Chapter 2 - Unit & Dimensions Vectors - Physics, Solution by D C Pandey, NEET NEET Notes | EduRev

DC Pandey (Questions & Solutions) of Physics: NEET

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NEET : Chapter 2 - Unit & Dimensions Vectors - Physics, Solution by D C Pandey, NEET NEET Notes | EduRev

 Page 1


Section I
Single Correct Option
1. Pressure ( ) p =
Force
Area
\       [p] =
-
[MLT ]
[L ]
2
2
          =
- -
[ ] ML T
1 2
Option (d) is correct.
2.       W I Rt =
2
\ [R] = =
- - -
[ML T ]
[A T]
[ML T A ]
T
2
2
2 2 2 2
[ ]
…(i)
      V L
dI
dt
= and W Vq =
\      L
W
q
dt
dI
=
     [L] =
-
[ML T ][T]
[A T]
2
2
2
 =
- -
[ML T A ]
2 2 2
Using Eq. (i)
[R] =
[ ]
[ ]
L
T
i.e., [ ] T
L
R
=
é
ë
ê
ù
û
ú
Option (c) is correct.
3.         F av =6ph
\        [ ]
[ ]
[ ]
h =
F
av
           =
-
-
[MLT ]
LLT
2
[ ]
1
          =
- -
[ML T ]
1 1
Option (d) is correct.
4.    f=Li
\    [f ] = [ ][ L i]
        =
- -
[ML T A ] A
2 2 2
[ ]
        =
- -
[ML T A ]
2 2 1
Option (a) is correct.
5. Linear impulse ( ) I F t = × D
         [I] =
-
[MLT ][T]
2
            =
-
[MLT ]
1
Option (c) is correct.
6.  F G
m m
d
=
1 2
2
[G] = =
-
[ ][ ]
[ ]
[ML T ] [L ]
[M ]
2
1 2
2
2
F d
m m
2
   =
- -
[M L T ]
3 1 2
Option (c) is correct.
7.     F
i i
d
=
× m
p
0 1 2
4
\  [ m
0
] =
-
[ML T ][L]
[A ]
2
2
 =
- -
[ML T A ]
2 2 2
Option (c) is correct.
8. [ ]
[L]
[L T ]
[T] k = =
-1
Option (c) is correct.
9. [ ] a = =
-
-
[ML T ]
[ T ]
[ML T ]
2
3
[ ] b = =
-
-
[ML T ]
[ T ]
[ML T ]
2
2
4
Option (c) is correct.
Units & Dimensions
Vectors
2
Page 2


Section I
Single Correct Option
1. Pressure ( ) p =
Force
Area
\       [p] =
-
[MLT ]
[L ]
2
2
          =
- -
[ ] ML T
1 2
Option (d) is correct.
2.       W I Rt =
2
\ [R] = =
- - -
[ML T ]
[A T]
[ML T A ]
T
2
2
2 2 2 2
[ ]
…(i)
      V L
dI
dt
= and W Vq =
\      L
W
q
dt
dI
=
     [L] =
-
[ML T ][T]
[A T]
2
2
2
 =
- -
[ML T A ]
2 2 2
Using Eq. (i)
[R] =
[ ]
[ ]
L
T
i.e., [ ] T
L
R
=
é
ë
ê
ù
û
ú
Option (c) is correct.
3.         F av =6ph
\        [ ]
[ ]
[ ]
h =
F
av
           =
-
-
[MLT ]
LLT
2
[ ]
1
          =
- -
[ML T ]
1 1
Option (d) is correct.
4.    f=Li
\    [f ] = [ ][ L i]
        =
- -
[ML T A ] A
2 2 2
[ ]
        =
- -
[ML T A ]
2 2 1
Option (a) is correct.
5. Linear impulse ( ) I F t = × D
         [I] =
-
[MLT ][T]
2
            =
-
[MLT ]
1
Option (c) is correct.
6.  F G
m m
d
=
1 2
2
[G] = =
-
[ ][ ]
[ ]
[ML T ] [L ]
[M ]
2
1 2
2
2
F d
m m
2
   =
- -
[M L T ]
3 1 2
Option (c) is correct.
7.     F
i i
d
=
× m
p
0 1 2
4
\  [ m
0
] =
-
[ML T ][L]
[A ]
2
2
 =
- -
[ML T A ]
2 2 2
Option (c) is correct.
8. [ ]
[L]
[L T ]
[T] k = =
-1
Option (c) is correct.
9. [ ] a = =
-
-
[ML T ]
[ T ]
[ML T ]
2
3
[ ] b = =
-
-
[ML T ]
[ T ]
[ML T ]
2
2
4
Option (c) is correct.
Units & Dimensions
Vectors
2
10. E h = n
\     [ ]
[ML T ]
[T ]
[ML T ]
2
2
h = =
-
-
-
2
1
1
Angular momentum ( ) J
nh
=
2p
[ ] J h = =
-
[ ] [ML T ]
2 1
Option (b) is correct.
11. [Energy] =
-
[ML T ]
2 2
        =
-
[M][LT ]
2 1
\ [Mass] =
-
[ ] Ev
2
Option (c) is correct.
12.
1
2
0
2
e = E Energy density =
Energy
Volume
\        
1
2
0
2
2
3
e
é
ë
ê
ù
û
ú
=
-
E
[ML T ]
L
2
[ ]
                =
- -
[ML T ]
1 2
Option (b) is correct.
13.       [ ] a =[T ]
2
       [ ]
[ ]
[ ][ ]
b =
- -
T
L ML T
2
1 2
\      
a
b
é
ë
ê
ù
û
ú
=
-
[MT ]
2
Option (b) is correct.
14. Velocity gradient =
dv
dx
[Velocity gradient] =
-
[L T ]
[L]
1
   =
-
[T ]
1
        =
-
[ ] M L T
0 0 1
Option (a) is correct.
15.           [Force] =
-
[MLT ]
2
\ [Mass] =
-
[F]
[L T ]
2
       =
-
[FL T ]
2 1
Option (a) is correct.
16. Coefficient of friction (m)
=
Limitting frictional force
Normal force
\    [m] =[M L T ]
0 0 0
Option (b) is correct.
17.          q CV =
and V iR =    
\ q iCR =
i t iCR = 
Þ  [CR] = = [ ] t [M L T A ]
0 0 0
Option (a) is correct.
18. F
q q
r
=
1
4
0
1 2
2
pe
\ Unit of e
0
= Newton-metre
2
/coulomb
2
.
Option (b) is correct.
19. Angular momentum ( ) J
nh
=
2p
        I mr = S
2
\ 
h
I
J n
mr
=
2
2
p /
S
 =
mvr
mr S
2
h
I
é
ë
ê
ù
û
ú
=
é
ë
ê
ù
û
ú
=
-
-
[LT ]
L
[T ]
1
1
= Frequency
Option (a) is correct.
20.    v at
b
t c
= +
+
      [ ] [ ] c T =
  
