Superposition of Waves JEE Advance - Physics, Solution by DC Pandey Notes | Study DC Pandey Solutions for JEE Physics - JEE

 Page 1


For JEE Advanced 
Objective Questions 
  Single Correct Option 
Q 1.  When tension of a string is increased by 2.5 N, the initial frequency is altered in the ratio of 3 : 2. 
The initial tension in the string is 
  (a) 6 N   (b) 5 N   (c) 4 N   (d) 2 N 
Q 2.  The lengths of two wires of same material are in the ratio 1 : 2, their tensions are in the ratio 1 :2 
and their diameters are in the ratio 1:3. The ratio of the notes they emit when sounded together by 
the same source is 
 (a) 2   (b) 3   (c) 23  (d) 32     
Q 3.  If f 1, f 2 and f 3 are the fundamental frequencies of three segments into which a string is divided, 
then the original fundamental frequency f 0 of the whole string is 
  (a) f 0 = f 1 + f 2 + f 3  (b) 
0 1 2 3
1 1 1 1
f f f f
? ? ?
 
(c) 
0 1 2 3
1 1 1 1
f f f f
? ? ? (d) None of these 
Q 4.  Two identical harmonic pulses travelling in opposite directions in a taut string approach each 
other. At the instant when they completely overlap, the total energy of the string will be 
 
  (a) zero      (b) partly kinetic and partly potential 
  (c) purely kinetic     (d) purely potential 
Q 5.  Two transverse waves A and B superimpose to produce a node at x = 0. If the equation of wave A 
is y = a cos (kx + ?t), then the equation of wave B is 
  (a) +a cos (kx - ?t)  (b) -acos (kx + ?t)  (c) -acos (kx - ?t)  (d) +a cos ( ?t - kx) 
Q 6.  In a standing wave, node is a point of 
  (a) maximum strain  (b) maximum pressure (c) maximum density  (d) All of these 
Q 7.  In a stationary wave system, all the particles of the medium  
  (a) have zero displacement simultaneously at some instant 
  (b) have maximum displacement simultaneously at some instant 
  (c) are at rest simultaneously at some instant 
  (d) All of the above 
Q 8.  Three one dimensional mechanical waves in an elastic medium is given as 
   y 1 = 3A sin ( ?t - kx), y 2=A sin( ?t – kx + ?) and y 3 = 2A sin ( ?t + kx) 
are superimposed with each other. The maximum displacement amplitude of the medium particle 
would be  
(a) 44    (b) 3A    (c) 2A    (d) A 
Page 2


For JEE Advanced 
Objective Questions 
  Single Correct Option 
Q 1.  When tension of a string is increased by 2.5 N, the initial frequency is altered in the ratio of 3 : 2. 
The initial tension in the string is 
  (a) 6 N   (b) 5 N   (c) 4 N   (d) 2 N 
Q 2.  The lengths of two wires of same material are in the ratio 1 : 2, their tensions are in the ratio 1 :2 
and their diameters are in the ratio 1:3. The ratio of the notes they emit when sounded together by 
the same source is 
 (a) 2   (b) 3   (c) 23  (d) 32     
Q 3.  If f 1, f 2 and f 3 are the fundamental frequencies of three segments into which a string is divided, 
then the original fundamental frequency f 0 of the whole string is 
  (a) f 0 = f 1 + f 2 + f 3  (b) 
0 1 2 3
1 1 1 1
f f f f
? ? ?
 
(c) 
0 1 2 3
1 1 1 1
f f f f
? ? ? (d) None of these 
Q 4.  Two identical harmonic pulses travelling in opposite directions in a taut string approach each 
other. At the instant when they completely overlap, the total energy of the string will be 
 
