You can not enter Recovery mode if your Xiaomi Redmi Note 3 bootloader is locked. It will help you How to unlock the Redmi Note 3 Bootloader. Make sure you have working MI-Account that verified and activated SIM Card and download unlocking tools.
Before starting to unlock the bootloader you needs:
Download and install Developer ROM:
- Download Xiaomi Redmi Note 3 Developer ROM
- Place that .zip file at the root of the internal storage.
- Go to Updater App (you will find it in Homescreen > Tools (folder) > Update)
- Tap on the 3 dots (right corner of the screen) > Select .zip file downloaded previously.
- The phone will reboot and you will be on Developer ROM!
Warning! It will erase all data in your device
Working MI-Account that verified and activated SIM Card
Download unlocking tools at http://www.smiui.net/files/MiFlashUnlock_hennessy.zip to your Computer.
Follow these steps:
- Make sure the Phone running in developer ROM
- Started the Setup Wizard
- Connect your Phone to your verified Mi-Account (Settings -> Accounts)
- Login to i.mi.com and make sure that your new RedMi Note 3 is connected to that Account.
- Reboot the Phone to Fastboot Mode ( press and hold volume down key and power key for a few seconds, then release all keys when Mi logo appears )
- Connect your Phone via USB to your Computer
- Open the MiFlashUnlock.exe
- Click the left Button (chinese text)
- Login with your Mi-Account you used to link your new Phone
- Press the Button with chinese Text and watch the show
- The Bootloader is unlocked when the progress bar shows 100%.
- Then Reboot your the phone.
If the phone bootloader is unlocked, you can root the phone. Read how to root and flash TWRP Recovery Redmi Note 3
Source and troubleshooting pls visit :
http://xiaomi.eu/community/threads/ultimate-guide-how-to-unlock-bootloader-root-and-flash-recovery-on-redmi-note-3.29213/