b
t c
v
+
é
ë
ê
ù
û
ú
=[ ]
or [ ] b = =
-
[LT ][T] [L]
1
         [ ] [ ] at v = =
-
[LT ]
1
Þ         [a] =
-
[LT ]
2
Option (a) is correct.
21.     y A ct x = -
é
ë
ê
ù
û
ú
sin ( )
2p
l
           = -
é
ë
ê
ù
û
ú
A ct
x
sin
2 2 p
l
p
l
       
2p
l
q
x
= (angle)
\       [ ] [ ] [ ] x = = l L
Further, y A = sinq
\ [ ] [ ] [ ] A y = = L
Option (a) is correct.
22.   [ ] X =
- -
[M L T A ]
3 1 3 2
         =
-
[TA ]
[ML T ]
2
2 2
6 | Mechanics-1
Page 3


Section I
Single Correct Option
1. Pressure ( ) p =
Force
Area
\       [p] =
-
[MLT ]
[L ]
2
2
          =
- -
[ ] ML T
1 2
Option (d) is correct.
2.       W I Rt =
2
\ [R] = =
- - -
[ML T ]
[A T]
[ML T A ]
T
2
2
2 2 2 2
[ ]
…(i)
      V L
dI
dt
= and W Vq =
\      L
W
q
dt
dI
=
     [L] =
-
[ML T ][T]
[A T]
2
2
2
 =
- -
[ML T A ]
2 2 2
Using Eq. (i)
[R] =
[ ]
[ ]
L
T
i.e., [ ] T
L
R
=
é
ë
ê
ù
û
ú
Option (c) is correct.
3.         F av =6ph
\        [ ]
[ ]
[ ]
h =
F
av
           =
-
-
[MLT ]
LLT
2
[ ]
1
          =
- -
[ML T ]
1 1
Option (d) is correct.
4.    f=Li
\    [f ] = [ ][ L i]
        =
- -
[ML T A ] A
2 2 2
[ ]
        =
- -
[ML T A ]
2 2 1
Option (a) is correct.
5. Linear impulse ( ) I F t = × D
         [I] =
-
[MLT ][T]
2
            =
-
[MLT ]
1
Option (c) is correct.
6.  F G
m m
d
=
1 2
2
[G] = =
-
[ ][ ]
[ ]
[ML T ] [L ]
[M ]
2
1 2
2
2
F d
m m
2
   =
- -
[M L T ]
3 1 2
Option (c) is correct.
7.     F
i i
d
=
× m
p
0 1 2
4
\  [ m
0
] =
-
[ML T ][L]
[A ]
2
2
 =
- -
[ML T A ]
2 2 2
Option (c) is correct.
8. [ ]
[L]
[L T ]
[T] k = =
-1
Option (c) is correct.
9. [ ] a = =
-
-
[ML T ]
[ T ]
[ML T ]
2
3
[ ] b = =
-
-
[ML T ]
[ T ]
[ML T ]
2
2
4
Option (c) is correct.
Units & Dimensions
Vectors
2
10. E h = n
\     [ ]
[ML T ]
[T ]
[ML T ]
2
2
h = =
-
-
-
2
1
1
Angular momentum ( ) J
nh
=
2p
[ ] J h = =
-
[ ] [ML T ]
2 1
Option (b) is correct.
11. [Energy] =
-
[ML T ]
2 2
        =
-
[M][LT ]
2 1
\ [Mass] =
-
[ ] Ev
2
Option (c) is correct.
12.
1
2
0
2
e = E Energy density =
Energy
Volume
\        
1
2
0
2
2
3
e
é
ë
ê
ù
û
ú
=
-
E
[ML T ]
L
2
[ ]
                =
- -
[ML T ]
1 2
Option (b) is correct.
13.       [ ] a =[T ]
2
       [ ]
[ ]
[ ][ ]
b =
- -
T
L ML T
2
1 2
\      
a
b
é
ë
ê
ù
û
ú
=
-
[MT ]
2
Option (b) is correct.
14. Velocity gradient =
dv
dx
[Velocity gradient] =
-
[L T ]
[L]
1
   =
-
[T ]
1
        =
-
[ ] M L T
0 0 1
Option (a) is correct.
15.           [Force] =
-
[MLT ]
2
\ [Mass] =
-
[F]
[L T ]
2
       =
-
[FL T ]
2 1
Option (a) is correct.
16. Coefficient of friction (m)
=
Limitting frictional force
Normal force
\    [m] =[M L T ]
0 0 0
Option (b) is correct.
17.          q CV =
and V iR =    
\ q iCR =
i t iCR = 
Þ  [CR] = = [ ] t [M L T A ]
0 0 0
Option (a) is correct.
18. F
q q
r
=
1
4
0
1 2
2
pe
\ Unit of e
0
= Newton-metre
2
/coulomb
2
.
Option (b) is correct.
19. Angular momentum ( ) J
nh
=
2p
        I mr = S
2
\ 
h
I
J n
mr
=
2
2
p /
S
 =
mvr
mr S
2
h
I
é
ë
ê
ù
û
ú
=
é
ë
ê
ù
û
ú
=
-
-
[LT ]
L
[T ]
1
1
= Frequency
Option (a) is correct.
20.    v at
b
t c
= +
+
      [ ] [ ] c T =
  
