Jee 2014 Booklet6 Hwt Vectors

Vidyamandir Classes DATE :

TIME : 40 Minutes

MARKS : [ ___ /10]

TEST CODE : VEC [1]

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.

2.

3.

 



6.

7.

x

1 or x  0 2

No such value for x

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



1 ˆ ˆ ˆ i jk 3



(B)



1 ˆ 2i  2 ˆj  kˆ 3



(C)

1 6

 2ˆi  2 ˆj  2kˆ  (D)

None of these

 When a right handed rectangular Cartesian system OXYZ is rotated about the Z-axis through an angle in the counter-clockwise 4   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) (B) (C) (D) None of these 5,  1, 4 2 1,  5, 4 2 5,  1, 4        Given three vectors a  6ˆi  3 ˆj , b  2ˆi  6 ˆj and c   2ˆi  21 ˆj such that   a  b  c . Then the resolution of the vector     into components with respect to a and b is given by :       (A) (B) (C) (D) None of these 3 a  2b 2 a  3b 3 b  2a

      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 n

8.

 48 a

The values of x for which the angle between the vectors 2 x 2 i  4 x j  k and 7i  2 j  xk are obtuse and the angle between the

(A)

5.

(D)

      Let a b  c be three unit vectors such that 3a  4b  5c  0 . Then which of the following statements is true ?     (A) (B) a is parallel to b a is perpendicular to b   (C) (D) None of the above a is neither parallel nor perpendicular to b

 Z-axis and 7i  2 j  xk is acute and less than is given by : 6 1 0x (A) (B) 2 1  x  15 (C) (D) 2

4.



        If | a |  2 and | b|  3 and a . b  0 , then  a  a  a  a  b  is equal to :      (A) (B) (C) 16 b  16 b 48 a

If



 ai  0 where

(B)

QRS

 ai  1  i , then the value of

VMC/Vectors

 n/2



PRS

(D)

PQS

(D)

n

 

 ai . a j is :

1i  j  n

i 1

(A)

(C)

(B)

n

(C)

125

n/2

HWT-6/Mathematics

Vidyamandir Classes 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 and r . c  3 .    If  a b c   1, then r is equal to :            (A) (B) a  2b  3 c b  c  2c  a  3a  b          (C) (D) None of the above b .c a  2 c .a b  3 a .b c



10.



The two vectors (A)

VMC/Vectors

2













 a  2ˆi  ˆj  3kˆ , b  4ˆi   ˆj  6 kˆ  are parallel if  is equal to : (B)

3

(C)

126

3

(D)

2

HWT-6/Mathematics

Vidyamandir Classes DATE :

TIME : 40 Minutes

MARKS : [ ___ /10]

TEST CODE : VEC [2]

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.

2.

                 If       a  and       b  and  ,  , are non-coplanar and  is not parallel to  , then        equals :     (A) (B) (C) (D) 0 a b a  b

   with ˆi  ˆj , then minimum and maximum values of ˆi  ˆj . c respectively are : If unit vector c makes an angle 3



(A) 3.

0,

3 2

8.

225

(B)

0

(C)

275

(B)

c0

32

(C)

 

  bc   ca

 

 b.   c.

1

(B)



(D) 



325

(D)

 c.   a.

  ba   is equal to : bc

(C)

2

 

 





  a  3b  3 a  b 

  ca   ab

 

48 2

is equal to :

300

 

(D)

3

0  c  4/3

(C)

 a

 c holds iff.:





 b

4 / 3  c  0

(D)

c0

    b . c  0, c . a  0       a .b  b .c  c .a  0

(B) (D)

            If a  2 b  3 c  0 and a  b  b  c  c  a is equal to  b  c then  is equal to :



3

(B)

4



(C)

5

(D)

6

(D)

0

(D)

None of these

    2 2 2 If a any vector, then a  ˆi  a  ˆj  a  kˆ is equal to : (A)

10.

54

      For non-zero vectors a, b, c a  b .c     (A) a . b  0, b . c  0     (C) c . a  0, a . b  0

(A) 9.

None of these

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

7.

(B)

 a.    If a, b, c are non-coplanar vectors, then  b. (A)

6.

(D)

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

5.

3 2

1,

(C)

        A parallelogram is constructed on 3 a  b and a  4b, where a  6 and b  8 and a and b are anti-parallel, then the length of the longer diagonal is : (A) 40

4.

