Should I put family responsibility after my father's death in us application?

Yes, I think this really needs to go somewhere on your application. That sounds like it would really affect your high school journey. On most college applications, there is an Additional Information section where you would put an explanation.

Write something brief that states the facts of the situation. Something like this: "My father passed away during my sophomore year. After that, it was necessary for me to work and care for my mother and sibling. Despite the challenges, I (write something the highlights the positives, like maintaining high grades or balancing academics and responsibilities well.)

I hope this is helpful to you. Please lmk if you have any questions. I am sorry for your loss.

If I'm being honest I wouldn't. I would try and outline how that death impacted you without necessarily trauma dumping if that makes sense. Use it to highlight how you as a person have grown or what you have learned. The best thing I can tell you is to make it seem positive despite it being such a hardship in your life. I'm sorry for your loss as well.