b
t c
v
+
é
ë
ê
ù
û
ú
=[ ]
or [ ] b = =
-
[LT ][T] [L]
1
         [ ] [ ] at v = =
-
[LT ]
1
Þ         [a] =
-
[LT ]
2
Option (a) is correct.
21.     y A ct x = -
é
ë
ê
ù
û
ú
sin ( )
2p
l
           = -
é
ë
ê
ù
û
ú
A ct
x
sin
2 2 p
l
p
l
       
2p
l
q
x
= (angle)
\       [ ] [ ] [ ] x = = l L
Further, y A = sinq
\ [ ] [ ] [ ] A y = = L
Option (a) is correct.
22.   [ ] X =
- -
[M L T A ]
3 1 3 2
         =
-
[TA ]
[ML T ]
2
2 2
6 | Mechanics-1
         =
[ ][ ]
[ ]
t i
2
Work
\  X is resistance. [ Q W i Rt =
2
]
23.   F i j k
®
= - + 2 3 4
^ ^ ^
     r i j k
®
= + + 3 2 3
^ ^ ^
\      t
® ® ®
= ´ r F =
-
½
½
½
½
½ ½
½
½
½
½
½ ½
i j k
^ ^ ^
3 2 3
2 3 4
or   t
®
= - - 17 6 13 i j k
^ ^ ^
24. ( ) ( ) ( ) 0.5 0.8
2 2 2
1 + + = c
or   0.25 0.64 + + = c
2
1
or      c
2
1 = -0.89
     c = 0.11
Option (b) is correct.
25. | | | | A B A B
® ® ® ®
+ = -
( ) ( ) ( ) ( ) A B A B A B A B
® ® ® ® ® ® ® ®
+ × + = - × -
A B A B
2 2 2 2
2 2 + + × = + - ×
® ® ® ®
A B A B
i.e.,       A B
® ®
× =0
\    Angle between A
®
 and B
®
 = ° 90
26. ( ) ( ) A B A B
® ® ® ®
+ × - = 0
A A B A B B A B
® ® ® ® ® ® ® ®
× + × - × - × = 0
 A B
2 2
0 - =
        A B = ±
       | | | | A B
® ®
=
Option (d) is correct.
27. Work ( ) = ×
® ®
F s is a scalar quantity.
Option (d) is correct.
28. Speed =
®
| | v
Option (d) is correct.
29. | | A
®
= 3, | | B
®
= 5 and angle between A
®
 and B
®
is 60°.
\ A B A B
® ® ® ®
× = ° | || |cos 60
=
æ
è
ç
ö
ø
÷ ( ) ( ) 3 5
1
2
= 7.5            
Option (b) is correct.
30. A B C
® ® ®
+ =
\       ( ) ( ) A B A B CC
® ® ® ® ®®
+ × + = ×
or AA AB BB CC
® ® ® ® ®® ®®
× + × + × = × 2
or   A B C
2 2 2
2 + × + =
® ®
A B
or          A B
® ®
× =0
or   | || |cos A B
® ®
= q 0
or         cosq =0
or            q
p
=
2
Option (d) is correct.
31. Magnetic field intensity.
Option (d) is correct.
32. P Q R
® ® ®
+ =
( ) ( ) P Q P Q R R
® ® ® ® ® ®
+ × + = ×
P Q R
2 2 2
2 + + × =
® ®
P Q
12 5 2 13
2 2 2
+ + × =
® ®
P Q
                P Q
® ®
× = 0
\    Angle between P
®
 and Q
®
 =
p
2
Option (b) is correct.
33. Option (b) is correct.
34. P Q R
® ® ®
+ + = 0
\     P Q R
® ® ®
+ = -
or               ( ) ( ) ( ) ( ) P Q P Q R R
® ® ® ® ® ®
+ × + = - × -
or P P Q Q P Q R R
® ® ® ® ® ® ® ®
× + × + × = × 2
or  P Q R
2 2 2
2 + + × =
® ®
P Q …(i)
Let Q P
2 2
= and R P = 2
Thus, Eq. (i) takes the form
P P PQ P
2 2 2
2 2 + + = cos q
or 2 0 PQcos q =
Units & Dimensions Vectors 7
Page 4


