JEE Advanced Numericals Errors - Physics for JEE Main & Advanced

 Page 1


?
 
    
 
 
   
1  
ERRORS 
 
1. In an experiment for determination of the focal length of a thin convex lens, the distance of the object 
from the lens is 10 ± 0.1 cm and the distance of its real image from the lens is 20 ± 0.2 cm. The error in 
the determination of focal length of the lens is n %. The value of n is _______.          
2. Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is  
0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale 
shifts by two divisions. The measured readings are listed below.   
Measurement condition Main scale reading Circular scale reading 
Two arms of gauge touching 
each other without wire 
0 division 4 division 
Attempt-1: With wire 4 divisions 20 divisions 
Attempt-2: With wire 4 divisions 16 divisions 
  
 What are the diameter and cross-sectional area of the wire measured using the screw gauge? 
 (A) 2.22 ± 0.02 mm, p(1.23 ± 0.02) mm
2 
(B) 2.22 ± 0.01 mm, p(1.23 ± 0.01) mm
2
 
 (C) 2.14 ± 0.02 mm, p(1.14 ± 0.02) mm
2 
(D) 2.14 ± 0.01 mm, p(1.14 ± 0.01) mm
2
 
3. The smallest division on the main scale of a Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale 
correspond to nine divisions of the main scale. The figure below on the left shows the reading of this 
calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere 
held between the jaws. The correct diameter of the sphere is   
 
0 1 
10 0 
main scale 
Vernier scale 
                
 
3 4 
10 0 
main scale 
Vernier scale 
 
 (A) 3.07 cm (B) 3.11 cm (C) 3.15 cm (D) 3.17 cm 
4. Two capacitors with capacitance values C
1
 = 2000 ± 10 pF and C
2 
= 3000 ± 15 pF are connected in series. 
The voltage applied across this combination is V = 5.00 ± 0.02 V. The percentage error in the calculation 
of the energy stored in this combination of capacitors is _______.   
5. An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal 
length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm 
mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at  
135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens 
is________.     
Page 2


?
 
    
 
 
   
1  
ERRORS 
 
1. In an experiment for determination of the focal length of a thin convex lens, the distance of the object 
from the lens is 10 ± 0.1 cm and the distance of its real image from the lens is 20 ± 0.2 cm. The error in 
the determination of focal length of the lens is n %. The value of n is _______.          
2. Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is  
0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale 
shifts by two divisions. The measured readings are listed below.   
Measurement condition Main scale reading Circular scale reading 
Two arms of gauge touching 
each other without wire 
0 division 4 division 
Attempt-1: With wire 4 divisions 20 divisions 
Attempt-2: With wire 4 divisions 16 divisions 
  
 What are the diameter and cross-sectional area of the wire measured using the screw gauge? 
 (A) 2.22 ± 0.02 mm, p(1.23 ± 0.02) mm
2 
(B) 2.22 ± 0.01 mm, p(1.23 ± 0.01) mm
2
 
 (C) 2.14 ± 0.02 mm, p(1.14 ± 0.02) mm
2 
(D) 2.14 ± 0.01 mm, p(1.14 ± 0.01) mm
2
 
3. The smallest division on the main scale of a Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale 
correspond to nine divisions of the main scale. The figure below on the left shows the reading of this 
calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere 
held between the jaws. The correct diameter of the sphere is   
 
0 1 
10 0 
main scale 
Vernier scale 
                
 
3 4 
10 0 
main scale 
Vernier scale 
 
 (A) 3.07 cm (B) 3.11 cm (C) 3.15 cm (D) 3.17 cm 
4. Two capacitors with capacitance values C
1
 = 2000 ± 10 pF and C
2 
= 3000 ± 15 pF are connected in series. 
The voltage applied across this combination is V = 5.00 ± 0.02 V. The percentage error in the calculation 
of the energy stored in this combination of capacitors is _______.   
5. An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal 
length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm 
mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at  
135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens 
is________.     
 
