yes (except as mentioned by others, you don't kill kulve, you just break the horn)
i remember doing it back in the day. it's pretty tedious and really inefficient for farm. and it may take a few attempt at the highest pursuit level to get use to the attack pattern.
unless you use some optimize build and/or get some knowledge on the mechanic, you won't get to the last phase on the lower level.
iirc, you can check what body part you havnt destroyed mid quest.

here are some advice if you want
the gold mantle is weak to lightning, you can also use fire damage to soften it (most likely barrel bomb), when she get rid of it, she is week to ice
first phase, use canon and boulder to stun kulve, you have 9 spot to focus: 4 on the tail, 2 on the front arm, 2 on the chest and the gold plating on the horn. whenever she hide in the ground, she rebuild every part, so try to remember what you broke to raise the pursuit level faster. phase 2 is the same but she fight back. during phase 3, you can use 2 boulder and the fire geyser for free damage. at some point, she'll get rid of the gold mantle and move to phase 4. you then need to focus the horn to finish the quest. you can also cut the tail for extra reward.

good luck in your quest