I have a couple of questions about English usage. I am going to make a vocabulary test for some research purpose, so I just want to make sure that some example sentences are acceptable to native speakers.

1) Water conducts with heat faster than with air

2) He asked if he might conduct us to the ball

3) Candidates will be examined on thier written and oral language skills

4) Police are still investigating about how the accident happened vs. Police are still investigate how the accident happened.

Thank you all for your help in advance!

1.Water conducts heat faster than air does.

2. He asked if he might ? us to the ball. conduct is an odd choice here. Accompany?

3. ok apart from spelling: their

4) Police are still investigating how the accident happened.