These are the instructions to root Samsung Galaxy Core GT-I8262 (running on android 4.1.2) using Framaroot Application on Phone itself (does not require computer).
Requirements: Your Samsung Galaxy Core GT-I8262 smartphone should have at-least 10-20 percent of phone battery to perform the rooting process.
Step 1: Enable installation of 3 party Apps on your Samsung Galaxy Core GT-I8262. To Enable, Open Settings > Security > Device Administration >Unknown Sources (tick to enable it).
Step 2: Download and install Framaroot Application on your Samsung Galaxy Core GT-I8262 smartphone. Once installation is completed you will be able to see the Framaroot Icon in the App Menu.
Step 3: Now, Tap on the Framaroot App Icon to Open it.
Step 4: Once Framaroot Application is launched you will be able to see the following screen on your Device:
Step 5: In Framaroot, You will be able to see 2 options i.e Aragorn and Gandalf. Tap on the Gandalf Option.
Step 6: Now, you will be able to see the Framaroot success Message on the Phone Screen.
Step 7: Now, you have to restart your Device.
Step 8: Now your Samsung Galaxy Core GT-I8262 smartphone is Rooted. To check whether your phone is rooted correctly, open Applications Menu, there you will be able to see a new app installed, called SuperSU. If this app exist then it means you have successfully rooted your device.
Optional: You can also verify the root status of your device by using the Root Checker Application.
[*] No data loss process: Your Samsung Galaxy Core GT-I8262 Phone Data like contacts, pictures, videos, music and etc will be not affected with this method.
[*] Device Warranty: Once you full unroot your device through SuperSU, you can claim for the phone warranty. In short no need to worry for phone warranty.
[*] How to unroot? Open SuperSU > Settings > Full Unroot.
[*] Credits: Framaroot Application is developed by alephzain (XDA Developer).