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Q15.12(i)(b):
For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (b) phase? Explain your answers.
Solution :
1. Theoretical Foundation of Phase in Traveling Waves
The displacement of a transverse wave traveling along a string is defined by the mathematical function: $y(x, t) = A \sin(kx - \omega t + \phi)$ where:
- $A$ is the amplitude.
- $k$ is the angular wave number ($\frac{2\pi}{\lambda}$).
- $\omega$ is the angular frequency ($2\pi f$).
- $(kx - \omega t + \phi)$ represents the phase ($\Phi$) of the wave at position $x$ and time $t$.
2. Analysis of Phase Variation
To determine if all points on the string oscillate with the same phase, we examine the phase difference ($\Delta\Phi$) between two arbitrary points on the string, $x_1$ and $x_2$, at any fixed instant of time $t$.
The phase at $x_1$ is $\Phi_1 = kx_1 - \omega t + \phi$.
The phase at $x_2$ is $\Phi_2 = kx_2 - \omega t + \phi$.
The phase difference is given by: $\Delta\Phi = \Phi_2 - \Phi_1 = (kx_2 - \omega t + \phi) - (kx_1 - \omega t + \phi)$ $\Delta\Phi = k(x_2 - x_1) = k\Delta x$
3. Mathematical Justification
Since $k = \frac{2\pi}{\lambda}$, the phase difference depends explicitly on the spatial separation $\Delta x$ between the points: $\Delta\Phi = \frac{2\pi}{\lambda} \Delta x$
For all points to oscillate with the same phase, the condition $\Delta\Phi = 0$ (or an integer multiple of $2\pi$) must hold for any arbitrary distance $\Delta x$. However, for a traveling wave where $k \neq 0$, the phase difference is a linear function of $x$. Therefore, points separated by any distance that is not an integer multiple of the wavelength $\lambda$ will necessarily have different phases.
4. Visual Representation of Phase Shift
Conclusion
In a traveling wave, the phase is a function of the spatial coordinate $x$. Points separated by a distance $\Delta x$ exhibit a phase difference of $\frac{2\pi}{\lambda} \Delta x$. Because this value is non-zero for arbitrary points, the points on the string do not oscillate in phase. They oscillate with the same frequency $\omega$, but with a phase lag or lead relative to one another depending on their spatial positions.
Final Solution: No, all points on the string do not oscillate with the same phase. The phase of each point depends on its position $x$ along the string, specifically defined by the term $(kx - \omega t + \phi)$.
More Questions from Class 11 Physics Waves EXERCISES
- Q14.1: A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?
- Q14.10(a): For the travelling harmonic wave (y(x, t)) = (2.0 \cos 2\pi (10t – 0.0080 x + 0.35)) where (x) and (y) are in cm and (t) in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of 4 m,
- Q14.10(b): For the travelling harmonic wave (y(x, t)) = (2.0 \cos 2\pi (10t – 0.0080 x + 0.35)) where (x) and (y) are in cm and (t) in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of 0.5 m,
- Q14.10(c): For the travelling harmonic wave (y(x, t)) = (2.0 \cos 2\pi (10t – 0.0080 x + 0.35)) where (x) and (y) are in cm and (t) in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of (\frac{\lambda}{2}),
- Q14.10(d): For the travelling harmonic wave (y(x, t)) = (2.0 \cos 2\pi (10t – 0.0080 x + 0.35)) where (x) and (y) are in cm and (t) in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of (\frac{3\lambda}{4})
- Q14.11(a): The transverse displacement of a string (clamped at its both ends) is given by (y(x, t)) = (0.06 \sin(\frac{2\pi x}{3}) \cos (120 \pi t)) where (x) and (y) are in m and (t) in s. The length of the string is 1.5 m and its mass is 3.0 (\times)10^{-2} kg. Does the function represent a travelling wave or a stationary wave?
- Q14.11(b): The transverse displacement of a string (clamped at its both ends) is given by (y(x, t)) = (0.06 \sin(\frac{2\pi x}{3}) \cos (120 \pi t)) where (x) and (y) are in m and (t) in s. The length of the string is 1.5 m and its mass is 3.0 (\times)10^{-2} kg. Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave ?
- Q14.11(c): The transverse displacement of a string (clamped at its both ends) is given by (y(x, t)) = (0.06 \sin(\frac{2\pi x}{3}) \cos (120 \pi t)) where (x) and (y) are in m and (t) in s. The length of the string is 1.5 m and its mass is 3.0 (\times)10^{-2} kg. Determine the tension in the string.
- Q14.12(i)(a): For the wave on a string described, do all the points on the string oscillate with the same frequency? Explain your answers.
- Q14.12(i)(b): For the wave on a string described, do all the points on the string oscillate with the same phase? Explain your answers.
