Question

a given liquid is heated in two different pans liquid in one pan is boiling gently while in another Pan liquid is boiling vigorously in temperature of liquid in both the pans same or different​

Answer 1

The bottom line is that for all practical purposes turning up the heat won't make any difference. However there are a few subtleties to consider.

Water boils when the chemical potential of the water is equal to the chemical potential of the water vapour at the same temperature. One of the factors that determine the chemical potential of the water vapour is the pressure, or more precisely the partial pressure of the water vapour in the air.

If you take a bowl of water at room temperature and put it in a sealed chamber then the water evaporates until the partial pressure of the water vapour is about 0.03 atmospheres. If you increase the temperature to say 50ºC then the chemical potential of the water increases so the partial pressure of the water vapour rises to match - to about 0.12 atmospheres. As long as the partial pressure of the water vapour can keep rising it will stop the water boiling.

But when heating water in the open the partial pressure of the water vapour can't rise above 1 atmosphere so when the temperature reaches 100ºC the partial pressure of the water vapour can no longer increase to stop the water boiling, and heating the water more vigorously just causes increased evaporation and doesn't raise the temperature. The temperature of the water remains locked at 100ºC.

However this assumes the water and the vapour are in equilibrium, and in real life the water/steam system may be near equilibrium but won't actually be at equilibrium. We inevitably get some superheating and the water temperature can rise slightly above 100ºC. The more vigorously you heat the water the greater will be the degree of superheating and the higher will be the temperature of the water.

So turning up the gas will increase the water temperature slightly above 100ºC. However we are talking about a very small increase - a few degrees at most - and this is unlikely to make any significant difference to the cooking time. Hence my opening comment that for all practical purposes turning up the heat will make no difference.