Jee 2014 Booklet6 Hwt Vectors

Choose the correct alternative. Only one choice is correct.

1. If | a | 2 and| b| 3 anda . b 0, then _a



a



a

 

ab





_ is equal to :

(A) 16 b (B) 16 b (C) 48 a (D) 48 a

2. Let a b c    be three unit vectors such that 3a4b5c0. Then which of the following statements is true ? (A) a is parallel to

b



(B) a is perpendicular to

b



(C) a is neither parallel nor perpendicular to

b



(D) None of the above

3. The values of x for which the angle between the vectors 2x i24x jk and 7i2j xk are obtuse and the angle between the Z-axis and 7i2j  xk is acute and less than

6  is given by : (A) 0 1 2 x   (B) 1or 0 2 x x (C) 1 15

2 x (D) No such value for x

4. A unit tangent vector at t = 2 on the curve xt22, y4t5,z2t2 6t is :

(A) 1





3 ˆ ˆ ˆ i  j k (B) 1



2 2



3 ˆ ˆ_i ˆ_{jk (C)} 1



_{2 2 2}



6 ˆ ˆ_i ˆ_{j k (D) None of these}

5. When a right handed rectangular Cartesian system OXYZ is rotated about the Z-axis through an angle 4

 _{in the counter-clockwise}

direction it is found that a vector a has the component 2 3, 3 2 and 4 . The components of a in the OXYZ coordinate system are :

(A) 5,1, 4 (B) 5,1, 4 2 (C)  1, 5, 4 2 (D) None of these

6. Given three vectors a6iˆ 3 ,ˆj b2iˆ 6ˆjandc 2iˆ 21 such thatˆj      a b c. Then the resolution of the vector

into components with respect to aandbis given by :

(A) 3a2b (B) 2a3b (C) 3b2a (D) None of these

7. For any four points P, Q, R, S ; PQ     RS QRPSRPQS is equal to 4 times the area of the triangle :

(A) PQR (B) QRS (C) PRS (D) PQS 8. If 1 0 where 1 n i _i i a a i    



  , then the value of

1 i j i j n a . a   





  is :

9. Let a b c  , , be three non-coplanar vectors and rbe any vector in space such that  r .a1, r . b2 andr . c 3. If _a b c   _ 1 then, ris equal to :

(A) a2b3c (B) b  c 2c  a 3a b

(C)

 

b . c a  2

 

c . a b  3

 

a . b c   (D) None of the above 10. The two vectors



a2ˆi ˆj 3 ,k bˆ 4ˆiˆj6kˆ



are parallel if



is equal to :

Choose the correct alternative. Only one choice is correct.

1. If     aand      band    , , are non-coplanar and is not parallel to , then       equals :

(A) a (B) b (C) 0 (D)



a b





2. If unit vector cmakes an angle with 3

ˆ ˆ

i j

 _{ , then minimum and maximum values of}

_{ }

_{ˆ ˆ}

i j . crespectively are : (A) 0, 3 2 (B) 3 3 , 2 2  (C) 1, 3 2  (D) None of these

3. A parallelogram is constructed on 3a banda4 , whereb a 6 and b 8 andaandbare anti-parallel, then the length of the longer diagonal is :

(A) 40 (B) 54 (C) 32 (D) 48

4. Vectors aandbare inclined at an angle 120. If a 1 and b 2, then





a3b

 

 3a b





2 is equal to :

(A) 225 (B) 275 (C) 325 (D) 300

5. If a b c  , , are non-coplanar vectors, then

 

 



 





 



a . b c b . c a c . b a b . c a c . a b a . b c                           is equal to : (A) 0 (B) 1 (C) 2 (D) 3

6. The value of c so that for all real x, the vectors cxiˆ6ˆj3 ,k xiˆ ˆ2ˆj2cxkˆ make an obtuse angle are :

(A) c0 (B) 0 c 4 3/ (C) 4 3/  c 0 (D) c0

7. For non-zero vectors a b c     , ,



ab .c



 a b c holds iff.:

(A) a . b 0,b . c 0 (B) b . c 0, c . a 0 (C) c . a 0,a . b 0 (D) a . b  b . c   c . a0 8. If a2b3 c0 anda         b b c c a is equal to 

 

b c thenis equal to :

(A) 3 (B) 4 (C) 5 (D) 6

9. If a any vector, then aˆi 2 aˆj2  akˆ 2 is equal to :

(A)

 

a 2 (B) 2 a

 

 2 (C) 3 a

 

2 (D) 0

10. For any vector A, the value ofˆi



Aˆi



 ˆj



Aˆj



 kˆ



Akˆ



is equal to :

DATE : TIME : 40 Minutes MARKS : [ ___ /10] TEST CODE : VEC [3]

START TIME : END TIME : TIME TAKEN: PARENT’S SIGNATURE :

 This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct.

