 ˆ 2 |ai | + |a j| + |ak| is (A) a2 _{(B) 2a}2 (C) 3a2 (D) None of these

15. If a iˆ jˆ,b2jˆkˆ and raba,rbab, then a unit vector in the direction of r is; (A)



iˆ 3jˆ kˆ



11 1   (B)



iˆ 3jˆ kˆ



11 1   (C)



iˆ jˆ kˆ



3 1   (D) none of these

16.

        

a.iˆ aiˆ  a.jˆ ajˆ  a.kˆ akˆ is equal to;

(A) 3 a (B) r

17. If the vertices of a tetrahedron have the position vectors 0, iˆ jˆ,2jˆkˆ and iˆ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 is ___________________________________ 1

19. If a,b,c are given non-coplanar unit vectors such that

2 c b ) c b (

a    , then the angle 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2j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 iˆ jˆkˆ on the line whose equation is r = (3 + ) iˆ + (2 -1) jˆ + 3 kˆ ,  being the scalar parameter is;

(A) 14 1 (B) 6 (C) 14 6 (D) none of these

3. If p,q are two non-collinear and non-zero vectors such that (b –c)p  +(c –a) p + (a –b) q = 0 q 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. B 2. A 3. A 4. D 5. C 6. B 7. A 8. 9. C 10. D 11. B 12. B 13. C 14. A 15. B 16. A 17. A 18. A 19. C 20. C 21. B 22. C 23. D 24. A 25. B 26. C 27. B 28. A 29. A 30. A 31. D 32. C 33. C 34. C 35. O 36. O 37. B 38. A L−II 1. A 2. B 3. B 4. C 5. A 6. A 7. A 8. B 9. D 10. C 11. C 12. D 13. D 14. B 15. A 16. D 17. A 18. K 19.   / 3 20. 5 5 / 2

21. − J K 2     ON J K 2    L−III 1. 2. C 3. C