First resonating length:

λ/4=L1+e Here,λ is wavelength L1 is length of first resonating column
e is end correction
λ/4=32 +e  ...... (1)

Second resonating length:

3λ/4=L2+e Here,λ is wavelength L2 is length of second resonating column 
e is end correction
3λ/4=100 +e  ...... (2)

Solve the equations (1) and (2):

(2) - (1) gives

λ/2=100−32
λ/2=68
λ=136 cm
Hence,
λ/4=32+e
e=136/4−32
=34−32= 2 cm
Therefore, the end correction is equal to 2 cm.

Introduction:
The end correction in a resonance tube refers to the adjustment made to the measured length of the air column in order to account for the presence of open ends. Since the air column in a resonance tube is open at both ends, the actual length of the column differs from the measured length. This difference is known as the end correction.

Given Information:
- Frequency of the tuning fork: 256 Hz
- Length of the resonating air columns: 32 cm and 100 cm

Determination of End Correction:
To determine the end correction, we need to calculate the wavelength of the sound produced by the tuning fork. We can then compare this wavelength to the measured lengths of the air columns.

The formula to calculate the wavelength (λ) is given by:

λ = 2L/n

Where:
- λ is the wavelength
- L is the measured length of the air column
- n is the harmonic number

For the 32 cm air column:
Since the air column is open at both ends, the harmonic number (n) for the fundamental frequency is 1.
Using the formula, we can calculate the wavelength for the 32 cm air column:

λ = 2 * 32 cm / 1
= 64 cm

For the 100 cm air column:
Again, the air column is open at both ends, so the harmonic number (n) for the fundamental frequency is 1.
Calculating the wavelength for the 100 cm air column:

λ = 2 * 100 cm / 1
= 200 cm

Calculating the End Correction:
The actual length of the air column is equal to the measured length (L) plus the end correction (EC). Thus, we can write the equation as:

Actual length = Measured length + End correction

To find the end correction, we rearrange the equation as:

End correction = Actual length - Measured length

For the 32 cm air column:
The actual length of the air column is equal to the wavelength (λ):

Actual length = λ
= 64 cm

Using the equation, we can calculate the end correction:

End correction = 64 cm - 32 cm
= 32 cm

For the 100 cm air column:
The actual length of the air column is again equal to the wavelength (λ):

Actual length = λ
= 200 cm

Calculating the end correction:

End correction = 200 cm - 100 cm
= 100 cm

Conclusion:
The end correction for the 32 cm air column is 32 cm, and for the 100 cm air column, it is 100 cm. The end correction accounts for the presence of open ends in the resonance tube, allowing us to accurately measure the wavelength and determine the frequency of the tuning fork.