1. If a b c  , , and  p q r, , are reciprocal system of vectors, then a      p   b q c r equals :

(A) _a b c  _ (B)   p q r (C) 0 (D) a   b c

2. If a b , andcare unit coplanar vectors then the scalar triple product _2a     b b2 c c2 a_is equal to :

(A) 0 (B) 1 (C)  3 (D) 3

3.

     

r . i ˆ riˆ  r . jˆ

 

rj 

   

r . k ˆ rkˆ is equal to :

(A) 3r (B) r (C) 0 (D) None of these

4. a b c  , , are non-coplanar vectors and   p q r, , are defined as p b c ,q c a ,r a b b c a c a b a b c                                     then



a  b . p







b  c .q

 

   ca . r



is equal to : (A) 0 (B) 1 (C) 2 (D) 3

5. If| a|  3| b |  1 | c | 4 and a     b c 0, find the value of a . b     b . cc . a.

(A) 13 2/ (B) 26 (C) 13 (D) None of these

6. Let  4i3 andj  be two vectors perpendicular to each other in the XY-plane. Find all the vector in the same plane having the projections 1 and 2 along andrespectively.

(A)  2i j (B)  i 2j (C) i2j (D) 2i j

7. Which of the following expressions is not meaningful ?

(A) u . v  



w



(B)

 

u . v . w   (C)

 

u . v w   (D) u



v w



8. If a  4 b  4 and c 5 such that a



b  c



 b



 ca



andc



a b



, then | a   b c | is :

(A) 7 (B) 5 (C) 13 (D) 57

9. The work done by the force F 2i  j k in moving an object along the vector 3i2j5k is :

(A)  units9 (B) 15 units (C) 9 units (D) None of these

10. If a b c    are vectors such that a . b 0and a   b c then :

Choose the correct alternative. Only one choice is correct. 1. If



    

and 1, then 8 d ab  bc v ca _a b c_   v           is equal to : (A) d . a   



 b c



(B) 2d . a   



 b c



(C) 4d . a   



 b c



(D) 8d . a   



 b c



2. If aandbare two vectors making angle with each other, then unit vectors along bisector of anda b is : (A) 2 ˆ ˆab  (B) 2 ˆ ˆa b cos   (C) 2 2 ˆ ˆa b cos/   (D) None of these

3. If a b c  , , be three vectors such that a   b c and b   c a then :

(A) a b c  , , are orthogonal in pairs (B) a  b  c 1

(C) a  b  c 1 (D) a  b  c

4. A vector a has components 2p and 1 with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the clockwise sense. If with respect to new system, a has components p + 1 and 1, then :

(A) p = 0 (B) p1 or p 1 3/

(C) p 1 or p1 3/ (D) p1 or p 1

5. If a b  andcbe three non-parallel unit vectors such that

 

1 2

a b c  b, then the angle which vector amakes with b is : (A) 3  (B) 4  (C) 2  (D) 6 

6. If u         a b v a b and|a| |b|  2, then | u v | is :

(A) 2 16

 

a . b  2 (B) 2 4

 

a . b  2 (C) 16

 

a . b  2 (D) 4

 

a . b  2

7. The moment of the couple formed by the forces 5ikand5ik acting at the points (9,  , 2) and (3, 21  , 1) respectively, is: (A)   i j 5k (B) i11j5k (C)  i 11j5k (D) i j 5k

8. If a . b . c  are three non-zero vectors such thata     b c 0 and ma .b       b c c . a , then :

9. ABCD is a parallelogram with AC  i 2j kandBD  i 2j5k. Area of this parallelogram is equal to : (A) 5 2/ sq. units (B) 2 5 sq. units (C) 4 5 sq. units (D) 5 sq. units

10. If the non-zero vectorsaandb are perpendicular to each other, then the solution of the equation, r   a b is given by : (where x is any scalar)

(A) 2 a b r xa | a |         (B) 2 a b r xb | a |         (C) r x a  



b



(D) rx b



 a



Choose the correct alternative. Only one choice is correct.

