Verb + (으)ㄹ까요?
1.’asking someone’s opinion if this person want to do’.
창문 닫을까요? Do you want me to close the window?
불 켤까요? Do you want me to turn on the light?
물 좀 드릴까요? Would you like to have some water?
뭐 드릴까요? What would you like to have?
내일 제가 갈까요? Do you want me to come tomorrow?
2.’suggesting to do something together‘. In this case, you can add 같이.
이번 주말에 영화 같이 볼까요? Shall we watch a movie this weekend?
이번 휴가 때 뭐 할까요? What should we do for this vacation?
오늘 저녁에 뭐 먹을까요? What should we eat for dinner?
뭐 시킬까요? What should we order?
이제 집에 갈까요? Shall we go home?
여기 잠깐 앉을까요? Shall we sit here for a second?
3.-(으)ㄹ까요 is also used for questions, thoughts or guesses. (Intermediate level)
민수 씨가 왜 아직 안 올까요? Why hasn’t he come yet?
내일 날씨가 좋을까요? Is it going to be nice tomorrow?
서울의 집 값이 비쌀까요? Are house prices expensive?
운동을 하면 살이 빠질까요? Can I lose weight if I exercise?