Section I
Single Correct Option
1. Pressure ( ) p =
Force
Area
\       [p] =
-
[MLT ]
[L ]
2
2
          =
- -
[ ] ML T
1 2
Option (d) is correct.
2.       W I Rt =
2
\ [R] = =
- - -
[ML T ]
[A T]
[ML T A ]
T
2
2
2 2 2 2
[ ]
…(i)
      V L
dI
dt
= and W Vq =
\      L
W
q
dt
dI
=
     [L] =
-
[ML T ][T]
[A T]
2
2
2
 =
- -
[ML T A ]
2 2 2
Using Eq. (i)
[R] =
[ ]
[ ]
L
T
i.e., [ ] T
L
R
=
é
ë
ê
ù
û
ú
Option (c) is correct.
3.         F av =6ph
\        [ ]
[ ]
[ ]
h =
F
av
           =
-
-
[MLT ]
LLT
2
[ ]
1
          =
- -
[ML T ]
1 1
Option (d) is correct.
4.    f=Li
\    [f ] = [ ][ L i]
        =
- -
[ML T A ] A
2 2 2
[ ]
        =
- -
[ML T A ]
2 2 1
Option (a) is correct.
5. Linear impulse ( ) I F t = × D
         [I] =
-
[MLT ][T]
2
            =
-
[MLT ]
1
Option (c) is correct.
6.  F G
m m
d
=
1 2
2
[G] = =
-
[ ][ ]
[ ]
[ML T ] [L ]
[M ]
2
1 2
2
2
F d
m m
2
   =
- -
[M L T ]
3 1 2
Option (c) is correct.
7.     F
i i
d
=
× m
p
0 1 2
4
\  [ m
0
] =
-
[ML T ][L]
[A ]
2
2
 =
- -
[ML T A ]
2 2 2
Option (c) is correct.
8. [ ]
[L]
[L T ]
[T] k = =
-1
Option (c) is correct.
9. [ ] a = =
-
-
[ML T ]
[ T ]
[ML T ]
2
3
[ ] b = =
-
-
[ML T ]
[ T ]
[ML T ]
2
2
4
Option (c) is correct.
Units & Dimensions
Vectors
2
10. E h = n
\     [ ]
[ML T ]
[T ]
[ML T ]
2
2
h = =
-
-
-
2
1
1
Angular momentum ( ) J
nh
=
2p
[ ] J h = =
-
[ ] [ML T ]
2 1
Option (b) is correct.
11. [Energy] =
-
[ML T ]
2 2
        =
-
[M][LT ]
2 1
\ [Mass] =
-
[ ] Ev
2
Option (c) is correct.
12.
1
2
0
2
e = E Energy density =
Energy
Volume
\        
1
2
0
2
2
3
e
é
ë
ê
ù
û
ú
=
-
E
[ML T ]
L
2
[ ]
                =
- -
[ML T ]
1 2
Option (b) is correct.
13.       [ ] a =[T ]
2
       [ ]
[ ]
[ ][ ]
b =
- -
T
L ML T
2
1 2
\      
a
b
é
ë
ê
ù
û
ú
=
-
[MT ]
2
Option (b) is correct.
14. Velocity gradient =
dv
dx
[Velocity gradient] =
-
[L T ]
[L]
1
   =
-
[T ]
1
        =
-
[ ] M L T
0 0 1
Option (a) is correct.
15.           [Force] =
-
[MLT ]
2
\ [Mass] =
-
[F]
[L T ]
2
       =
-
[FL T ]
2 1
Option (a) is correct.
16. Coefficient of friction (m)
=
Limitting frictional force
Normal force
\    [m] =[M L T ]
0 0 0
Option (b) is correct.
17.          q CV =
and V iR =    
\ q iCR =
i t iCR = 
Þ  [CR] = = [ ] t [M L T A ]
0 0 0
Option (a) is correct.
18. F
q q
r
=
1
4
0
1 2
2
pe
\ Unit of e
0
= Newton-metre
2
/coulomb
2
.
Option (b) is correct.
19. Angular momentum ( ) J
nh
=
2p
        I mr = S
2
\ 
h
I
J n
mr
=
2
2
p /
S
 =
mvr
mr S
2
h
I
é
ë
ê
ù
û
ú
=
é
ë
ê
ù
û
ú
=
-
-
[LT ]
L
[T ]
1
1
= Frequency
Option (a) is correct.
20.    v at
b
t c
= +
+
      [ ] [ ] c T =
  
b
t c
v
+
é
ë
ê
ù
û
ú
=[ ]
or [ ] b = =
-
[LT ][T] [L]
1
         [ ] [ ] at v = =
-
[LT ]
1
Þ         [a] =
-
[LT ]
2
Option (a) is correct.
21.     y A ct x = -
é
ë
ê
ù
û
ú
sin ( )
2p
l
           = -
é
ë
ê
ù
û
ú
A ct
x
sin
2 2 p
l
p
l
       