  (a) zero      (b) partly kinetic and partly potential 
  (c) purely kinetic     (d) purely potential 
Q 5.  Two transverse waves A and B superimpose to produce a node at x = 0. If the equation of wave A 
is y = a cos (kx + ?t), then the equation of wave B is 
  (a) +a cos (kx - ?t)  (b) -acos (kx + ?t)  (c) -acos (kx - ?t)  (d) +a cos ( ?t - kx) 
Q 6.  In a standing wave, node is a point of 
  (a) maximum strain  (b) maximum pressure (c) maximum density  (d) All of these 
Q 7.  In a stationary wave system, all the particles of the medium  
  (a) have zero displacement simultaneously at some instant 
  (b) have maximum displacement simultaneously at some instant 
  (c) are at rest simultaneously at some instant 
  (d) All of the above 
Q 8.  Three one dimensional mechanical waves in an elastic medium is given as 
   y 1 = 3A sin ( ?t - kx), y 2=A sin( ?t – kx + ?) and y 3 = 2A sin ( ?t + kx) 
are superimposed with each other. The maximum displacement amplitude of the medium particle 
would be  
(a) 44    (b) 3A    (c) 2A    (d) A 
Q 9.  A string is stretched so that its length is increased by 
1
?
of its original length. The ratio of 
fundamental frequency of transverse vibration to that of fundamental frequency of longitudinal 
vibration will be 
 (a) ? : 1  (b) 1 : ?  (c) ? : 1  (d) 1 : ? 
Q 10.  A string of length 1 m and linear mass density 0.01 kg/m is stretched to a tension of 100 N. When 
both ends of the string are fixed, the three lowest frequencies for standing wave are f1, f 2 and f 3. 
When only one end of the string is fixed, the three lowest frequencies for standing wave are n 1, n 2 
and n3. Then  
  (a) n3 = 5n1 = f 3 = 125 Hz    (b) f 3 = 5f 1 =n2= 125 Hz 
  (c) f 3 =n2 =3f 1 =150Hz    (d) 
12
2
ff
n
2
?
? = 75 Hz 
Q 11.  In a standing wave on a string 
  (a) In one time period all the particles are simultaneously at rest twice 
  (b) All the particles must be at their positive extremes simultaneously once in a time period 
  (c) All the particles may be at their positive extremes simultaneously once in a time period 
  (d) All the particles are never at rest simultaneously 
Q 12.  Three resonant frequencies of string with both rigid ends are 90, 150 and 210 Hz. If the length of 
the string is 80 cm, what is the speed of the transverse wave in the string? 
  (a) 45 m/s   (b) 75 m/s   (c) 48 m/s   (d) 80 m/s  
Q 13.  Three coherent waves having amplitudes 12 mm, 6 mm and 4 mm arrive at a given point with 
successive phase difference of ?/2. Then the amplitude of the resultant wave is 
  (a) 7 mm   (b) 10 mm   (c) 5 mm   (d) 4.8 mm 
Q 14.  For a certain stretched string, three consecutive resonance frequencies are observed as 105, 175 
and 245 Hz respectively. Then the fundamental frequency is 
  (a) 30 Hz   (b) 45 Hz   (c) 35 Hz   (d) None of these 
Q 15.  A sonometer wire has a length 114 cm between two fixed ends. Where should two bridges be 
placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio 
1:3:4 
  (a) l 1 = 72cm, l 2 = 24cm, l 3 = 18cm   (b) l 1 = 60cm, l 2 = 40cm, l 3 = 14 cm 
  (c) l 1 = 52cm, l 2 = 30cm, l 3 = 32cm   (d) l 1 = 65cm, l 2 = 30cm, l 3 = 19cm 
Q 16.  The frequency of a sonometer wire is f. The frequency becomes f/2 when the mass producing the 
tension is completely immersed in water and on immersing the mass in a certain liquid, frequency 
becomes f/3. The relative density of the liquid is 
  (a) 1.32   (b) 1.67   (c) 1.41   (d) 1.18 
Q 17.  A string of length 1.5 m with its two ends clamped is vibrating in fundamental mode. Amplitude at 
the centre of the string is 4 mm. Distance between the two points having amplitude 2 mm is 
  (a) 1 m   (b) 75 cm   (c) 60 cm   (d) 50 cm 
Page 3


For JEE Advanced 
Objective Questions 
  Single Correct Option 
Q 1.  When tension of a string is increased by 2.5 N, the initial frequency is altered in the ratio of 3 : 2. 
The initial tension in the string is 
  (a) 6 N   (b) 5 N   (c) 4 N   (d) 2 N 
Q 2.  The lengths of two wires of same material are in the ratio 1 : 2, their tensions are in the ratio 1 :2 
and their diameters are in the ratio 1:3. The ratio of the notes they emit when sounded together by 
the same source is 
 (a) 2   (b) 3   (c) 23  (d) 32     
Q 3.  If f 1, f 2 and f 3 are the fundamental frequencies of three segments into which a string is divided, 
then the original fundamental frequency f 0 of the whole string is 
  (a) f 0 = f 1 + f 2 + f 3  (b) 
0 1 2 3
1 1 1 1
f f f f
? ? ?
 
(c) 
0 1 2 3
1 1 1 1
f f f f
? ? ? (d) None of these 
Q 4.  Two identical harmonic pulses travelling in opposite directions in a taut string approach each 
other. At the instant when they completely overlap, the total energy of the string will be 
 
