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**Hint:**Here, we will find the gain time of the clock. In this question, it is given that the clock gains 12 minutes in 12 hours. And also we have to find the total hours that the clock travels. Based on this relation we will get actual time.

**Complete step-by-step solution:**

Here, in the question given that the clock gains 12 minutes in 12 hours.

Therefore, we can say that a clock will gain 1 minute in 1 hour.

The difference in time that the clock indicates is the difference between today 11:00 a.m. to 5:30 p.m. the next day.

So, let us calculate the total hour that the cock travels between today 11:00 a.m. to 5:30 p.m. the next day. Today 11:00 a.m. to 11:00 a.m. the next day is 24 hours and 11:00 am to 5:30 p.m. is another 6.5 hours.

That is equal to 30.5 hours.

Therefore, the time gained

$ \Rightarrow \dfrac{{30.5}}{1}$

$ \Rightarrow $ 30 minutes

Hence, the actual time is $5:30$ p.m. + $30$ minutes.

That is $6:00$ p.m.

**Option D is the correct answer.**

**Note:**Clocks do not show the correct time. They show the interval of that particular clock’s movement, which may or may not be accepted and agreed upon the interval of movement and display. If every clock vanished tomorrow, time would still move forward and exist. But once everyone was given another clock they would all have to agree on what time to set it. Therefore, the true time is the one that is universally accepted. For daylight saving, the clocks rearrange according to the need. The typical implementation of DST is to set clocks forward or backward to return standard time.

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