- Q14.12(i)(c): For the wave on a string described, do all the points on the string oscillate with the same amplitude? Explain your answers.
- Q14.12(ii): For the wave on a string described, what is the amplitude of a point 0.375 m away from one end?
- Q14.13(a): Given below are some functions of (x) and (t) to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (y) = (2 \cos (3x) \sin (10t))
- Q14.13(b): Given below are some functions of (x) and (t) to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (y) = (2\sqrt{x - vt})
- Q14.13(c): Given below are some functions of (x) and (t) to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (y) = (3 \sin (5x – 0.5t) + 4 \cos (5x – 0.5t))
- Q14.13(d): Given below are some functions of (x) and (t) to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (y) = (\cos x \sin t + \cos 2x \sin 2t)
- Q14.14(a): A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 (\times) 10^{-2} kg and its linear mass density is 4.0 (\times) 10^{-2} kg (m^{-1}). What is the speed of a transverse wave on the string?
- Q14.14(b): A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 (\times) 10^{-2} kg and its linear mass density is 4.0 (\times) 10^{-2} kg (m^{-1}). What is the tension in the string?
- Q14.15: A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
- Q14.16: A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of sound in steel?
- Q14.17: A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source ? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is 340 m (s^{-1})).
- Q14.18: Two sitar strings (A) and (B) playing the note ‘Ga’ are slightly out of tune and produce beats of frequency 6 Hz. The tension in the string (A) is slightly reduced and the beat frequency is found to reduce to 3 Hz. If the original frequency of (A) is 324 Hz, what is the frequency of (B)?
- Q14.19(a): Explain why (or how): in a sound wave, a displacement node is a pressure antinode and vice versa,
- Q14.19(b): Explain why (or how): bats can ascertain distances, directions, nature, and sizes of the obstacles without any “eyes”,
- Q14.19(c): Explain why (or how): a violin note and sitar note may have the same frequency, yet we can distinguish between the two notes,
- Q14.19(d): Explain why (or how): solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases, and
- Q14.19(e): Explain why (or how): the shape of a pulse gets distorted during propagation in a dispersive medium.
- Q14.2: A stone dropped from the top of a tower of height 300 m splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m (s^{-1}) ? ((g) = 9.8 m (s^{-2}))
- Q14.3: A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at 20 (^{\circ})C = 343 m (s^{-1}).
- Q14.4(a): Use the formula (v) = (\sqrt{\frac{\gamma P}{\rho}}) to explain why the speed of sound in air is independent of pressure,
- Q14.4(b): Use the formula (v) = (\sqrt{\frac{\gamma P}{\rho}}) to explain why the speed of sound in air increases with temperature,
- Q14.4(c): Use the formula (v) = (\sqrt{\frac{\gamma P}{\rho}}) to explain why the speed of sound in air increases with humidity.
- Q14.5(a): You have learnt that a travelling wave in one dimension is represented by a function (y) = (f (x, t)) where (x) and (t) must appear in the combination (x – v t) or (x + v t), i.e. (y) = (f (x \pm v t)). Is the converse true? Examine if the following functions for (y) can possibly represent a travelling wave : ((x – vt)^{2})
- Q14.5(b): You have learnt that a travelling wave in one dimension is represented by a function (y) = (f (x, t)) where (x) and (t) must appear in the combination (x – v t) or (x + v t), i.e. (y) = (f (x \pm v t)). Is the converse true? Examine if the following functions for (y) can possibly represent a travelling wave : (\log [\frac{(x + vt)}{x_{0}}])
- Q14.5(c): You have learnt that a travelling wave in one dimension is represented by a function (y) = (f (x, t)) where (x) and (t) must appear in the combination (x – v t) or (x + v t), i.e. (y) = (f (x \pm v t)). Is the converse true? Examine if the following functions for (y) can possibly represent a travelling wave : (\frac{1}{(x + vt)})
- Q14.6(a): A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of the reflected sound? Speed of sound in air is 340 m (s^{-1}) and in water 1486 m (s^{-1}).
- Q14.6(b): A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of the transmitted sound? Speed of sound in air is 340 m (s^{-1}) and in water 1486 m (s^{-1}).
- Q14.7: A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km (s^{-1}) ? The operating frequency of the scanner is 4.2 MHz.
- Q14.8(a): A transverse harmonic wave on a string is described by (y(x, t)) = (3.0 \sin (36 t + 0.018 x + \frac{\pi}{4})) where (x) and (y) are in cm and (t) in s. The positive direction of (x) is from left to right. Is this a travelling wave or a stationary wave ? If it is travelling, what are the speed and direction of its propagation ?
- Q14.8(b): A transverse harmonic wave on a string is described by (y(x, t)) = (3.0 \sin (36 t + 0.018 x + \frac{\pi}{4})) where (x) and (y) are in cm and (t) in s. The positive direction of (x) is from left to right. What are its amplitude and frequency ?