2p
l
q
x
= (angle)
\       [ ] [ ] [ ] x = = l L
Further, y A = sinq
\ [ ] [ ] [ ] A y = = L
Option (a) is correct.
22.   [ ] X =
- -
[M L T A ]
3 1 3 2
         =
-
[TA ]
[ML T ]
2
2 2
6 | Mechanics-1
         =
[ ][ ]
[ ]
t i
2
Work
\  X is resistance. [ Q W i Rt =
2
]
23.   F i j k
®
= - + 2 3 4
^ ^ ^
     r i j k
®
= + + 3 2 3
^ ^ ^
\      t
® ® ®
= ´ r F =
-
½
½
½
½
½ ½
½
½
½
½
½ ½
i j k
^ ^ ^
3 2 3
2 3 4
or   t
®
= - - 17 6 13 i j k
^ ^ ^
24. ( ) ( ) ( ) 0.5 0.8
2 2 2
1 + + = c
or   0.25 0.64 + + = c
2
1
or      c
2
1 = -0.89
     c = 0.11
Option (b) is correct.
25. | | | | A B A B
® ® ® ®
+ = -
( ) ( ) ( ) ( ) A B A B A B A B
® ® ® ® ® ® ® ®
+ × + = - × -
A B A B
2 2 2 2
2 2 + + × = + - ×
® ® ® ®
A B A B
i.e.,       A B
® ®
× =0
\    Angle between A
®
 and B
®
 = ° 90
26. ( ) ( ) A B A B
® ® ® ®
+ × - = 0
A A B A B B A B
® ® ® ® ® ® ® ®
× + × - × - × = 0
 A B
2 2
0 - =
        A B = ±
       | | | | A B
® ®
=
Option (d) is correct.
27. Work ( ) = ×
® ®
F s is a scalar quantity.
Option (d) is correct.
28. Speed =
®
| | v
Option (d) is correct.
29. | | A
®
= 3, | | B
®
= 5 and angle between A
®
 and B
®
is 60°.
\ A B A B
® ® ® ®
× = ° | || |cos 60
=
æ
è
ç
ö
ø
÷ ( ) ( ) 3 5
1
2
= 7.5            
Option (b) is correct.
30. A B C
® ® ®
+ =
\       ( ) ( ) A B A B CC
® ® ® ® ®®
+ × + = ×
or AA AB BB CC
® ® ® ® ®® ®®
× + × + × = × 2
or   A B C
2 2 2
2 + × + =
® ®
A B
or          A B
® ®
× =0
or   | || |cos A B
® ®
= q 0
or         cosq =0
or            q
p
=
2
Option (d) is correct.
31. Magnetic field intensity.
Option (d) is correct.
32. P Q R
® ® ®
+ =
( ) ( ) P Q P Q R R
® ® ® ® ® ®
+ × + = ×
P Q R
2 2 2
2 + + × =
® ®
P Q
12 5 2 13
2 2 2
+ + × =
® ®
P Q
                P Q
® ®
× = 0
\    Angle between P
®
 and Q
®
 =
p
2
Option (b) is correct.
33. Option (b) is correct.
34. P Q R
® ® ®
+ + = 0
\     P Q R
® ® ®
+ = -
or               ( ) ( ) ( ) ( ) P Q P Q R R
® ® ® ® ® ®
+ × + = - × -
or P P Q Q P Q R R
® ® ® ® ® ® ® ®
× + × + × = × 2
or  P Q R
2 2 2
2 + + × =
® ®
P Q …(i)
Let Q P
2 2
= and R P = 2
Thus, Eq. (i) takes the form
P P PQ P
2 2 2
2 2 + + = cos q
or 2 0 PQcos q =
Units & Dimensions Vectors 7
or    cosq =0
or         q = ° 90
\  Angle between P
®
 and Q
®
 is 90°
P Q R
® ® ®
+ + = 0
\       P R Q
® ® ®
+ = -
or ( ) ( ) ( ) ( ) P R P R Q Q
® ® ® ® ® ®
+ × + = - × -
or P R PR Q
2 2 2
2 + + f = cos
or 2
2 2 2
PR Q P R cosf = - -
or       2
2
PR R cosf = -
or        2P R cosf = -
or       2 2 P P cosf = -
or          cosf = -
1
2
\             f = ° 135
\  Angle between P
®
 and R
®
 is 135°.
Option (a) is correct.
35. Angle ( ) f between P Q
® ®
+ and P Q
® ®
-
tan
sin
cos
f =
+
Q
P Q
q
q
Angle f¢ between P Q
® ®
- and P
®
tan
sin ( )
cos ( )
f¢ =
+
+ +
Q
P Q
p q
p q
  
