Question

under what condition,the time period of a pendulum does not depend upon the amplitudeof oscillation

Answer 1

The amplitude certainly does affect the period, but for sufficiently small amplitudes, say less than 5°, it is a good approximation to say that the period is does not depend on the amplitude. In this small-amplitude limit, the period is

T=2πℓ/g−−−√T=2πℓ/g

Notice that this equation makes no reference to amplitude. For larger amplitudes, the period will be larger than this equation predicts.