Learn Icelandic: Voruð þið að rannsaka glæpinn? - Were you investigating the crime?

Just like in English, Icelandic forms the past continuous using the past tense of the verb vera (to be) followed by að and the infinitive verb. Here, voruð is the past tense of to be for you (plural), making voruð þið að rannsaka mean were you investigating.

Why does the sentence start with the verb Voruð?

To ask a yes/no question in Icelandic, you swap the subject and the verb. A regular statement would be Þið voruð að rannsaka glæpinn (You were investigating the crime), but for a question, the verb Voruð jumps to the front.

Could I use Varst þú instead of Voruð þið here?

Only if you are talking to one person! Þið means you in the plural, so voruð þið is asking a group of people. If you were asking just one person, you would use the singular you and say Varst þú að rannsaka glæpinn?

Why is the word for the crime glæpinn instead of its dictionary form glæpurinn?

The verb rannsaka (to investigate) forces the object being investigated into the accusative case. The base word is the masculine noun glæpur, which drops its -ur ending in the accusative to become glæp. When you add the accusative definite article (the) to the end, it becomes glæpinn.