Somehow, Ratul managed _____ the locked door without _____ it.

The correct answer is: to open, damaging. 

Infinitive: 
- An infinitive is the word “to + a verb’s root". 
- We use the to-infinitive after certain verbs (verbs followed by to-infinitive), particularly verbs of thinking and feeling:

Present Participle 
A present participle is a word that (1) ends "-ing," (2) is formed from a verb, and (3) is used as an adjective or to form verb tense.
For example: My mother is next to the lady wearing the red hat.

মনে রাখতে হবে, 
- Gerund এবং Present Participle উভয়েরই গঠন ( V+ing) হলেও Gerund  বাক্যে Verb+Noun এর কাজ করে। অপরদিকে, Present Participle একই সাথে verb+adjective এর কাজ করে।  
- এছাড়া, Gerund কখনো কখনো শুধু Noun এর কাজ করে; তখন একে Verbal noun বলা হয়।
যেমন: The reading of newspaper is a good habit. 

Source: British Council Learning website