The displacement of a damped harmonic oscillator is given by
x(t ) = e^–0.1t cos (10$$\pi $$t + f).
Here t is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to :

Two radioactive materials A and B have decay constants 10$$\lambda $$ and $$\lambda $$, respectively. It initially they have the same number of nuclei, then the ratio of the number of nuclei of A to that of B will be 1/e after a time :

A stationary source emits sound waves of frequency 500 Hz. Two observers moving along a line passing through the source detect sound to be of frequencies 480 Hz and 530 Hz. Their respective speeds are, in ms–1, (Given speed of sound = 300 m/s)

An n-p-n transistor operates as a common emitter amplifier, with a power gain of 60 dB. The input circuit resistance is 100$$\Omega $$ and the output load resistance is 10 k$$\Omega $$. The common emitter current gain $$\beta $$ is :