  (a) zero      (b) partly kinetic and partly potential 
  (c) purely kinetic     (d) purely potential 
Q 5.  Two transverse waves A and B superimpose to produce a node at x = 0. If the equation of wave A 
is y = a cos (kx + ?t), then the equation of wave B is 
  (a) +a cos (kx - ?t)  (b) -acos (kx + ?t)  (c) -acos (kx - ?t)  (d) +a cos ( ?t - kx) 
Q 6.  In a standing wave, node is a point of 
  (a) maximum strain  (b) maximum pressure (c) maximum density  (d) All of these 
Q 7.  In a stationary wave system, all the particles of the medium  
  (a) have zero displacement simultaneously at some instant 
  (b) have maximum displacement simultaneously at some instant 
  (c) are at rest simultaneously at some instant 
  (d) All of the above 
Q 8.  Three one dimensional mechanical waves in an elastic medium is given as 
   y 1 = 3A sin ( ?t - kx), y 2=A sin( ?t – kx + ?) and y 3 = 2A sin ( ?t + kx) 
are superimposed with each other. The maximum displacement amplitude of the medium particle 
would be  
(a) 44    (b) 3A    (c) 2A    (d) A 
Q 9.  A string is stretched so that its length is increased by 
1
?
of its original length. The ratio of 
fundamental frequency of transverse vibration to that of fundamental frequency of longitudinal 
vibration will be 
 (a) ? : 1  (b) 1 : ?  (c) ? : 1  (d) 1 : ? 
Q 10.  A string of length 1 m and linear mass density 0.01 kg/m is stretched to a tension of 100 N. When 
both ends of the string are fixed, the three lowest frequencies for standing wave are f1, f 2 and f 3. 
When only one end of the string is fixed, the three lowest frequencies for standing wave are n 1, n 2 
and n3. Then  
  (a) n3 = 5n1 = f 3 = 125 Hz    (b) f 3 = 5f 1 =n2= 125 Hz 
  (c) f 3 =n2 =3f 1 =150Hz    (d) 
12
2
ff
n
2
?
? = 75 Hz 
Q 11.  In a standing wave on a string 
  (a) In one time period all the particles are simultaneously at rest twice 
  (b) All the particles must be at their positive extremes simultaneously once in a time period 
  (c) All the particles may be at their positive extremes simultaneously once in a time period 
  (d) All the particles are never at rest simultaneously 
Q 12.  Three resonant frequencies of string with both rigid ends are 90, 150 and 210 Hz. If the length of 
the string is 80 cm, what is the speed of the transverse wave in the string? 
  (a) 45 m/s   (b) 75 m/s   (c) 48 m/s   (d) 80 m/s  
Q 13.  Three coherent waves having amplitudes 12 mm, 6 mm and 4 mm arrive at a given point with 
successive phase difference of ?/2. Then the amplitude of the resultant wave is 
  (a) 7 mm   (b) 10 mm   (c) 5 mm   (d) 4.8 mm 
Q 14.  For a certain stretched string, three consecutive resonance frequencies are observed as 105, 175 
and 245 Hz respectively. Then the fundamental frequency is 
  (a) 30 Hz   (b) 45 Hz   (c) 35 Hz   (d) None of these 
Q 15.  A sonometer wire has a length 114 cm between two fixed ends. Where should two bridges be 
placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio 
1:3:4 
  (a) l 1 = 72cm, l 2 = 24cm, l 3 = 18cm   (b) l 1 = 60cm, l 2 = 40cm, l 3 = 14 cm 
  (c) l 1 = 52cm, l 2 = 30cm, l 3 = 32cm   (d) l 1 = 65cm, l 2 = 30cm, l 3 = 19cm 
Q 16.  The frequency of a sonometer wire is f. The frequency becomes f/2 when the mass producing the 
tension is completely immersed in water and on immersing the mass in a certain liquid, frequency 
becomes f/3. The relative density of the liquid is 
  (a) 1.32   (b) 1.67   (c) 1.41   (d) 1.18 
Q 17.  A string of length 1.5 m with its two ends clamped is vibrating in fundamental mode. Amplitude at 
the centre of the string is 4 mm. Distance between the two points having amplitude 2 mm is 
  (a) 1 m   (b) 75 cm   (c) 60 cm   (d) 50 cm 
Q 18.  A man generates a symmetrical pulse in a string by moving his hand up and down. At t = 0, the 
point in his hand moves downwards from mean position. The pulse travels with speed 3 m/s on the 
string and his hand passes 6 times in each second from the mean position. Then the point on the 
string at a distance 3 m will reach its upper extreme first time at t =  
  (a) 1.25 s   (b) 1s    (c) 11/12 s   (d) 
23
s
24
 
Q 19.  Among two interfering sources, let S1 be ahead of the phase by 90° relative to S2 . If an 
observation point P is such that PS1 - PS2 = 1.5 ?, the phase difference between the waves from S1 
and S2 reaching P is 
 (a) 3 ?   (b) 
5
2
?
   (c) 
7
2
?
   (d) 4 ? 
Q 20.  A wire having a linear density 0.1 kg/m is kept under a tension of 490 N. It is observed that it  
resonates at a frequency of 400 Hz. The next higher frequency is 450 Hz. Find the length of the 
wire 
  (a) 0.4 m   (b) 0.7 m   (c) 0.6 m   (d) 0.49 m  
Q 21.  A string 1 m long is drawn by a 300 Hz vibrator attached to its end. The string vibrates in three 
segments. The speed of transverse waves in the string is equal to 
  (a) 100 m/s   (b) 200 m/s   (c) 300 m/s   (d) 400 m/s 
Q 22.  If ?1, ?2, ?3
 
are the wavelength of the waves giving resonance to the fundamental, first and second 
overtone modes respectively in a string fixed at both ends. The ratio of the wavelengths ?1 : ?2 : ?3
 