1. Let r a b  , , andcbe four non zero vectors such that r .a 0,  rb  r b , r  c  r c then _a b c  _ is equal to :

(A) 1 (B) 0 (C) 1 (D) 2

2. Given a cube ABCDA1B1C1D1 with lower base ABCD, upper base A1B1C1D1 and the lateral edges AA1, BB1, CC1 and DD1; M and M1 are the centres of the faces ABCD and A1B1C1D1 respectively. O is a point on the line MM1 such that

1, OA OB OC OD  OM

    

then OMOM₁ifis equal to :

(A) 1/16 (B) 1/8 (C) 1/4 (D) 1/2

3. Let a b , andcbe three non-zero and non-coplanar vectors and  p q, andr be three vectors given by

2 , 3 2 and 4 2

p  a b c q  a bc r  a b c

           

If the volume of the parallelepiped determined by a b , andc is V1 and that of the parallelepiped determined by p q, and r

  

is V2 then V2 : V1 is equal to:

(A) 3 : 1 (B) 7 : 1 (C) 11 : 1 (D) 15 : 1

4. Let  A B, andCbe unit vectors. Suppose that  A .B  A . C  0and that the angle between BandC is 6  , then





Ak BC    and k is equal to : (A) 2 (B) 4 (C) 6 (D) 0

5. A vector a has components a1, a2, a3, in a right handed rectangular cartesian system OXYZ. The coordinate system is rotated about Z-axis through angle/2 in anti-clockwise direction. The components of ain the new system is :

(A)



  a₂ a₁ a₃



(B)



a₂  a₁ a₃



(C)



a₂ a₁ a₃



(D)



a₃ a₁ a₂



6. If aandb are unit vectors and is the angle between them, then 2 ab   is : (A) 2

sin (B) sin (C) 2 sin (D) sin2

7. If a b  andcare unit vectors such that a . b  0



a c . b

 

 c



0 and cabw a



 b



then : (where   and w are scalars).

(A)  2 w21 (B) 2 2 w2 1

(C) 2  



1



w2 0 (D) None of these

8. The volume of the tetrahedron whose vertices have position vectors i6j10k  i 3j 7k 5i j k and 7i4j7k is 11 cubic units if  equals :

9. Let bandcbe non-collinear vectors. If ais a vector such that a . b  



c



4and a

 

b c 



x2 2x6



bsin y . c, then (x, y) lies on the line :

(A) x + y = 0 (B) x y0 (C) x = 1 (D) y

10. Let O be the centre of a regular pentagon ABCDE and OA a. Then AB2BC3CD4DE5EA equals :

Choose the correct alternative. Only one choice is correct.

1. If I be the incentre of the triangle ABC and a, b, c be the lengths of the sides then the force a IAbIBc IC is equal to : (A)   AB ABCA (B) a AB bBC cCA   (C) 0 (D) None of these

2. A vector a



x y z, ,



makes an obtuse angle with Y-axis, and make equal angles with b



y,2 , 3z x



and



2 , 3 ,



and c z xy a

 

is perpendicular to d



1,1, 2 if



| a| 2 3then vector ais :

(A) (1, 2, 3) (B)



2,2,2



(C)



1, 2, 4



(D) None of these 3. In a parallelogram ABCD,| AB | a | AD |,   b and| AC |  c, thenDB . AB has the value :

(A) 2 2 2 3 2 a b c (B) 2 ₃ 2 2 2 a  b c (C) 2 2 ₃ 2 2 a b  c (D) 2 ₃ 2 2 2 a  b c

4. A vector a has components a1, a2, a3 in the right handed rectangular cartesian system OXYZ. The coordinate system is rotated about the X-axis through an angle

4



in the anticlockwise direction. the components of a in the new system are :

(A) _{1 2 3} 3 2 2 a a a a a   (B) ₁ 2 3 _{3 2} 2 a a a  a a (C) ₁ 2 3 3 2 2 2 a a a a a    (D) None of these 5. If a



    i j k



a . b1 and a   b   thenj k b is :

(A) i j k (B) 2 j k (C) i (D) 2i

6. If a b  andcare unit vectors, then|a b|2| b c|2|c a|2 does not exceed:

(A) 4 (B) 9 (C) 8 (D) 6

7. If



a b



  c a

 

b c where a b  andcare any three vectors such that a . b  0 b . c  0 then aandc are : (A) inclined at an angle of /6between them (B) Perpendicular

(C) Parallel (D) inclined at an angle of /3 between them

8. Let a b  andc be non-zero vectors such that





1 2

a b  c | b | | c | a   . If  is the acute angle between the vectors andb c, then sin equals :

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

9. Let u    i j v i j and w  i 2j3k If n is a unit vector such thatu . n  =v . n 0, then| w . n|  is equal to :

(A) 0 (B) 1 (C) 2 (D) 3

10. If aandb are mutually perpendicular unit vectors, r is a vector satisfying r . a  0 r . b 1 and_r a b    _ 1 thenr: (A) a



a b



(B) b



a b



(C) a



a b



(D) a  b



a b



Choose the correct alternative. Only one choice is correct.

