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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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FAQs on DC Pandey Solutions: Unit & Dimensions Vectors - DC Pandey Solutions for JEE Physics
| 1. What are units and dimensions in physics? |  |
Ans. Units and dimensions in physics refer to the measurement and description of physical quantities. Units are used to measure and express the magnitude of a physical quantity, such as length, time, or mass. Dimensions, on the other hand, represent the nature of the physical quantity being measured, such as whether it is a scalar or a vector quantity.
| 2. How do we convert units in physics? | |
Ans. To convert units in physics, you can use conversion factors. A conversion factor is a ratio of two equivalent quantities expressed in different units. By multiplying the given quantity by the appropriate conversion factor, you can convert it into the desired unit. For example, to convert meters to kilometers, you can use the conversion factor 1 kilometer = 1000 meters.
| 3. What are vectors in physics? | |
Ans. Vectors in physics are quantities that have both magnitude and direction. They are used to represent physical quantities that require both a numerical value and a specific direction to fully describe them. Examples of vector quantities include displacement, velocity, force, and acceleration.
| 4. How do we perform vector addition and subtraction? | |
Ans. Vector addition and subtraction involve combining or separating vectors to find their resultant or difference, respectively. To perform vector addition, you can use the head-to-tail method or the parallelogram method. In the head-to-tail method, you place the tail of one vector at the head of another vector and draw a vector from the tail of the first vector to the head of the second vector. The resultant vector is the vector that connects the tail of the first vector to the head of the second vector. To perform vector subtraction, you can add the negative of the vector you want to subtract to the original vector.
| 5. How do we resolve vectors into their components? | |
Ans. Resolving vectors into their components involves breaking down a vector into its perpendicular components along two or more axes. This is often done to simplify vector calculations or to analyze vector quantities in different directions. To resolve a vector, you can use trigonometry. For example, if you have a vector at an angle θ with the horizontal axis, you can find its horizontal and vertical components by multiplying the magnitude of the vector by the cosine and sine of θ, respectively.
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