  
    
 
   
 
2 
 
Paragraph for Q.6 & Q.7 
 If the measurement errors in all the independent quantities are known, then it is possible to determine the 
error in any dependent quantity. This is done by the use of series expansion and truncating the expansion 
at the first power of the error. For example, consider the relation z = x/y. If the errors in x, y and z are  
?x, ?y and ?z, respectively, then  
  
xx
z z
yy
±?
±? =
±?
 = 
1
xx y
11
yx y
-
?? ????
±±
??
??
??
??
. 
 The series expansion for 
1
y
1
y
-
? ??
±
??
??
, to first power in ?y/y, is ( ) 1 y /y ? ? . The relative errors in 
independent variables are always added. So the error in z will be 
xy
zz
xy
?? ??
?= +
??
??
. 
 The above derivation makes the assumption that 
xy
1, 1
xy
? ?
< < < < . Therefore, the higher powers of these 
quantities are neglected. 
6. Consider the ratio 
( )
( )
1a
r
1a
-
=
+
 to be determined by measuring a dimensionless quantity a. If the error in the 
measurement of a is ?a( ?a/a << 1), then what is the error ?r in determining r ?
 (A) 
( )
2
a
1a
?
+
 (B) 
( )
2
2a
1a
?
+
 (C) 
( )
2
2a
1a
?
-
 (D) 
( )
2
2a a
1a
?
-
 
7. In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40 nuclei decayed 
in the first 1.0 s. For 1 << x , In (1 + x) = x up to first power in x. The error ??, in the determination of 
the decay constant ?, in s
–1
 is :-    
 (A) 0.04 (B) 0.03 (C) 0.02 (D) 0.01 
8. A steel wire of diameter 0.5 mm and Young's modulus 2 × 10
11
 N m
–2
 carries a load of mass M. The 
length of the wire with the load is 1.0 m. A vernier scale with 10 divisions is attached to the end of this 
wire. Next to the steel wire is a reference wire to which a main scale, of least count  
1.0 mm, is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. 
Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is 
increased by 1.2 kg, the vernier scale division which coincides with a main scale division is_______. 
 (Take g = 10 ms
–2
 and p = 3.2)    
9. A person measures the depth of a well by measuring the time interval between dropping a stone and 
receiving the sound of impact with the bottom of the well. The error in his measurement of time is  
dT = 0.01 second and he measures the depth of the well to be L = 20 meters. Take the acceleration due to 
gravity g = 10 ms
–2 
and the velocity of sound is 300 ms
–1
. Then the fractional error in the measurement, 
dL/L, is closest to      
 (A) 0.2% (B) 5% (C) 3% (D) 1% 
Page 3


?
 
    
 
 
   
1  
ERRORS 
 
1. In an experiment for determination of the focal length of a thin convex lens, the distance of the object 
from the lens is 10 ± 0.1 cm and the distance of its real image from the lens is 20 ± 0.2 cm. The error in 
the determination of focal length of the lens is n %. The value of n is _______.          
2. Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is  
0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale 
shifts by two divisions. The measured readings are listed below.   
Measurement condition Main scale reading Circular scale reading 
Two arms of gauge touching 
each other without wire 
0 division 4 division 
Attempt-1: With wire 4 divisions 20 divisions 
Attempt-2: With wire 4 divisions 16 divisions 
  
 What are the diameter and cross-sectional area of the wire measured using the screw gauge? 
 (A) 2.22 ± 0.02 mm, p(1.23 ± 0.02) mm
2 
(B) 2.22 ± 0.01 mm, p(1.23 ± 0.01) mm
2
 
 (C) 2.14 ± 0.02 mm, p(1.14 ± 0.02) mm
2 
(D) 2.14 ± 0.01 mm, p(1.14 ± 0.01) mm
2
 
3. The smallest division on the main scale of a Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale 
correspond to nine divisions of the main scale. The figure below on the left shows the reading of this 
calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere 
held between the jaws. The correct diameter of the sphere is   
 
0 1 
10 0 
main scale 
Vernier scale 
                
 
3 4 
10 0 
main scale 
Vernier scale 
 
 (A) 3.07 cm (B) 3.11 cm (C) 3.15 cm (D) 3.17 cm 
4. Two capacitors with capacitance values C
1
 = 2000 ± 10 pF and C
2 
= 3000 ± 15 pF are connected in series. 
The voltage applied across this combination is V = 5.00 ± 0.02 V. The percentage error in the calculation 
of the energy stored in this combination of capacitors is _______.   
5. An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal 
length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm 
mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at  
135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens 
is________.     
 