- Q14.8(c): A transverse harmonic wave on a string is described by (y(x, t)) = (3.0 \sin (36 t + 0.018 x + \frac{\pi}{4})) where (x) and (y) are in cm and (t) in s. The positive direction of (x) is from left to right. What is the initial phase at the origin ?
- Q14.8(d): A transverse harmonic wave on a string is described by (y(x, t)) = (3.0 \sin (36 t + 0.018 x + \frac{\pi}{4})) where (x) and (y) are in cm and (t) in s. The positive direction of (x) is from left to right. What is the least distance between two successive crests in the wave ?
- Q14.9: For the wave described, plot the displacement ((y)) versus ((t)) graphs for (x) = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase ?
- Q15.1: A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?
- Q15.10(a): For the travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi (10t – 0.0080 x + 0.35)$ where $x$ and $y$ are in cm and $t$ in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of (a) 4 m,
- Q15.10(b): For the travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi (10t – 0.0080 x + 0.35)$ where $x$ and $y$ are in cm and $t$ in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of (b) 0.5 m,
- Q15.10(c): For the travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi (10t – 0.0080 x + 0.35)$ where $x$ and $y$ are in cm and $t$ in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of (c) $\lambda/2$,
- Q15.10(d): For the travelling harmonic wave $y(x, t) = 2.0 \cos 2\pi (10t – 0.0080 x + 0.35)$ where $x$ and $y$ are in cm and $t$ in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of (d) $3\lambda/4$
- Q15.11(a): The transverse displacement of a string (clamped at its both ends) is given by $y(x, t) = 0.06 \sin(\frac{2\pi x}{3}) \cos(120 \pi t)$ where $x$ and $y$ are in m and $t$ in s. The length of the string is 1.5 m and its mass is $3.0 \times 10^{-2}$ kg. Answer the following : (a) Does the function represent a travelling wave or a stationary wave?
- Q15.11(b): The transverse displacement of a string (clamped at its both ends) is given by $y(x, t) = 0.06 \sin(\frac{2\pi x}{3}) \cos(120 \pi t)$ where $x$ and $y$ are in m and $t$ in s. The length of the string is 1.5 m and its mass is $3.0 \times 10^{-2}$ kg. Answer the following : (b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave ?
- Q15.11(c): The transverse displacement of a string (clamped at its both ends) is given by $y(x, t) = 0.06 \sin(\frac{2\pi x}{3}) \cos(120 \pi t)$ where $x$ and $y$ are in m and $t$ in s. The length of the string is 1.5 m and its mass is $3.0 \times 10^{-2}$ kg. Answer the following : (c) Determine the tension in the string.
- Q15.12(i)(a): For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (a) frequency? Explain your answers.
- Q15.12(i)(c): For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (c) amplitude? Explain your answers.
- Q15.12(ii): What is the amplitude of a point 0.375 m away from one end?
- Q15.13(a): Given below are some functions of $x$ and $t$ to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (a) $y = 2 \cos(3x) \sin(10t)$
- Q15.13(b): Given below are some functions of $x$ and $t$ to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (b) $y = 2 \sqrt{x-vt}$
- Q15.13(c): Given below are some functions of $x$ and $t$ to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (c) $y = 3 \sin(5x - 0.5t) + 4 \cos(5x - 0.5t)$
- Q15.13(d): Given below are some functions of $x$ and $t$ to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (d) $y = \cos x \sin t + \cos 2x \sin 2t$
- Q15.14(a): A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is $3.5 \times 10^{-2}$ kg and its linear mass density is $4.0 \times 10^{-2}$ kg $m^{-1}$. What is (a) the speed of a transverse wave on the string, and
- Q15.14(b): A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is $3.5 \times 10^{-2}$ kg and its linear mass density is $4.0 \times 10^{-2}$ kg $m^{-1}$. What is (b) the tension in the string?
- Q15.15: A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
- Q15.16: A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of sound in steel?
- Q15.17: A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source ? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is 340 m $s^{-1}$).
- Q15.18: Two sitar strings A and B playing the note ‘Ga’ are slightly out of tune and produce beats of frequency 6 Hz. The tension in the string A is slightly reduced and the beat frequency is found to reduce to 3 Hz. If the original frequency of A is 324 Hz, what is the frequency of B?
- Q15.19(a): Explain why (or how): (a) in a sound wave, a displacement node is a pressure antinode and vice versa,
- Q15.19(b): Explain why (or how): (b) bats can ascertain distances, directions, nature, and sizes of the obstacles without any “eyes”,
- Q15.19(c): Explain why (or how): (c) a violin note and sitar note may have the same frequency, yet we can distinguish between the two notes,
- Q15.19(d): Explain why (or how): (d) solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases, and
- Q15.19(e): Explain why (or how): (e) the shape of a pulse gets distorted during propagation in a dispersive medium.
