A clock is set right at 11 am. The clock gains 12 minutes in a day. What will be the true time when the watch indicates 5 pm on the next day?

A clock is set right at 11 am. The clock gains 12 minutes in a day. 

Clock gaining 12 mins a day implies it is gaining 0.5 minutes for every 1 hour. 
                                        $$\rightarrow$$ Clock gains $$\dfrac{\ 0.5}{60}$$ hours per every hour. 

After 24 hrs, the time in the clock would be 11:12 am the next day. After this time, lets assume "t" hours it takes to show 5pm : 

                                    $$\rightarrow$$ $$11:12+t+\dfrac{\ 0.5\left(t\right)}{60}=17:00$$

                                                    $$t\left(1+\dfrac{\ 1}{120}\right)=5\ hr\ 48\ \min=5.8\ hr$$

                                    $$\rightarrow$$ $$t=\dfrac{\ 5.8}{1+0.0083}=5.752\ hrs$$

Therefore, the correct time should have been = 11:00 am + 5.752 hours = 16:45 hrs. 
Hence, the answer is 45 mins past 4 pm .