  
    
 
   
 
2 
 
Paragraph for Q.6 & Q.7 
 If the measurement errors in all the independent quantities are known, then it is possible to determine the 
error in any dependent quantity. This is done by the use of series expansion and truncating the expansion 
at the first power of the error. For example, consider the relation z = x/y. If the errors in x, y and z are  
?x, ?y and ?z, respectively, then  
  
xx
z z
yy
±?
±? =
±?
 = 
1
xx y
11
yx y
-
?? ????
±±
??
??
??
??
. 
 The series expansion for 
1
y
1
y
-
? ??
±
??
??
, to first power in ?y/y, is ( ) 1 y /y ? ? . The relative errors in 
independent variables are always added. So the error in z will be 
xy
zz
xy
?? ??
?= +
??
??
. 
 The above derivation makes the assumption that 
xy
1, 1
xy
? ?
< < < < . Therefore, the higher powers of these 
quantities are neglected. 
6. Consider the ratio 
( )
( )
1a
r
1a
-
=
+
 to be determined by measuring a dimensionless quantity a. If the error in the 
measurement of a is ?a( ?a/a << 1), then what is the error ?r in determining r ?
 (A) 
( )
2
a
1a
?
+
 (B) 
( )
2
2a
1a
?
+
 (C) 
( )
2
2a
1a
?
-
 (D) 
( )
2
2a a
1a
?
-
 
7. In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40 nuclei decayed 
in the first 1.0 s. For 1 << x , In (1 + x) = x up to first power in x. The error ??, in the determination of 
the decay constant ?, in s
–1
 is :-    
 (A) 0.04 (B) 0.03 (C) 0.02 (D) 0.01 
8. A steel wire of diameter 0.5 mm and Young's modulus 2 × 10
11
 N m
–2
 carries a load of mass M. The 
length of the wire with the load is 1.0 m. A vernier scale with 10 divisions is attached to the end of this 
wire. Next to the steel wire is a reference wire to which a main scale, of least count  
1.0 mm, is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. 
Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is 
increased by 1.2 kg, the vernier scale division which coincides with a main scale division is_______. 
 (Take g = 10 ms
–2
 and p = 3.2)    
9. A person measures the depth of a well by measuring the time interval between dropping a stone and 
receiving the sound of impact with the bottom of the well. The error in his measurement of time is  
dT = 0.01 second and he measures the depth of the well to be L = 20 meters. Take the acceleration due to 
gravity g = 10 ms
–2 
and the velocity of sound is 300 ms
–1
. Then the fractional error in the measurement, 
dL/L, is closest to      
 (A) 0.2% (B) 5% (C) 3% (D) 1% 
 
    
 
 
   
3  
10. There are two vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. 
The Vernier scale of one of the calipers (C
1
) has 10 equal divisions that correspond to 9 main scale 
divisions. The Vernier scale of the other caliper (C
2
) has 10 equal divisions that correspond to 11 main 
scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by 
calipers C
1
 and C
2
 respectively, are    
 
C
1
2 3 4
0
5
10
C
2
2 3 4
0
5
10
C
1
 
C
2
 
 
 (A) 2.87 and 2.86 (B) 2.87 and 2.87 (C) 2.87 and 2.83 (D) 2.85 and 2.82 
11. In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a 
periodic motion is 
( ) 7R r
T2
5g
-
= p . The values of R and r are measured to be (60 ± 1) mm and (10 ± 1) mm, 
respectively. In five successive measurements, the time period is found to be 0.52 s, 0.56 s, 0.57s, 0.54s 
and 0.59s. The least count of the watch used for the measurement of time period is 0.01s. Which of the 
following statement(s) is(are) true ?    
 (A) The error in the measurement of r is 10% (B) The error in the measurement of T is 3.57 % 
 (C) The error in the measurement of T is 2% (D) The error in the determined value of g is 11% 
12. Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a 
screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier 
scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the 
circular scale moves it by two divisions on the linear scale. Then :   
 (A) If the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the 
screw gauge is 0.01 mm. 
 (B) If the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the 
screw gauge is 0.005 mm. 
 (C) If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, 
the least count of the screw gauge is 0.01 mm. 
 (D) If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, 
the least count of the screw gauge is 0.005 mm. 
Page 4


?
 