=
-
-
Q
P Q
sin
cos
q
q
 tan[ ( )]
tan tan
tan tan
f + -f¢ =
f - f¢
+ f f¢ 1
    =
+
-
-
-
-
+
×
-
Q
P Q
Q
P Q
Q
P Q
Q
sin
cos
( sin )
cos
sin
( cos )
( s
q
q
q
q
q
q
1
in )
( cos )
q
q P Q -
    =
+
2
2
2 2
PQ
P Q
sin
cos
q
q
This implies that angle between P Q
® ®
+ and 
P Q
® ®
- will vary from 0 to p.
Option (b) is correct.
36. R P Q PQ
2 2 2
2 = + + cos q
for R P Q = =
P P P PP
2 2 2
2 = + + cos q
or    cosq = -
1
2
or       q = ° 120
Option (b) is correct.
37.      W = ×
® ®
F s
    = + × + ( ) ( )
^ ^ ^ ^
3 4 3 4 i j i j
    =25 J
Option (b) is correct.
38. P Q i j k i j k
® ®
× = + + × - - ( ) ( )
^ ^ ^ ^ ^ ^
a a a 3 2
    = - - a a
2
2 3
For P Q
® ®
^ , P Q
® ®
× =0
i.e.,     a a
2
2 3 0 - - =
or  ( )( ) a a - + = 3 1 0
Þ          a =3
Other value is - ive.
Option (d) is correct.
39. If a vector makes angles a, b and g with
the co-ordinate axes, then
  cos cos cos
2 2 2
1 a b g + + =
Now,
3
7
9
49
2
æ
è
ç
ö
ø
÷ = , 
6
7
36
49
2
æ
è
ç
ö
ø
÷ = , 
2
7
4
49
2
æ
è
ç
ö
ø
÷ =
and          
9
49
36
49
4
49
1 + + =
\ Option (a) is correct.
40. A i j
®
= - 4 3
^
     and     B i j
®
= + 8 8
^
$
\ A B C i j
® ® ®
+ = = + 12 5
^ ^
8 | Mechanics-1
P
90°
Q
R
135°
135°
®
®
®
– Q
Q
Q P  –
Q P  +
P
q
f
f'
®
®
®
®
®
®
®
Page 5


Section I
Single Correct Option
1. Pressure ( ) p =
Force
Area
\       [p] =
-
[MLT ]
[L ]
2
2
          =
- -
[ ] ML T
1 2
Option (d) is correct.
2.       W I Rt =
2
\ [R] = =
- - -
[ML T ]
[A T]
[ML T A ]
T
2
2
2 2 2 2
[ ]
…(i)
      V L
dI
dt
= and W Vq =
\      L
W
q
dt
dI
=
     [L] =
-
[ML T ][T]
[A T]
2
2
2
 =
- -
[ML T A ]
2 2 2
Using Eq. (i)
[R] =
[ ]
[ ]
L
T
i.e., [ ] T
L
R
=
é
ë
ê
ù
û
ú
Option (c) is correct.
3.         F av =6ph
\        [ ]
[ ]
[ ]
h =
F
av
           =
-
-
[MLT ]
LLT
2
[ ]
1
          =
- -
[ML T ]
1 1
Option (d) is correct.
4.    f=Li
\    [f ] = [ ][ L i]
        =
- -
[ML T A ] A
2 2 2
[ ]
        =
- -
[ML T A ]
2 2 1
Option (a) is correct.
5. Linear impulse ( ) I F t = × D
         [I] =
-
[MLT ][T]
2
            =
-
[MLT ]
1
Option (c) is correct.
6.  F G
m m
d
=
1 2
2
[G] = =
-
[ ][ ]
[ ]
[ML T ] [L ]
[M ]
2
1 2
2
2
F d
m m
2
   =
- -
[M L T ]
3 1 2
Option (c) is correct.
7.     F
i i
d
=
× m
p
0 1 2
4
\  [ m
0
] =
-
[ML T ][L]
[A ]
2
2
 =
- -
[ML T A ]
2 2 2
Option (c) is correct.
8. [ ]
[L]
[L T ]
[T] k = =
-1
Option (c) is correct.
9. [ ] a = =
-
-
[ML T ]
[ T ]
[ML T ]
2
3
[ ] b = =
-
-
[ML T ]
[ T ]
[ML T ]
2
2
4
Option (c) is correct.
Units & Dimensions
Vectors
2
10. E h = n
\     [ ]
[ML T ]
[T ]
[ML T ]
2
2
h = =
-
-
-
2
1
1
Angular momentum ( ) J
nh
=
2p
[ ] J h = =
-
[ ] [ML T ]
2 1
Option (b) is correct.
11. [Energy] =
-
[ML T ]
2 2
        =
-
[M][LT ]
2 1
\ [Mass] =
-
[ ] Ev
2
Option (c) is correct.
12.
1
2
0
2
e = E Energy density =
Energy
Volume
\        
1
2
0
2
2
3
e
é
ë
ê
ù
û
ú
=
-
E
[ML T ]
L
2
[ ]
                =
- -
[ML T ]
1 2
Option (b) is correct.
13.       [ ] a =[T ]
2
       [ ]
[ ]
[ ][ ]
b =
- -
T
L ML T
2
1 2
\      
a
b
é
ë
ê
ù
û
ú
=
-
[MT ]
2
Option (b) is correct.
14. Velocity gradient =
dv
dx
[Velocity gradient] =
-
[L T ]
[L]
1
   =
-
[T ]
1
        =
-
[ ] M L T
0 0 1
Option (a) is correct.
15.           [Force] =
-
[MLT ]
2
\ [Mass] =
-
[F]
[L T ]
2
       =
-
[FL T ]
2 1
Option (a) is correct.
16. Coefficient of friction (m)
=
Limitting frictional force
Normal force
\    [m] =[M L T ]
0 0 0
Option (b) is correct.
17.          q CV =
and V iR =    
\ q iCR =
i t iCR = 
Þ  [CR] = = [ ] t [M L T A ]
0 0 0
Option (a) is correct.
18. F
q q
r
=
1
4
0
1 2
2
pe
\ Unit of e
0
= Newton-metre
2
/coulomb
2
.
Option (b) is correct.
19. Angular momentum ( ) J
nh
=
2p
        I mr = S
2
\ 
h
I
J n
mr
=
2
2
p /
S
 =
mvr
mr S
2
h
I
é
ë
ê
ù
û
ú
=
é
ë
ê
ù
û
ú
=
-
-
[LT ]
L
[T ]
1
1
= Frequency
Option (a) is correct.
20.    v at
b
t c
= +
+
      [ ] [ ] c T =
  
