Where does the battle with Grendels mother take place and what is the outcome?

The battle takes place in an underwater cave; Beowulf wins.

Explanation:

After killing Grendel, Beowulf moves on to kill another monster - Grendel's mother. She lives in a cave under the lake and Beowulf goes there to fight her. Ultimately, after a very long fight, he manages to grab hold of a giant sword within the cave and pierces the monster's skin. As a result, Grendel's mother dies and Beowulf emerges victorious.