    
 
 
   
1  
ERRORS 
 
1. In an experiment for determination of the focal length of a thin convex lens, the distance of the object 
from the lens is 10 ± 0.1 cm and the distance of its real image from the lens is 20 ± 0.2 cm. The error in 
the determination of focal length of the lens is n %. The value of n is _______.          
2. Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is  
0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale 
shifts by two divisions. The measured readings are listed below.   
Measurement condition Main scale reading Circular scale reading 
Two arms of gauge touching 
each other without wire 
0 division 4 division 
Attempt-1: With wire 4 divisions 20 divisions 
Attempt-2: With wire 4 divisions 16 divisions 
  
 What are the diameter and cross-sectional area of the wire measured using the screw gauge? 
 (A) 2.22 ± 0.02 mm, p(1.23 ± 0.02) mm
2 
(B) 2.22 ± 0.01 mm, p(1.23 ± 0.01) mm
2
 
 (C) 2.14 ± 0.02 mm, p(1.14 ± 0.02) mm
2 
(D) 2.14 ± 0.01 mm, p(1.14 ± 0.01) mm
2
 
3. The smallest division on the main scale of a Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale 
correspond to nine divisions of the main scale. The figure below on the left shows the reading of this 
calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere 
held between the jaws. The correct diameter of the sphere is   
 
0 1 
10 0 
main scale 
Vernier scale 
                
 
3 4 
10 0 
main scale 
Vernier scale 
 
 (A) 3.07 cm (B) 3.11 cm (C) 3.15 cm (D) 3.17 cm 
4. Two capacitors with capacitance values C
1
 = 2000 ± 10 pF and C
2 
= 3000 ± 15 pF are connected in series. 
The voltage applied across this combination is V = 5.00 ± 0.02 V. The percentage error in the calculation 
of the energy stored in this combination of capacitors is _______.   
5. An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal 
length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm 
mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at  
135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens 
is________.     
 
  
    
 
   
 
2 
 
Paragraph for Q.6 & Q.7 
 If the measurement errors in all the independent quantities are known, then it is possible to determine the 
error in any dependent quantity. This is done by the use of series expansion and truncating the expansion 
at the first power of the error. For example, consider the relation z = x/y. If the errors in x, y and z are  
?x, ?y and ?z, respectively, then  
  
xx
z z
yy
±?
±? =
±?
 = 
1
xx y
11
yx y
-
?? ????
±±
??
??
??
??
. 
 The series expansion for 
1
y
1
y
-
? ??
±
??
??
, to first power in ?y/y, is ( ) 1 y /y ? ? . The relative errors in 
independent variables are always added. So the error in z will be 
xy
zz
xy
?? ??
?= +
??
??
. 
 The above derivation makes the assumption that 
xy
1, 1
xy
? ?
< < < < . Therefore, the higher powers of these 
quantities are neglected. 
6. Consider the ratio 
( )
( )
1a
r
1a
-
=
+
 to be determined by measuring a dimensionless quantity a. If the error in the 
measurement of a is ?a( ?a/a << 1), then what is the error ?r in determining r ?
 (A) 
( )
2
a
1a
?
+
 (B) 
( )
2
2a
1a
?
+
 (C) 
( )
2
2a
1a
?
-
 (D) 
( )
2
2a a
1a
?
-
 
7. In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40 nuclei decayed 
in the first 1.0 s. For 1 << x , In (1 + x) = x up to first power in x. The error ??, in the determination of 
the decay constant ?, in s
–1
 is :-    
 (A) 0.04 (B) 0.03 (C) 0.02 (D) 0.01 
8. A steel wire of diameter 0.5 mm and Young's modulus 2 × 10
11
 N m
–2
 carries a load of mass M. The 
length of the wire with the load is 1.0 m. A vernier scale with 10 divisions is attached to the end of this 
wire. Next to the steel wire is a reference wire to which a main scale, of least count  
1.0 mm, is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. 
Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is 
increased by 1.2 kg, the vernier scale division which coincides with a main scale division is_______. 
 (Take g = 10 ms
–2
 and p = 3.2)    
9. A person measures the depth of a well by measuring the time interval between dropping a stone and 
receiving the sound of impact with the bottom of the well. The error in his measurement of time is  
dT = 0.01 second and he measures the depth of the well to be L = 20 meters. Take the acceleration due to 
gravity g = 10 ms
–2 
and the velocity of sound is 300 ms
–1
. Then the fractional error in the measurement, 
dL/L, is closest to      
 (A) 0.2% (B) 5% (C) 3% (D) 1% 
 
    
 
 
   