3 3 , 2 2



(B)



 2

a

 2 2 a



(B)

 2 3 a



(C)

    For any vector A , the value of ˆi  A  ˆi  ˆj  A  ˆj  kˆ  A  kˆ is equal to :    (A) (B) (C) 0 2A 2A

VMC/Vectors









127





HWT-6/Mathematics

Vidyamandir Classes 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.

2.

            If a, b, c and p, q, r are reciprocal system of vectors, then a  p  b  q  c  r equals :      a b c  (A) (B) (C) (D) 0 pqr  

         If a, b and c are unit coplanar vectors then the scalar triple product  2a  b 2b  c 2c  a  is equal to :   (A)

3.

0



(B)







1



(C)

 3

(D)

 3r

(C)

 0

(D)

 r

(B)











(A)

0 (B) 1 (C) 2              If | a|  3 | b |  1 | c |  4 and a  b  c  0 , find the value of a . b  b . c  c . a . (A)

6.

8.

2i  j



13

(C)

 i  2 j

(B)



  If a  4  b  4 and

3

(D)

None of these

7





(C)

i  2 j

(C)

u . v  w

  

(D)

2i  j

(D)

   u vw





            c  5 such that a  b  c  b  c  a and c  a  b , then | a  b  c | is :



(B)

5





(C)





13



57

(D)

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

10.

26

(B)

Which of the following expressions is not meaningful ?       (A) (B) u. vw u .v .w

(A) 9.

13 / 2

(D)

 Let   4i  3 j and  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)

7.

None of these

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

 5.

3



 r . ˆi   r  ˆi    r . ˆj   r  j    r . kˆ   r  kˆ  is equal to : (A)

4.

   a b c

9 units

(B)

15 units

(C)

9 units

(D)

None of these

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

(A)

VMC/Vectors

  2  2 a 2  b  c (B)

  2  2 a 2  b  c (C)

128

  2  2 b 2  a  c (D)

None of these

HWT-6/Mathematics

Vidyamandir Classes DATE :

TIME : 40 Minutes

MARKS : [ ___ /10]

TEST CODE : VEC [4]

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.

         1 If d   a  b   b  c  v c  a and  a b c   , then     v is equal to : 8         (A) (B) d. a  b  c 2d . a  b  c         (C) (D) 4d . a  b  c 8d . a  b  c







2.

4.

6.



 



aˆ  bˆ 2

(B)



aˆ  bˆ 2 cos



(C)

aˆ  bˆ 2 cos  / 2

  a  b    a  b 

(D)

None of these

 c 1  c

p   1 or p  1 / 3

p  1 or p   1

(D)

        1 If a  b and c be three non-parallel unit vectors such that a  b  c  b , then the angle which vector a makes with b is : 2     (A) (B) (C) (D) 3 4 2 6





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

  2 2 16  a . b





(B)

  2 2 4  a .b





(C)

  2 16  a . b





  2 4  a .b



(D)



The moment of the couple formed by the forces 5i  k and  5i  k acting at the points (9, 1 , 2) and (3, 2 , 1) respectively, is: (A)

8.



 

 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

(A)

7.





         If a, b, c be three vectors such that a  b  c and b  c  a then :    (A) (B) a, b, c are orthogonal in pairs    (C) (D) a  b  c 1

(C)

5.

 

    If a and b are two vectors making angle  with each other, then unit vectors along bisector of a and b is :

(A)

3.



i  j  5k

(B)

i  11j  5k

(C)

i  11j  5k

i  j  5k

(D)

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

(A)

VMC/Vectors

m0

(B)

m0

(C)

129

m=0

(D)

m=3

HWT-6/Mathematics

Vidyamandir Classes 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)

10.

5 / 2 sq. units

(B)

2 5 sq. units

(C)

4 5 sq. units

(D)

5 sq. units

     If the non-zero vectors a and b are perpendicular to each other, then the solution of the equation, r  a  b is given by :

(where x is any scalar)     ab r  xa   (A) | a |2

VMC/Vectors

(B)

    ab r  xb   | a |2

(C)

130

   r  x ab





(D)

   r  x ba





HWT-6/Mathematics

Vidyamandir Classes DATE :

TIME : 40 Minutes

MARKS : [ ___ /10]

TEST CODE : VEC [5]

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.

         Let r , a, b and c be four non zero vectors such that r .a  0, r  b  r

1

(A) 2.

4.

0

(C)

1

 c

  then  a b c  is equal to :   2

(D)

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        OA  OB OC  OD  OM1 , then OM   OM1 if  is equal to : (A)

3.

