In medium where E. coli was growing lactose was added, which induced the lac operon. But why does lac operon shut down after some time after addition of lactose in the medium?

Which molecule does act as inducer molecule in lac operon?

The rope shown at an instant is carrying a wave travelling towards right, created by a source vibrating at a frequency n . Consider the following statements I. The speed of the wave is 4n xx ab II. The medium at a will be in the same phase as d after (4)/(3n)s III. The phase difference between b and e is (3pi)/(2) Which of these statements are correct