3  
10. There are two vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. 
The Vernier scale of one of the calipers (C
1
) has 10 equal divisions that correspond to 9 main scale 
divisions. The Vernier scale of the other caliper (C
2
) has 10 equal divisions that correspond to 11 main 
scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by 
calipers C
1
 and C
2
 respectively, are    
 
C
1
2 3 4
0
5
10
C
2
2 3 4
0
5
10
C
1
 
C
2
 
 
 (A) 2.87 and 2.86 (B) 2.87 and 2.87 (C) 2.87 and 2.83 (D) 2.85 and 2.82 
11. In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a 
periodic motion is 
( ) 7R r
T2
5g
-
= p . The values of R and r are measured to be (60 ± 1) mm and (10 ± 1) mm, 
respectively. In five successive measurements, the time period is found to be 0.52 s, 0.56 s, 0.57s, 0.54s 
and 0.59s. The least count of the watch used for the measurement of time period is 0.01s. Which of the 
following statement(s) is(are) true ?    
 (A) The error in the measurement of r is 10% (B) The error in the measurement of T is 3.57 % 
 (C) The error in the measurement of T is 2% (D) The error in the determined value of g is 11% 
12. Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a 
screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier 
scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the 
circular scale moves it by two divisions on the linear scale. Then :   
 (A) If the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the 
screw gauge is 0.01 mm. 
 (B) If the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the 
screw gauge is 0.005 mm. 
 (C) If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, 
the least count of the screw gauge is 0.01 mm. 
 (D) If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, 
the least count of the screw gauge is 0.005 mm. 
 
  
    
 
   
 
4 
 
13. The energy of a system as a function of time t is given as E(t) = A
2
exp (–at), where a = 0.2s
–1
. The 
measurement of A has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage 
error in the value of E(t) at t = 5 s is.    
14. During Searle's experiment, zero of the Vernier scale lies between 3.20 × 10
–2
 m and 3.25 × 10
–2 
m of the 
main scale. The 20
th
 division of the Vernier scale exactly coincides with one of the main scale divisions. 
When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between  
3.20 × 10
–2
 m and 3.25 × 10
–2
 m of the main scale but now the 45
th
 division of Vernier scale coincides 
with one of the main scale divisions. The length of the thin metallic wire is 2m and its cross-sectional area 
is 8 × 10
–7
 m
2
. The least count of the Vernier scale is 1.0 × 10
–5
 m. The maximum percentage error in the 
Young's modulus of the wire is.    
Page 5


?
 
    
 
 
   
1  
ERRORS 
 
1. In an experiment for determination of the focal length of a thin convex lens, the distance of the object 
from the lens is 10 ± 0.1 cm and the distance of its real image from the lens is 20 ± 0.2 cm. The error in 
the determination of focal length of the lens is n %. The value of n is _______.          
2. Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is  
0.5 mm. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale 
shifts by two divisions. The measured readings are listed below.   
Measurement condition Main scale reading Circular scale reading 
Two arms of gauge touching 
each other without wire 
0 division 4 division 
Attempt-1: With wire 4 divisions 20 divisions 
Attempt-2: With wire 4 divisions 16 divisions 
  
 What are the diameter and cross-sectional area of the wire measured using the screw gauge? 
 (A) 2.22 ± 0.02 mm, p(1.23 ± 0.02) mm
2 
(B) 2.22 ± 0.01 mm, p(1.23 ± 0.01) mm
2
 
 (C) 2.14 ± 0.02 mm, p(1.14 ± 0.02) mm
2 
(D) 2.14 ± 0.01 mm, p(1.14 ± 0.01) mm
2
 
3. The smallest division on the main scale of a Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale 
correspond to nine divisions of the main scale. The figure below on the left shows the reading of this 
calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere 
held between the jaws. The correct diameter of the sphere is   
 
0 1 
10 0 
main scale 
Vernier scale 
                
 
3 4 
10 0 
main scale 
Vernier scale 
 
 (A) 3.07 cm (B) 3.11 cm (C) 3.15 cm (D) 3.17 cm 
4. Two capacitors with capacitance values C
1
 = 2000 ± 10 pF and C
2 
= 3000 ± 15 pF are connected in series. 
The voltage applied across this combination is V = 5.00 ± 0.02 V. The percentage error in the calculation 
of the energy stored in this combination of capacitors is _______.   
5. An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal 
length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm 
mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at  
135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens 
is________.     
 