b
t c
v
+
é
ë
ê
ù
û
ú
=[ ]
or [ ] b = =
-
[LT ][T] [L]
1
         [ ] [ ] at v = =
-
[LT ]
1
Þ         [a] =
-
[LT ]
2
Option (a) is correct.
21.     y A ct x = -
é
ë
ê
ù
û
ú
sin ( )
2p
l
           = -
é
ë
ê
ù
û
ú
A ct
x
sin
2 2 p
l
p
l
       
2p
l
q
x
= (angle)
\       [ ] [ ] [ ] x = = l L
Further, y A = sinq
\ [ ] [ ] [ ] A y = = L
Option (a) is correct.
22.   [ ] X =
- -
[M L T A ]
3 1 3 2
         =
-
[TA ]
[ML T ]
2
2 2
6 | Mechanics-1
         =
[ ][ ]
[ ]
t i
2
Work
\  X is resistance. [ Q W i Rt =
2
]
23.   F i j k
®
= - + 2 3 4
^ ^ ^
     r i j k
®
= + + 3 2 3
^ ^ ^
\      t
® ® ®
= ´ r F =
-
½
½
½
½
½ ½
½
½
½
½
½ ½
i j k
^ ^ ^
3 2 3
2 3 4
or   t
®
= - - 17 6 13 i j k
^ ^ ^
24. ( ) ( ) ( ) 0.5 0.8
2 2 2
1 + + = c
or   0.25 0.64 + + = c
2
1
or      c
2
1 = -0.89
     c = 0.11
Option (b) is correct.
25. | | | | A B A B
® ® ® ®
+ = -
( ) ( ) ( ) ( ) A B A B A B A B
® ® ® ® ® ® ® ®
+ × + = - × -
A B A B
2 2 2 2
2 2 + + × = + - ×
® ® ® ®
A B A B
i.e.,       A B
® ®
× =0
\    Angle between A
®
 and B
®
 = ° 90
26. ( ) ( ) A B A B
® ® ® ®
+ × - = 0
A A B A B B A B
® ® ® ® ® ® ® ®
× + × - × - × = 0
 A B
2 2
0 - =
        A B = ±
       | | | | A B
® ®
=
Option (d) is correct.
27. Work ( ) = ×
® ®
F s is a scalar quantity.
Option (d) is correct.
28. Speed =
®
| | v
Option (d) is correct.
29. | | A
®
= 3, | | B
®
= 5 and angle between A
®
 and B
®
is 60°.
\ A B A B
® ® ® ®
× = ° | || |cos 60
=
æ
è
ç
ö
ø
÷ ( ) ( ) 3 5
1
2
= 7.5            
Option (b) is correct.
30. A B C
® ® ®
+ =
\       ( ) ( ) A B A B CC
® ® ® ® ®®
+ × + = ×
or AA AB BB CC
® ® ® ® ®® ®®
× + × + × = × 2
or   A B C
2 2 2
2 + × + =
® ®
A B
or          A B
® ®
× =0
or   | || |cos A B
® ®
= q 0
or         cosq =0
or            q
p
=
2
Option (d) is correct.
31. Magnetic field intensity.
Option (d) is correct.
32. P Q R
® ® ®
+ =
( ) ( ) P Q P Q R R
® ® ® ® ® ®
+ × + = ×
P Q R
2 2 2
2 + + × =
® ®
P Q
12 5 2 13
2 2 2
+ + × =
® ®
P Q
                P Q
® ®
× = 0
\    Angle between P
®
 and Q
®
 =
p
2
Option (b) is correct.
33. Option (b) is correct.
34. P Q R
® ® ®
+ + = 0
\     P Q R
® ® ®
+ = -
or               ( ) ( ) ( ) ( ) P Q P Q R R
® ® ® ® ® ®
+ × + = - × -
or P P Q Q P Q R R
® ® ® ® ® ® ® ®
× + × + × = × 2
or  P Q R
2 2 2
2 + + × =
® ®
P Q …(i)
Let Q P
2 2
= and R P = 2
Thus, Eq. (i) takes the form
P P PQ P
2 2 2
2 2 + + = cos q
or 2 0 PQcos q =
Units & Dimensions Vectors 7
or    cosq =0
or         q = ° 90
\  Angle between P
®
 and Q
®
 is 90°
P Q R
® ® ®
+ + = 0
\       P R Q
® ® ®
+ = -
or ( ) ( ) ( ) ( ) P R P R Q Q
® ® ® ® ® ®
+ × + = - × -
or P R PR Q
2 2 2
2 + + f = cos
or 2
2 2 2
PR Q P R cosf = - -
or       2
2
PR R cosf = -
or        2P R cosf = -
or       2 2 P P cosf = -
or          cosf = -
1
2
\             f = ° 135
\  Angle between P
®
 and R
®
 is 135°.
Option (a) is correct.
35. Angle ( ) f between P Q
® ®
+ and P Q
® ®
-
tan
sin
cos
f =
+
Q
P Q
q
q
Angle f¢ between P Q
® ®
- and P
®
tan
sin ( )
cos ( )
f¢ =
+
+ +
Q
P Q
p q
p q
  