is  
  (a) 1:2:3   (b) 1 :3:5   (c) 
11
1: :
23
   (d) 
11
1: :
35
 
Q 23.  The period of oscillations of a point is 0.04 s and the velocity of propagation of oscillation is 300 
m/s. The difference of phases between the oscillations of two points at distance 10 m and 16 m 
respectively from the source of oscillations is 
 (a) 2 ?   (b) 
2
?
    (c) 
4
?
    (d) ? 
Q 24.  A transverse wave described by an equation y = 0.02 sin (x + 30t) where x and t are in metre and 
second, is traveling along a wire of area of cross-section 1 mm
2
 and density 8000 kgm
-3
. What is 
the tension in the string? 
  (a) 20 N   (b) 7.2 N   (c) 30 N   (d) 14.4 N 
Q 25.  A string vibrates in 5 segments to a frequency of 480 Hz. The frequency that will cause it to 
vibrate in 2 segments will be 
  (a) 96 Hz   (b) 192 Hz   (c) 1200 Hz   (d) 2400 Hz 
Q 26.  A wave travels on a light string. The equation of the waves is Y = A sin (kx - ?t + 30° ) It is 
reflected from a heavy string tied to an end of the light string at x = 0. If 64% of the incident 
energy is reflected then the equation of the reflected wave is 
Page 4


For JEE Advanced 
Objective Questions 
  Single Correct Option 
Q 1.  When tension of a string is increased by 2.5 N, the initial frequency is altered in the ratio of 3 : 2. 
The initial tension in the string is 
  (a) 6 N   (b) 5 N   (c) 4 N   (d) 2 N 
Q 2.  The lengths of two wires of same material are in the ratio 1 : 2, their tensions are in the ratio 1 :2 
and their diameters are in the ratio 1:3. The ratio of the notes they emit when sounded together by 
the same source is 
 (a) 2   (b) 3   (c) 23  (d) 32     
Q 3.  If f 1, f 2 and f 3 are the fundamental frequencies of three segments into which a string is divided, 
then the original fundamental frequency f 0 of the whole string is 
  (a) f 0 = f 1 + f 2 + f 3  (b) 
0 1 2 3
1 1 1 1
f f f f
? ? ?
 
(c) 
0 1 2 3
1 1 1 1
f f f f
? ? ? (d) None of these 
Q 4.  Two identical harmonic pulses travelling in opposite directions in a taut string approach each 
other. At the instant when they completely overlap, the total energy of the string will be 
 
  (a) zero      (b) partly kinetic and partly potential 
  (c) purely kinetic     (d) purely potential 
Q 5.  Two transverse waves A and B superimpose to produce a node at x = 0. If the equation of wave A 
is y = a cos (kx + ?t), then the equation of wave B is 
  (a) +a cos (kx - ?t)  (b) -acos (kx + ?t)  (c) -acos (kx - ?t)  (d) +a cos ( ?t - kx) 
Q 6.  In a standing wave, node is a point of 
  (a) maximum strain  (b) maximum pressure (c) maximum density  (d) All of these 
Q 7.  In a stationary wave system, all the particles of the medium  
  (a) have zero displacement simultaneously at some instant 
  (b) have maximum displacement simultaneously at some instant 
  (c) are at rest simultaneously at some instant 
  (d) All of the above 
Q 8.  Three one dimensional mechanical waves in an elastic medium is given as 
   y 1 = 3A sin ( ?t - kx), y 2=A sin( ?t – kx + ?) and y 3 = 2A sin ( ?t + kx) 
are superimposed with each other. The maximum displacement amplitude of the medium particle 