  
    
 
   
 
2 
 
Paragraph for Q.6 & Q.7 
 If the measurement errors in all the independent quantities are known, then it is possible to determine the 
error in any dependent quantity. This is done by the use of series expansion and truncating the expansion 
at the first power of the error. For example, consider the relation z = x/y. If the errors in x, y and z are  
?x, ?y and ?z, respectively, then  
  
xx
z z
yy
±?
±? =
±?
 = 
1
xx y
11
yx y
-
?? ????
±±
??
??
??
??
. 
 The series expansion for 
1
y
1
y
-
? ??
±
??
??
, to first power in ?y/y, is ( ) 1 y /y ? ? . The relative errors in 
independent variables are always added. So the error in z will be 
xy
zz
xy
?? ??
?= +
??
??
. 
 The above derivation makes the assumption that 
xy
1, 1
xy
? ?
< < < < . Therefore, the higher powers of these 
quantities are neglected. 
6. Consider the ratio 
( )
( )
1a
r
1a
-
=
+
 to be determined by measuring a dimensionless quantity a. If the error in the 
measurement of a is ?a( ?a/a << 1), then what is the error ?r in determining r ?
 (A) 
( )
2
a
1a
?
+
 (B) 
( )
2
2a
1a
?
+
 (C) 
( )
2
2a
1a
?
-
 (D) 
( )
2
2a a
1a
?
-
 
7. In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40 nuclei decayed 
in the first 1.0 s. For 1 << x , In (1 + x) = x up to first power in x. The error ??, in the determination of 
the decay constant ?, in s
–1
 is :-    
 (A) 0.04 (B) 0.03 (C) 0.02 (D) 0.01 
8. A steel wire of diameter 0.5 mm and Young's modulus 2 × 10
11
 N m
–2
 carries a load of mass M. The 
length of the wire with the load is 1.0 m. A vernier scale with 10 divisions is attached to the end of this 
wire. Next to the steel wire is a reference wire to which a main scale, of least count  
1.0 mm, is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. 
Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is 
increased by 1.2 kg, the vernier scale division which coincides with a main scale division is_______. 
 (Take g = 10 ms
–2
 and p = 3.2)    
9. A person measures the depth of a well by measuring the time interval between dropping a stone and 
receiving the sound of impact with the bottom of the well. The error in his measurement of time is  
dT = 0.01 second and he measures the depth of the well to be L = 20 meters. Take the acceleration due to 
gravity g = 10 ms
–2 
and the velocity of sound is 300 ms
–1
. Then the fractional error in the measurement, 
dL/L, is closest to      
 (A) 0.2% (B) 5% (C) 3% (D) 1% 
 
    
 
 
   
3  
10. There are two vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. 
The Vernier scale of one of the calipers (C
1
) has 10 equal divisions that correspond to 9 main scale 
divisions. The Vernier scale of the other caliper (C
2
) has 10 equal divisions that correspond to 11 main 
scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by 
calipers C
1
 and C
2
 respectively, are    
 
C
1
2 3 4
0
5
10
C
2
2 3 4
0
5
10
C
1
 
C
2
 
 
 (A) 2.87 and 2.86 (B) 2.87 and 2.87 (C) 2.87 and 2.83 (D) 2.85 and 2.82 
11. In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a 
periodic motion is 
( ) 7R r
T2
5g
-
= p . The values of R and r are measured to be (60 ± 1) mm and (10 ± 1) mm, 
respectively. In five successive measurements, the time period is found to be 0.52 s, 0.56 s, 0.57s, 0.54s 
and 0.59s. The least count of the watch used for the measurement of time period is 0.01s. Which of the 
following statement(s) is(are) true ?    
 (A) The error in the measurement of r is 10% (B) The error in the measurement of T is 3.57 % 
 (C) The error in the measurement of T is 2% (D) The error in the determined value of g is 11% 
12. Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a 
screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier 
scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the 
circular scale moves it by two divisions on the linear scale. Then :   
 (A) If the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the 
screw gauge is 0.01 mm. 
 (B) If the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the 
screw gauge is 0.005 mm. 
 (C) If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, 
the least count of the screw gauge is 0.01 mm. 
 (D) If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, 
the least count of the screw gauge is 0.005 mm. 
 
