Umesh would not have been able to check his exam result online if the power was down, because even though the laptop runs on battery, the Internet router (assuming) needs continuous power supply. Plus, his mother also confirms about the absence of an Internet connection in their house earlier in the film.
When Umesh is checking his mark result in laptop his branch is shown as electronics and communication engineering but later Aju Vargheese says his branch is Mechanical while explaining their story to Vineeth Srinivasan.