would be  
(a) 44    (b) 3A    (c) 2A    (d) A 
Q 9.  A string is stretched so that its length is increased by 
1
?
of its original length. The ratio of 
fundamental frequency of transverse vibration to that of fundamental frequency of longitudinal 
vibration will be 
 (a) ? : 1  (b) 1 : ?  (c) ? : 1  (d) 1 : ? 
Q 10.  A string of length 1 m and linear mass density 0.01 kg/m is stretched to a tension of 100 N. When 
both ends of the string are fixed, the three lowest frequencies for standing wave are f1, f 2 and f 3. 
When only one end of the string is fixed, the three lowest frequencies for standing wave are n 1, n 2 
and n3. Then  
  (a) n3 = 5n1 = f 3 = 125 Hz    (b) f 3 = 5f 1 =n2= 125 Hz 
  (c) f 3 =n2 =3f 1 =150Hz    (d) 
12
2
ff
n
2
?
? = 75 Hz 
Q 11.  In a standing wave on a string 
  (a) In one time period all the particles are simultaneously at rest twice 
  (b) All the particles must be at their positive extremes simultaneously once in a time period 
  (c) All the particles may be at their positive extremes simultaneously once in a time period 
  (d) All the particles are never at rest simultaneously 
Q 12.  Three resonant frequencies of string with both rigid ends are 90, 150 and 210 Hz. If the length of 
the string is 80 cm, what is the speed of the transverse wave in the string? 
  (a) 45 m/s   (b) 75 m/s   (c) 48 m/s   (d) 80 m/s  
Q 13.  Three coherent waves having amplitudes 12 mm, 6 mm and 4 mm arrive at a given point with 
successive phase difference of ?/2. Then the amplitude of the resultant wave is 
  (a) 7 mm   (b) 10 mm   (c) 5 mm   (d) 4.8 mm 
Q 14.  For a certain stretched string, three consecutive resonance frequencies are observed as 105, 175 
and 245 Hz respectively. Then the fundamental frequency is 
  (a) 30 Hz   (b) 45 Hz   (c) 35 Hz   (d) None of these 
Q 15.  A sonometer wire has a length 114 cm between two fixed ends. Where should two bridges be 
placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio 
1:3:4 
  (a) l 1 = 72cm, l 2 = 24cm, l 3 = 18cm   (b) l 1 = 60cm, l 2 = 40cm, l 3 = 14 cm 
  (c) l 1 = 52cm, l 2 = 30cm, l 3 = 32cm   (d) l 1 = 65cm, l 2 = 30cm, l 3 = 19cm 
Q 16.  The frequency of a sonometer wire is f. The frequency becomes f/2 when the mass producing the 
tension is completely immersed in water and on immersing the mass in a certain liquid, frequency 
becomes f/3. The relative density of the liquid is 
  (a) 1.32   (b) 1.67   (c) 1.41   (d) 1.18 
Q 17.  A string of length 1.5 m with its two ends clamped is vibrating in fundamental mode. Amplitude at 
the centre of the string is 4 mm. Distance between the two points having amplitude 2 mm is 
  (a) 1 m   (b) 75 cm   (c) 60 cm   (d) 50 cm 
Q 18.  A man generates a symmetrical pulse in a string by moving his hand up and down. At t = 0, the 
point in his hand moves downwards from mean position. The pulse travels with speed 3 m/s on the 
string and his hand passes 6 times in each second from the mean position. Then the point on the 
string at a distance 3 m will reach its upper extreme first time at t =  
  (a) 1.25 s   (b) 1s    (c) 11/12 s   (d) 
23
s
24
 