(B)

    b , rc  r

1/16

(B)

1/8

(C)

1/4

(D)

1/2

      Let a, b and c be three non-zero and non-coplanar vectors and be three vectors given by p, q and r             p  a  b  2 c, q  3a  2b  c and r  a  4b  2c       If the volume of the parallelepiped determined by a, b and c 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

Let

       A, B and C be unit vectors. Suppose that A.B  A . C  0 and that the angle between

   A  k B  C and k is equal to :



(A) 5.

6.

2

(B)

4

6

(C)

(D)

is

 , then 6

0

 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) (B) (C) (D)  a2  a1 a3   a2   a1 a3   a2  a1 a3   a3  a1 a2 

  If a and b are unit vectors and  is the angle between them, then

(A)

7.



  B and C

sin

 2

(B)

sin 

(C)

  ab 2

is :

2 sin

sin 2

(D)

              If a b and c 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).

8.

(A)

 2  w2  1

(B)

2  2  w2  1

(C)

2     1  w2  0

(D)

None of these

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

VMC/Vectors

3

(C)

131

7

(D)

1

HWT-6/Mathematics

Vidyamandir Classes 9.

           Let b and c be non-collinear vectors. If a is a vector such that a . b  c  4 and a  b  c  x 2  2 x  6 b  sin y . c , then



(x, y) lies on the line : (A) x+y=0 10.

(B)

x y0

(C)



x=1



 



(D)

y 

       Let O be the centre of a regular pentagon ABCDE and OA  a . Then AB  2 BC  3CD  4 DE  5 EA equals :     (A) (B) (C) (D) 6a 5a 4a 0

VMC/Vectors

132

HWT-6/Mathematics

Vidyamandir Classes DATE :

TIME : 40 Minutes

MARKS : [ ___ /15]

TEST CODE : VEC [6]

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.

2.

   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) (D) None of these 0 AB  AB  CA (B) a AB  bBC  cCA (C)

 A vector a   x, y, z  makes an obtuse angle with Y-axis, and make equal angles with      c   2 z , 3x,  y  and a is perpendicular to d  1,  1, 2  if | a|  2 3 then vector a is : (A)

3.

(1, 2, 3)

5.

3a 2  b 2  c 2 2

8.

 1, 2, 4 

(D)

None of these

(B)

a 2  3b 2  c 2 2

(C)

a 2  b 2  3c 2 2

(D)

a 2  3b 2  c 2 2

         If a  i  j  k  a . b  1 and a  b  j  k then b is :





i  j  k

(B)

2 j  k

(C)

i

(D)

2i

(D)

6

         If a b and c are unit vectors, then |a  b|2  | b  c|2  |c  a|2 does not exceed: (A)

7.

(C)

 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 in the anticlockwise direction. the components of a in the new system are : 4 a3  a2 a  a3 (A) (B) a1  a2  a3  a1  2  a3  a2 2 2 a  a3 a3  a2 (C) (D) None of these a1  2  2 2

(A)

6.

 2,  2,  2 

and

     In a parallelogram ABCD, | AB |  a, | AD |  b and | AC |  c, then DB . AB has the value :

(A)

4.

(B)

 b   y,  2 z, 3x 

4

(B)

9

(C)

8

               If a  b  c  a  b  c where a b and c are any three vectors such that a . b  0  b . c  0 then a and c are :









(A)

inclined at an angle of  / 6 between them

(B)

Perpendicular

(C)

Parallel

(D)

inclined at an angle of  / 3 between them

   1         Let a b and c be non-zero vectors such that a  b  c  | b | | c | a . If  is the acute angle between the vectors b and c , 2 then sin equals :



(A)

VMC/Vectors

1 3

(B)

2 3



(C)

133

3 2

(D)

2 2 3

HWT-6/Mathematics

Vidyamandir Classes 9.

10.

       is equal to : Let u  i  j v  i  j and w  i  2 j  3k If n is a unit vector such that u . n = v . n  0 , then | w . n| (A) 0 (B) 1 (C) 2 (D) 3           If a and b are mutually perpendicular unit vectors, r is a vector satisfying r . a  0 r . b  1 and  r a b   1 then r :                (A) (B) (C) (D) a  ab b ab a  ab ab ab



VMC/Vectors









134







HWT-6/Mathematics

Vidyamandir Classes DATE :

TIME : 40 Minutes

MARKS : [ ___ /15]

TEST CODE : VEC [7]

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.

2.

             The position vectors a, b, c and d 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)

(D)

None of these

Circumcentre of ΔABC

(B)

4

0

(D)



(B)

GM of a and b







None of these



(C)

HM of a and b

(D)







and cx  c y  b x  y ,

Equal to zero

                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 b c  r  

   2 a b c  r  

 

 

 

 

 



   (D) None of these 3 a b c  r      If V is the volume of the parallelopiped having three coterminus edges as a . b and c , 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)

5.