  
    
 
   
 
4 
 
13. The energy of a system as a function of time t is given as E(t) = A
2
exp (–at), where a = 0.2s
–1
. The 
measurement of A has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage 
error in the value of E(t) at t = 5 s is.    
14. During Searle's experiment, zero of the Vernier scale lies between 3.20 × 10
–2
 m and 3.25 × 10
–2 
m of the 
main scale. The 20
th
 division of the Vernier scale exactly coincides with one of the main scale divisions. 
When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between  
3.20 × 10
–2
 m and 3.25 × 10
–2
 m of the main scale but now the 45
th
 division of Vernier scale coincides 
with one of the main scale divisions. The length of the thin metallic wire is 2m and its cross-sectional area 
is 8 × 10
–7
 m
2
. The least count of the Vernier scale is 1.0 × 10
–5
 m. The maximum percentage error in the 
Young's modulus of the wire is.    
 
    
 
 
   
5  
SOLUTIONS 
 
1. Ans. (1) 
Sol. u = 10 ± 0.1 cm, v = 20 ± 0.2 cm 
 
22 2
11 1 1 1 1
dv du df
vu f vu f
- = ? + =- 
 
1 1 1 1 3 20
f cm
20 10 f f 20 3
+ = ? = ?= 
 
( )
( )
( )
( )
22
1 1 9
0.2 0.1 df
400
20 10
? + = 
 
1 400 400
df 0.2 0.1
9 400 100
? ?
= ×+ ×
? ?
? ?
 
 ( )
1 0.6
df 0.2 0.4 df
99
= + ?= 
 
df 0.6 3 1
f 9 20 100
= ×= 
 % error = 1 % 
Alternate Solution  
 
1 11
VU f
-= ;  
1 1 1
20 10 f
++ = 
 
2 22
1 dU df
dv
V uf
- +=- 
12 1
20 f
+
= ; 
20
f
3
= 
 
0.1 0.2 f%
100 400 f
+= 
 
0.4 0.2 f
20 400
f
3
+?
=
??
??
??
 
 
0.6 20 f
400 3 f
×?
=
×
 
 
1 f
100 f
?
= 
 % change in f is 1% 
2. Ans.  (Dropped) 
Sol. 
0.1
LC 0.001mm
100
= = 
 Zero error = 4 × 0.001 = 0.004 mm 
Reading 1 = 0.5 × 4 + 20 × 0.001 – 0.004 = 2.16 mm 
Reading 2 = 0.5 × 4 + 16 × 0.001 – 0.004 = 2.12 mm 
 Mean value = 2.14 mm 
 Mean absolute error = 
0.02 0.02
0.02
2
+
= 
 Diameter = 2.14 ± 0.02 
 Area = 
2
d
4
p
 
3. Ans.  (C) 
Sol. Given 10 VSD = 9 MSD Here    
MSD ? Main Scale division  
    1 VSD = 
9
10
MSD    
VSD? Vernier Scale division  
 Least count  = 1 MSD – 1 VSD 
    = 
9
1
10
? ?
-
? ?
? ?
 MSD 
    = 0.1 MSD 
    = 0.1 × 0.1 cm    
    = 0.01 cm 
 As '0' of V.S. lie before '0' of M.S. 
 Zero error  = –[10 – 6]L.C.  
   = – 4 × 0.01 cm 
   = –0.04 cm   
 Reading  = 3.1 cm + 1 × LC 
   = 3.4 cm + 1 × 0.01 cm 
   = 3.11 cm 
 True diameter = Reading – Zero error 
   = 3.11 – (–0.04) cm  = 3.15 cm 
4. Ans. (1.30) 
Sol. 
2 12
12
CC 1
UV
2C C
=
+
 
 Let 
12
eq
12
CC
C
CC
=
+
 
 
eq eq 1 1 2 2
1 11
C C C CC C
= +
±? ±? ±?
 
 
12 1 2 2 1
eq eq
12 1 2
C CC CC C
CC
CC C C
+? + ?
? ±?
+ +? +?
? 
 =
12
1200 1
1200
25
1
5000
? ?
±
? ?
? ?
??
±
??
??
 
 = 
11
1200 1
100 200
? ? ??
±-
?? ? ?
?? ? ?
  
 
eq
eq
C
U 2V
100 100 100
UC V
?
? ?
× = ×+ × 
 = 
1 0.02
100 2 100
200 5
× + × × 
 = 1.3 %