Q 19.  Among two interfering sources, let S1 be ahead of the phase by 90° relative to S2 . If an 
observation point P is such that PS1 - PS2 = 1.5 ?, the phase difference between the waves from S1 
and S2 reaching P is 
 (a) 3 ?   (b) 
5
2
?
   (c) 
7
2
?
   (d) 4 ? 
Q 20.  A wire having a linear density 0.1 kg/m is kept under a tension of 490 N. It is observed that it  
resonates at a frequency of 400 Hz. The next higher frequency is 450 Hz. Find the length of the 
wire 
  (a) 0.4 m   (b) 0.7 m   (c) 0.6 m   (d) 0.49 m  
Q 21.  A string 1 m long is drawn by a 300 Hz vibrator attached to its end. The string vibrates in three 
segments. The speed of transverse waves in the string is equal to 
  (a) 100 m/s   (b) 200 m/s   (c) 300 m/s   (d) 400 m/s 
Q 22.  If ?1, ?2, ?3
 
are the wavelength of the waves giving resonance to the fundamental, first and second 
overtone modes respectively in a string fixed at both ends. The ratio of the wavelengths ?1 : ?2 : ?3
 
is  
  (a) 1:2:3   (b) 1 :3:5   (c) 
11
1: :
23
   (d) 
11
1: :
35
 
Q 23.  The period of oscillations of a point is 0.04 s and the velocity of propagation of oscillation is 300 
m/s. The difference of phases between the oscillations of two points at distance 10 m and 16 m 
respectively from the source of oscillations is 
 (a) 2 ?   (b) 
2
?
    (c) 
4
?
    (d) ? 
Q 24.  A transverse wave described by an equation y = 0.02 sin (x + 30t) where x and t are in metre and 
second, is traveling along a wire of area of cross-section 1 mm
2
 and density 8000 kgm
-3
. What is 
the tension in the string? 
  (a) 20 N   (b) 7.2 N   (c) 30 N   (d) 14.4 N 
Q 25.  A string vibrates in 5 segments to a frequency of 480 Hz. The frequency that will cause it to 
vibrate in 2 segments will be 
  (a) 96 Hz   (b) 192 Hz   (c) 1200 Hz   (d) 2400 Hz 
Q 26.  A wave travels on a light string. The equation of the waves is Y = A sin (kx - ?t + 30° ) It is 
reflected from a heavy string tied to an end of the light string at x = 0. If 64% of the incident 
energy is reflected then the equation of the reflected wave is 
  (a) Y = 0.8 A sin (kx - ?t + 30°+ 180°)  (b) Y = 0.8 A sin (kx + ?t + 30° + 180°) 
  (c) Y = 0.8 A sin (kx - ?T -30°)   (d) Y = 0.8 A sin (kx - ?t +30°) 
Q 27.  The tension, length, diameter and density of a string B are double than that of another string A. 
Which of the following overtones of B is same as the fundamental frequency of A ? 
  (a) 1st    (b) 2nd   (c) 3rd    (d) 4th 
  Passage : (Q. No. 28 to 30) 
Incident wave y = A sin ax bt
2
? ??
??
??
??
is reflected by an obstacle at x=0 which reduces intensity of 
reflected wave by 36%. Due to superposition a resulting wave consist of standing wave and 
travelling wave given by 
   y = - 1.6 sin ax sin bt + cA cos (bt + ax) where A, a, b and c are positive constants. 
Q 28.  Amplitude of reflected wave is 
  (a) 0.6, 4   (b) 0.8, 4   (c) 0.4, 4   (d) 0.2, 4 
Q 29.  Value of c is 
  (a) 0.2    (b) 0.4    (c) 0.6    (d) 0.3 
Q 30.  Position of second antinode is 
 (a) x
3a
?
?   (b) 
3
x
a
?
?   (c) 
3
x
2a
?
?   (d) 
2
x
3a
?
? 
  More than One Correct Options 
Q 31.  If the tension in a stretched string fixed at both ends is increased by 21% the fundamental 
frequency is found to change by 15 Hz. Then the  
  (a) original frequency is 150 Hz   
  (b) velocity of propagation of the transverse wave along the string increases by 5% 
  (c) velocity of propagation of the transverse wave along the string increases by 10% 
  (d) fundamental wavelength on the string does not change 
Q 32.  For interference to take place 
  (a) sources must be coherent    (b) sources must have same amplitude 
  (c) waves should travel in opposite directions (d) sources must have same frequency 
Q 33.  Regarding stationary waves choose the correct options. 
  (a) This is an example of interference  (b) Amplitudes of waves may be different 
  (c) Particles at nodes are always at rest  (d) Energy is conserved 
Q 34.  When a wave travels from a denser to rarer medium 
  (a) speed of wave increases    (b) wavelength of wave decreases 
  (c) amplitude of wave increases   (d) there is no change in phase angle 
Q 35.  A wire is stretched and fixed at two ends. Transverse stationary waves are formed in it. It 
oscillates in its third overtone mode. The equation of stationary wave is 
   Y = A sin kx cos ?t Choose the correct options. 
Page 5


For JEE Advanced 
Objective Questions 
  Single Correct Option 
Q 1.  When tension of a string is increased by 2.5 N, the initial frequency is altered in the ratio of 3 : 2. 
The initial tension in the string is 
  (a) 6 N   (b) 5 N   (c) 4 N   (d) 2 N 
Q 2.  The lengths of two wires of same material are in the ratio 1 : 2, their tensions are in the ratio 1 :2 
and their diameters are in the ratio 1:3. The ratio of the notes they emit when sounded together by 
the same source is 
 (a) 2   (b) 3   (c) 23  (d) 32     
Q 3.  If f 1, f 2 and f 3 are the fundamental frequencies of three segments into which a string is divided, 
then the original fundamental frequency f 0 of the whole string is 
  (a) f 0 = f 1 + f 2 + f 3  (b) 
0 1 2 3
1 1 1 1
f f f f
? ? ?
 
(c) 
0 1 2 3
1 1 1 1
f f f f
? ? ? (d) None of these 
Q 4.  Two identical harmonic pulses travelling in opposite directions in a taut string approach each 
other. At the instant when they completely overlap, the total energy of the string will be 
 