(C)

x and y are two mutually perpendicular unit vectors, if the vectors ax  a y  c x  y  x  x  y lie in a plane then c is : (A) AM of a and b

4.

 

    Consider a tetrahedron with faces F1  F2  F3  F4 . Let V 1  V 2  V 3  V 4 be the vectors whose magnitudes are respectively equal to     areas of F1  F2  F3  F4 and whose directions are perpendicular to their faces in outward direction. Then | V 1  V 2  V 3  V 4 | ,

equals : (A) 1 3.



(B)



 

(C)

 









 



            a . c a  b . c b  c . c c , is :

     

(A) 6.

7.

V3

(B)

3V

V2

(D)

2V

Let G1, G2, G3 be the centroids of the triangular face OBC, OCA, OAB of a tetrahedron OABC. If V1 denote the volume of the tetrahedron OABC and V2 that of the parallelopied with OG1, OG2, OG3 as three concurrent edges, then: (A) (B) (C) (D) 4V1  9V2 9V1  4V2 3V1  2V2 3V2  2V1       a b c are non-coplanar vectors and a1  b1  c1 constitute the corresponding reciprocal system of vectors, then we have             a1  b1  b1  c1  c1  a1   a b c  a  b  c where ‘  ’ is equal to :  



(A) 8.

(C)

1

(B)

0



(C)

1

(D)

            If a  b c and p q r is reciprocal system of vectors, then a  p  b  q  c  r equals :        a b c  (A) (B) (C) (D) 0 pqr  

VMC/Vectors

135

2

   abc

HWT-6/Mathematics

Vidyamandir Classes 9.



   a b c   

(A)

10.

   a b d   

(B)



 

(C)

      If a  b  c is perpendicular to a  b  c , then we may have :     (A) (B) (C) b .c  0 a .b  0



VMC/Vectors







136

 

   b c d   

  a .c  0

 





  c  d  is always equal to :

              For any four vectors a b c d , the expression b  c . a  d  c  a . b  d  a  b . (D)

0

(D)

None of these

HWT-6/Mathematics

Vidyamandir Classes DATE :

TIME : 40 Minutes

MARKS : [ ___ /10]

TEST CODE : VEC [8]

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.

2.

  Let a and b are two vectors making angle  with each   other, then unit vectors along bisector of a and b is : a  b a  b   (A) (B) 2 2 cos a  b  (C) (D) None of these  2 cos 2    Let be three vectors such that a b c          a  b  c  b  c  a c  a  b then :       (A) (B) | a | | b | | c | | a |  | b | | c |       (C) (D) | a |  | b |  | c| |a||b||c|

6.

(A) (C) 7.

8. 3.

4.

If S is the circumcentre, G the centroid, O the orthocente    of a triangle ABC, then SA  SB  SC is :   (A) (B) 3SG OS   (C) (D) SG OG    Let a b and c 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) (C)

5.

1    abc 2 1    a  b c 3









(D)

1    2a  b  c 3  1   a  b  2c 2









   If are unit vectors a b c   2  2   2 | a  b |  | b  c |  | c  a | does not exceed.

(A) (C)

VMC/Vectors

4 8

(B) (D)

9 6

137

    a b c       2  a b c   

   2  a  b c  (D)     a  i  j  k  b  4i  3 j  4k ,

  1    1

(B)

  1    1

(C)

   1    1

(D)

   1   1

The volume of the tetrahedron whose vertices are with position vectors i  6 j  10k   i  3 j  7 k  5i  j   k

7i  4 j  7 k is 11 cubic unit if  equals : 3

(B)

3

1    Let a and b be non collinear vectors of which a is a (C)

7

(D)

unit vector. The angles of triangle whose two sides are       represented by 3 a  b and b  a . b a are :



10.

then

(B)

then

(A)

(A)

9.

0

vectors,

If and  c  ˆi   j   k are linearly dependent vectors and  |c|  3 , then :

and



(B)

   are three non-coplanar a b c        a  b b  c c  a  is equal to :  



 

(A)

     2 3 6

(B)

     2 4 4

(C)

     3 3 3

(D)

Data insufficient

  Let the unit vectors a and b 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 :





2

(A)

x  cos   y  sin   z  cos 2

(B)

x  sin   y  cos   z 2   cos 2

(C)

x  y  cos   z 2   cos 2

(D)

x  y  cos   z 2   cos 2

HWT-6/Mathematics