Your approach to learning about RTOS is just completely wrong. Asking if this or that example requires a mutex does not teach you anything about a mutex. You need to learn why a mutex is needed, then you can answer the question about any situation.

Find a good description of mutex use, read it and come back with questions if you do not understand. Include a link to what you have read. When you do think you understand it, make up an example, tell us how you see it and let us correct you if you are wrong.

 

Here is a real-world example of a RTOS project which uses mutexs.
Click the link above.

 

Does it mean that at a time we do not give full time to complete the task. In this case the task required 3 seconds but we are giving it 300 milliseconds per second.

But if task is completed in pieces, will it not be corrupt?

Do you not completely understand interrupts?