In fundamental mode, the time required for the sound wave to reach up to closed end of a pipe filled with air is 't' second. The frequency of vibration of air column is (Neglect end correction)

Fundamental frequency of a closed pipe

For a pipe closed at one end vibrating in its fundamental mode, the pipe length equals one quarter of the wavelength: $L = \frac{\lambda}{4}$.

The speed of sound relates to frequency and wavelength by $v = f\lambda$.

The sound wave travels the pipe length $L$ in time $t$, giving $L = vt$.

Combining these relationships yields $\lambda = 4L = 4vt$. Substituting into the wave equation: $v = f(4vt)$.

Solving for frequency: $f = \frac{1}{4t} = (4t)^{-1}$.

The expression matches option A.