- Q15.2: A stone dropped from the top of a tower of height 300 m splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m $s^{-1}$ ? ($g$ = 9.8 m $s^{-2}$)
- Q15.20(i)(a): A train, standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air. (i) What is the frequency of the whistle for a platform observer when the train (a) approaches the platform with a speed of 10 m $s^{-1}$,
- Q15.20(i)(b): A train, standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air. (i) What is the frequency of the whistle for a platform observer when the train (b) recedes from the platform with a speed of 10 m $s^{-1}$?
- Q15.20(ii): A train, standing at the outer signal of a railway station blows a whistle of frequency 400 Hz in still air. (ii) What is the speed of sound in each case ? The speed of sound in still air can be taken as 340 m $s^{-1}$.
- Q15.21: A train, standing in a station-yard, blows a whistle of frequency 400 Hz in still air. The wind starts blowing in the direction from the yard to the station with a speed of 10 m $s^{-1}$. What are the frequency, wavelength, and speed of sound for an observer standing on the station’s platform? Is the situation exactly identical to the case when the air is still and the observer runs towards the yard at a speed of 10 m $s^{-1}$? The speed of sound in still air can be taken as 340 m $s^{-1}$
- Q15.3: A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at 20 °C = 343 m $s^{-1}$.
- Q15.4(a): Use the formula $v = \sqrt{\frac{\gamma P}{\rho}}$ to explain why the speed of sound in air (a) is independent of pressure,
- Q15.4(b): Use the formula $v = \sqrt{\frac{\gamma P}{\rho}}$ to explain why the speed of sound in air (b) increases with temperature,
- Q15.4(c): Use the formula $v = \sqrt{\frac{\gamma P}{\rho}}$ to explain why the speed of sound in air (c) increases with humidity.
- Q15.5(a): You have learnt that a travelling wave in one dimension is represented by a function $y = f (x, t)$ where $x$ and $t$ must appear in the combination $x - v t$ or $x + v t$, i.e. $y = f (x \pm v t)$. Is the converse true? Examine if the following functions for $y$ can possibly represent a travelling wave : (a) $(x - vt)^2$
- Q15.5(b): You have learnt that a travelling wave in one dimension is represented by a function $y = f (x, t)$ where $x$ and $t$ must appear in the combination $x - v t$ or $x + v t$, i.e. $y = f (x \pm v t)$. Is the converse true? Examine if the following functions for $y$ can possibly represent a travelling wave : (b) $\log[\frac{(x + vt)}{x_0}]$
- Q15.5(c): You have learnt that a travelling wave in one dimension is represented by a function $y = f (x, t)$ where $x$ and $t$ must appear in the combination $x - v t$ or $x + v t$, i.e. $y = f (x \pm v t)$. Is the converse true? Examine if the following functions for $y$ can possibly represent a travelling wave : (c) $\frac{1}{(x + vt)}$
- Q15.6(a): A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted sound? Speed of sound in air is 340 m $s^{-1}$ and in water 1486 m $s^{-1}$.
- Q15.6(b): A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (b) the transmitted sound? Speed of sound in air is 340 m $s^{-1}$ and in water 1486 m $s^{-1}$.
- Q15.7: A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km $s^{-1}$ ? The operating frequency of the scanner is 4.2 MHz.
- Q15.8(a): A transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4)$ where $x$ and $y$ are in cm and $t$ in s. The positive direction of $x$ is from left to right. (a) Is this a travelling wave or a stationary wave ? If it is travelling, what are the speed and direction of its propagation ?
- Q15.8(b): A transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4)$ where $x$ and $y$ are in cm and $t$ in s. The positive direction of $x$ is from left to right. (b) What are its amplitude and frequency ?
- Q15.8(c): A transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4)$ where $x$ and $y$ are in cm and $t$ in s. The positive direction of $x$ is from left to right. (c) What is the initial phase at the origin ?
- Q15.8(d): A transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36 t + 0.018 x + \pi/4)$ where $x$ and $y$ are in cm and $t$ in s. The positive direction of $x$ is from left to right. (d) What is the least distance between two successive crests in the wave ?
- Q15.9: For the wave described in Exercise 15.8, plot the displacement ($y$) versus ($t$) graphs for $x$ = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase ?
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Mahipalpur, Delhi
14 yrs of
Exp
1 student
2 hrs of classes completed
I'm professional tutor, teaching maths for last 14 years. Maximizing potential of student through.
Student's Home
As an experienced teacher for 12 years, I have been teaching to make students interested in the subjects of Maths. My attitude is to help children improve in studies in the best possible way.
zeba
Highly recommended
Skills:
Mathematics
,
Applied Mathematics
Also teaches:
Class 9 Tuition, Engineering Entrance Coaching
and more
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