  (a) zero      (b) partly kinetic and partly potential 
  (c) purely kinetic     (d) purely potential 
Q 5.  Two transverse waves A and B superimpose to produce a node at x = 0. If the equation of wave A 
is y = a cos (kx + ?t), then the equation of wave B is 
  (a) +a cos (kx - ?t)  (b) -acos (kx + ?t)  (c) -acos (kx - ?t)  (d) +a cos ( ?t - kx) 
Q 6.  In a standing wave, node is a point of 
  (a) maximum strain  (b) maximum pressure (c) maximum density  (d) All of these 
Q 7.  In a stationary wave system, all the particles of the medium  
  (a) have zero displacement simultaneously at some instant 
  (b) have maximum displacement simultaneously at some instant 
  (c) are at rest simultaneously at some instant 
  (d) All of the above 
Q 8.  Three one dimensional mechanical waves in an elastic medium is given as 
   y 1 = 3A sin ( ?t - kx), y 2=A sin( ?t – kx + ?) and y 3 = 2A sin ( ?t + kx) 
are superimposed with each other. The maximum displacement amplitude of the medium particle 
would be  
(a) 44    (b) 3A    (c) 2A    (d) A 
Q 9.  A string is stretched so that its length is increased by 
1
?
of its original length. The ratio of 
fundamental frequency of transverse vibration to that of fundamental frequency of longitudinal 
vibration will be 
 (a) ? : 1  (b) 1 : ?  (c) ? : 1  (d) 1 : ? 
Q 10.  A string of length 1 m and linear mass density 0.01 kg/m is stretched to a tension of 100 N. When 
both ends of the string are fixed, the three lowest frequencies for standing wave are f1, f 2 and f 3. 
When only one end of the string is fixed, the three lowest frequencies for standing wave are n 1, n 2 
and n3. Then  
  (a) n3 = 5n1 = f 3 = 125 Hz    (b) f 3 = 5f 1 =n2= 125 Hz 
  (c) f 3 =n2 =3f 1 =150Hz    (d) 
12
2
ff
n
2
?
? = 75 Hz 
Q 11.  In a standing wave on a string 
  (a) In one time period all the particles are simultaneously at rest twice 
  (b) All the particles must be at their positive extremes simultaneously once in a time period 
  (c) All the particles may be at their positive extremes simultaneously once in a time period 
  (d) All the particles are never at rest simultaneously 
Q 12.  Three resonant frequencies of string with both rigid ends are 90, 150 and 210 Hz. If the length of 
the string is 80 cm, what is the speed of the transverse wave in the string? 
  (a) 45 m/s   (b) 75 m/s   (c) 48 m/s   (d) 80 m/s  
Q 13.  Three coherent waves having amplitudes 12 mm, 6 mm and 4 mm arrive at a given point with 
successive phase difference of ?/2. Then the amplitude of the resultant wave is 
  (a) 7 mm   (b) 10 mm   (c) 5 mm   (d) 4.8 mm 
Q 14.  For a certain stretched string, three consecutive resonance frequencies are observed as 105, 175 
and 245 Hz respectively. Then the fundamental frequency is 
  (a) 30 Hz   (b) 45 Hz   (c) 35 Hz   (d) None of these 
Q 15.  A sonometer wire has a length 114 cm between two fixed ends. Where should two bridges be 
placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio 
1:3:4 
  (a) l 1 = 72cm, l 2 = 24cm, l 3 = 18cm   (b) l 1 = 60cm, l 2 = 40cm, l 3 = 14 cm 
  (c) l 1 = 52cm, l 2 = 30cm, l 3 = 32cm   (d) l 1 = 65cm, l 2 = 30cm, l 3 = 19cm 
Q 16.  The frequency of a sonometer wire is f. The frequency becomes f/2 when the mass producing the 
tension is completely immersed in water and on immersing the mass in a certain liquid, frequency 
becomes f/3. The relative density of the liquid is 
  (a) 1.32   (b) 1.67   (c) 1.41   (d) 1.18 
Q 17.  A string of length 1.5 m with its two ends clamped is vibrating in fundamental mode. Amplitude at 
the centre of the string is 4 mm. Distance between the two points having amplitude 2 mm is 
  (a) 1 m   (b) 75 cm   (c) 60 cm   (d) 50 cm 
Q 18.  A man generates a symmetrical pulse in a string by moving his hand up and down. At t = 0, the 
point in his hand moves downwards from mean position. The pulse travels with speed 3 m/s on the 
string and his hand passes 6 times in each second from the mean position. Then the point on the 
string at a distance 3 m will reach its upper extreme first time at t =  
  (a) 1.25 s   (b) 1s    (c) 11/12 s   (d) 
23
s
24
 
Q 19.  Among two interfering sources, let S1 be ahead of the phase by 90° relative to S2 . If an 
observation point P is such that PS1 - PS2 = 1.5 ?, the phase difference between the waves from S1 
and S2 reaching P is 
 (a) 3 ?   (b) 
5
2
?
   (c) 
7
2
?
   (d) 4 ? 
Q 20.  A wire having a linear density 0.1 kg/m is kept under a tension of 490 N. It is observed that it  
resonates at a frequency of 400 Hz. The next higher frequency is 450 Hz. Find the length of the 
wire 
  (a) 0.4 m   (b) 0.7 m   (c) 0.6 m   (d) 0.49 m  
Q 21.  A string 1 m long is drawn by a 300 Hz vibrator attached to its end. The string vibrates in three 
segments. The speed of transverse waves in the string is equal to 
  (a) 100 m/s   (b) 200 m/s   (c) 300 m/s   (d) 400 m/s 
Q 22.  If ?1, ?2, ?3
 
are the wavelength of the waves giving resonance to the fundamental, first and second 
overtone modes respectively in a string fixed at both ends. The ratio of the wavelengths ?1 : ?2 : ?3
 