1. The position vectors a b c  , , anddof four points A, B, C and D on a plane are such that



a   d . b

 

c

 

     bd . c

 

a



0, then the point D is :

(A) Centroid of ΔABC (B) Orthocenter of ΔABC

(C) Circumcentre of ΔABC (D) None of these

2. Consider a tetrahedron with faces F F₁ ₂F₃F₄. Let V1V2V3V4

   

be the vectors whose magnitudes are respectively equal to areas of F F₁ ₂F₃F₄ and whose directions are perpendicular to their faces in outward direction. Then | V1V2V3V |4

    , equals :

(A) 1 (B) 4 (C) 0 (D) None of these

3. xand y are two mutually perpendicular unit vectors, if the vectors a x a yc x

 

  y  x

 

x y and cxc yb x

 

  y , lie in a plane then c is :

(A) AM of a and b (B) GM of a and b (C) HM of a and b (D) Equal to zero

4. a b c    be three non coplanar vector and rbe any arbitrary vector, then



a b

          

  rc   bc   ra  c a  r b is equal to :

(A) _a b c r   _ (B) 2 a b c r_   _ (C) 3 a b c r_   _ (D) None of these

5. If V is the volume of the parallelopiped having three coterminus edges as a . b andc , then the volume of the parallelepiped having three coterminus edge as 

     

a . a a   a . b b   a . c c    

     

a . b a    b . b b   b . c c  ,

     

a . c a b . c b c . c c

          , is :

(A) V3 (B) 3V (C) V2 _{(D) 2V}

6. Let G1, G2, G3 be the centroids of the triangular face OBC, OCA, OAB of a tetrahedron OABC. If V1denote the volume of the tetrahedron OABC and V2 that of the parallelopied with OG1, OG2, OG3 as three concurrent edges, then:

(A) 4V₁9V₂ (B) 9V₁4V₂ (C) 3V₁2V₂ (D) 3V₂2V₁

7. a b c    are non-coplanar vectors and a1 b1 c1

  

constitute the corresponding reciprocal system of vectors, then we have





1 1 1 1 1 1

a b b c c a _a b c  _   a b c where ‘



’ is equal to :

(A) 1 (B) 0 (C)



1

(D) 2

8. If a b c    and  p q r  is reciprocal system of vectors, then a      p   b q c requals :

9. For any four vectors a b c d      , the expression

  

b c . a d

   

 c a . b d

  

  ab .



c

 



d



is always equal to : (A) _a b c  _ (B) _a b d  _ (C) _b c d  _ (D) 0

10. If a

 

b c is perpendicular to



a b



c, then we may have :

Choose the correct alternative. Only one choice is correct. 1. Let aandb are two vectors making angle  with each

other, then unit vectors along bisector of aandb is :

(A)  2 ab   (B)  2 a b cos    (C)  2 2 a b cos    (D) None of these

2. Let a b c    be three vectors such that a               b c b c a c a b then :

(A) | a | | b | | c |     (B) | a | | b | | c |  

(C) | a | | b | | c| (D) | a | | b | | c |

3. If S is the circumcentre, G the centroid, O the orthocente of a triangle ABC, then SA  SBSCis :

(A) 3SG (B) OS

(C) SG (D) OG

4. Let a b  andc be mutually perpendicular vectors of the same magnitude. If a vector x satisfies the equation









a_ x b a_ b _  xc b_c



x a



c   _   _      = 0, then xis given by : (A) 1





2 a b c    (B) 1



2



3 a b c    (C) 1





3 a b c    (D) 1



2



2 a b c   

5. If a b c    are unit vectors then

2 2 2

| a b | | b c | | c a | does not exceed.

(A) 4 (B) 9

(C) 8 (D) 6

6. a b c    are three non-coplanar vectors, then a b b c c a              is equal to : (A)

0

(B) _a b c    _ (C) _  a b c  _2 (D) 2 a b c_    _ 7. If a    i j k b  4i3j4k, and   ˆ c i jk 

are linearly dependent vectors and 3

|c|  , then :

(A)   1   1 (B)   1  1 (C)    1  1 (D)    1 1 8. The volume of the tetrahedron whose vertices are with

position vectors

     

6 10 3 7 5

i j k  i j k i j k

  

and 7i4j7k is 11 cubic unit if  equals :

(A) 3 (B) 3

(C) 7 (D) 1

9. Let aandb be non collinear vectors of which a is a unit vector. The angles of triangle whose two sides are represented by 3

 

a b andb

 

a . b a   are : (A) 2 3 6   _{ } (B) 2 4 4   _{ } (C) 3 3 3   _{ } (D) Data insufficient 10. Let the unit vectors aandb be perpendicular to each

other and the unit vector c be inclined at an angle  to both aand b. If cxa yb z a



 b



, then : (A) xcos y sinz2 cos2 (B) xsin y cosz2  cos2 (C) x ycosz2 cos2 (D) x ycosz2  cos2