=
-
-
Q
P Q
sin
cos
q
q
 tan[ ( )]
tan tan
tan tan
f + -f¢ =
f - f¢
+ f f¢ 1
    =
+
-
-
-
-
+
×
-
Q
P Q
Q
P Q
Q
P Q
Q
sin
cos
( sin )
cos
sin
( cos )
( s
q
q
q
q
q
q
1
in )
( cos )
q
q P Q -
    =
+
2
2
2 2
PQ
P Q
sin
cos
q
q
This implies that angle between P Q
® ®
+ and 
P Q
® ®
- will vary from 0 to p.
Option (b) is correct.
36. R P Q PQ
2 2 2
2 = + + cos q
for R P Q = =
P P P PP
2 2 2
2 = + + cos q
or    cosq = -
1
2
or       q = ° 120
Option (b) is correct.
37.      W = ×
® ®
F s
    = + × + ( ) ( )
^ ^ ^ ^
3 4 3 4 i j i j
    =25 J
Option (b) is correct.
38. P Q i j k i j k
® ®
× = + + × - - ( ) ( )
^ ^ ^ ^ ^ ^
a a a 3 2
    = - - a a
2
2 3
For P Q
® ®
^ , P Q
® ®
× =0
i.e.,     a a
2
2 3 0 - - =
or  ( )( ) a a - + = 3 1 0
Þ          a =3
Other value is - ive.
Option (d) is correct.
39. If a vector makes angles a, b and g with
the co-ordinate axes, then
  cos cos cos
2 2 2
1 a b g + + =
Now,
3
7
9
49
2
æ
è
ç
ö
ø
÷ = , 
6
7
36
49
2
æ
è
ç
ö
ø
÷ = , 
2
7
4
49
2
æ
è
ç
ö
ø
÷ =
and          
9
49
36
49
4
49
1 + + =
\ Option (a) is correct.
40. A i j
®
= - 4 3
^
     and     B i j
®
= + 8 8
^
$
\ A B C i j
® ® ®
+ = = + 12 5
^ ^
8 | Mechanics-1
P
90°
Q
R
135°
135°
®
®
®
– Q
Q
Q P  –
Q P  +
P
q
f
f'
®
®
®
®
®
®
®
      C
C i j ^
^ ^
| |
= =
+
+
®
C
12 5
12 5
2 2
    = +
12
13
5
13
i j
^ ^
Option (b) is correct.
41. A i j k
®
= + - 2 3 2
^ ^
$
, B i j k
®
= + + 5
^ ^
$
n , 
C i j k
®
= - + +
^ ^
$
2 3
\ Vectors A B C
® ® ®
, ,and will be coplanar if
their scalar triple product is zero i.e.,
( ) A C B
® ® ®
´ × = 0
$ $ $
( )
^ ^ ^
i j k
i j k 2 3 2
1 2 3
5 0 -
-
½
½
½
½
½
½
½
½
½
½
× + + = n
or ( ) ( )
^ ^ ^ ^ ^ ^
13 4 7 5 0 i j k i j k - + × + + = n
or 65 4 7 0 - + = n
or                 n = 18
Option (a) is correct.
42. Option (a) is correct.
43. ( ) ( ) a b a b
® ® ® ®
+ ´ -
    = ´ + ´ - ´ - ´
® ® ® ® ® ® ® ®
a a b a a b b b
    = - ´ - ´ -
® ® ® ®
0 0 a b a b
    =- ´
® ®
2( ) a b = ´
® ®
2( ) b a
Option (a) is correct.
44.   A i j k
®
= + + 3 4 5
^ ^ ^
      B i j k
®
= + - 3 4 5
^ ^ ^
    cos
( )
| || |
q=
×
® ®
® ®
A B
A B
       =
+ -
+ +
9 16 25
3 4 5
2 2 2
       =0
Þ     q = ° 90
Option (c) is correct.
45.    A B + =7
         A B - =3
\            B=2 N 
Option (c) is correct.
46. Angle between A i j
®
= + 2 3
^ ^
 
and          B i j
®
= +
^ ^
    q=
×
® ®
® ®
A B
|A||B|
 =
+
+ ×
2 3
2 3 2
2 2
         =
×
5
13 2
         =
5
26
Component of A
®
 along i j
^ ^
+ 
C i j
®
= +
5
26
2 3 ( )
^ ^
         | | C
®
=
5
2
Option (a) is correct.
47. R P P P P
2 2 2
3 2 2 3 2 = + + ´ ´ ( ) ( ) cos q
or R P P
2 2 2
13 12 = + cos q …(i)
Further
 ( ) ( ) ( ) cos 2 6 2 2 6 2
2 2 2
R P P P P = + + ´ ´ q
or 4 40 24
2 2 2
R P P = + cos q …(ii)
Dividing Eq. (ii) by Eq. (i),
10 6 13 12
2 2 2 2
P P P P + = + cos cos q q
or         6cosq = - P
or  q = ° 120
Option (b) is correct.
48.   tan
sin
cos
q
a
a
=
+
Q
P Q
As      q = ° 90 , tana = ¥
\         P Q + = cosa 0
i.e.,   cosa = -
P
Q
        = -
P
P 2
Units & Dimensions Vectors 9
q = 90°
R
P
Q = 2P
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