is  
  (a) 1:2:3   (b) 1 :3:5   (c) 
11
1: :
23
   (d) 
11
1: :
35
 
Q 23.  The period of oscillations of a point is 0.04 s and the velocity of propagation of oscillation is 300 
m/s. The difference of phases between the oscillations of two points at distance 10 m and 16 m 
respectively from the source of oscillations is 
 (a) 2 ?   (b) 
2
?
    (c) 
4
?
    (d) ? 
Q 24.  A transverse wave described by an equation y = 0.02 sin (x + 30t) where x and t are in metre and 
second, is traveling along a wire of area of cross-section 1 mm
2
 and density 8000 kgm
-3
. What is 
the tension in the string? 
  (a) 20 N   (b) 7.2 N   (c) 30 N   (d) 14.4 N 
Q 25.  A string vibrates in 5 segments to a frequency of 480 Hz. The frequency that will cause it to 
vibrate in 2 segments will be 
  (a) 96 Hz   (b) 192 Hz   (c) 1200 Hz   (d) 2400 Hz 
Q 26.  A wave travels on a light string. The equation of the waves is Y = A sin (kx - ?t + 30° ) It is 
reflected from a heavy string tied to an end of the light string at x = 0. If 64% of the incident 
energy is reflected then the equation of the reflected wave is 
  (a) Y = 0.8 A sin (kx - ?t + 30°+ 180°)  (b) Y = 0.8 A sin (kx + ?t + 30° + 180°) 
  (c) Y = 0.8 A sin (kx - ?T -30°)   (d) Y = 0.8 A sin (kx - ?t +30°) 
Q 27.  The tension, length, diameter and density of a string B are double than that of another string A. 
Which of the following overtones of B is same as the fundamental frequency of A ? 
  (a) 1st    (b) 2nd   (c) 3rd    (d) 4th 
  Passage : (Q. No. 28 to 30) 
Incident wave y = A sin ax bt
2
? ??
??
??
??
is reflected by an obstacle at x=0 which reduces intensity of 
reflected wave by 36%. Due to superposition a resulting wave consist of standing wave and 
travelling wave given by 
   y = - 1.6 sin ax sin bt + cA cos (bt + ax) where A, a, b and c are positive constants. 
Q 28.  Amplitude of reflected wave is 
  (a) 0.6, 4   (b) 0.8, 4   (c) 0.4, 4   (d) 0.2, 4 
Q 29.  Value of c is 
  (a) 0.2    (b) 0.4    (c) 0.6    (d) 0.3 
Q 30.  Position of second antinode is 
 (a) x
3a
?
?   (b) 
3
x
a
?
?   (c) 
3
x
2a
?
?   (d) 
2
x
3a
?
? 
  More than One Correct Options 
Q 31.  If the tension in a stretched string fixed at both ends is increased by 21% the fundamental 
frequency is found to change by 15 Hz. Then the  
  (a) original frequency is 150 Hz   
  (b) velocity of propagation of the transverse wave along the string increases by 5% 
  (c) velocity of propagation of the transverse wave along the string increases by 10% 
  (d) fundamental wavelength on the string does not change 
Q 32.  For interference to take place 
  (a) sources must be coherent    (b) sources must have same amplitude 
  (c) waves should travel in opposite directions (d) sources must have same frequency 
Q 33.  Regarding stationary waves choose the correct options. 
  (a) This is an example of interference  (b) Amplitudes of waves may be different 
  (c) Particles at nodes are always at rest  (d) Energy is conserved 
Q 34.  When a wave travels from a denser to rarer medium 
  (a) speed of wave increases    (b) wavelength of wave decreases 
  (c) amplitude of wave increases   (d) there is no change in phase angle 
Q 35.  A wire is stretched and fixed at two ends. Transverse stationary waves are formed in it. It 
oscillates in its third overtone mode. The equation of stationary wave is 
   Y = A sin kx cos ?t Choose the correct options. 
 (a) Amplitude of constituent waves is 
A
2
  (b) The wire oscillates in three loops 
  (c) The length of the wire is 
4
k
?
   (d) Speed of stationary wave is 
k
?
 
Q 36.  Which of the following equations can form stationary waves? 
  (i) Y = A sin ( ?t - kx)    (ii)Y = Acos( ?t - kx) 
  (iii) Y = A sin ( ?t + kx)    (iv) Y = A cos ( ?t + kx) 
  (a) (i) and (ii)   (b) (i) and (iii)  (c) (iii) and (iv)  (d) (ii) and (iv) 
Q 37.  Two waves 
   y 1 = A sin ( ?t - kx) and y 2 = A sin ( ?t + kx) 
  superimpose to produce a stationary wave then, 
  (a) x = 0 is a node  (b) x = 0 is an antinode (c) x
k
?
? is a node (d) 
2
k
?
?? is an antinode 
Answers 
  1.(d) 2.(d) 3.(b) 4.(c) 5.(c) 6.(d) 7.(d) 8.(a) 9.(d) 10.(d) 11. (a) 12.(c) 13.(b) 14.(c) 15.(a) 16. (d) 
  17.(a) 18.(c) 19. (b) 20. (b) 21.(b) 22. (c) 23. (d) 24. (b) 25. (b) 26.(b) 27.(c) 28. (b) 29.(a) 30. (c) 
  31.(a,c,d) 32.(a,d) 33.(a,b,d) 34.(a,c,d) 35.(a,c) 36.(b,d) 37.(b,d) 
 
Solutions 
1.  
   
  Showing we get T = 2 N 
2.  
   
   
3.  
   
  Now                 l = l 1 +  l 2+ l 3