He answered the questions on / at the blackboard. Both versions are fine but have different meanings. He wrote his answers on the blackboard.
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Vincent Teo(a) He answered the questions on / at the blackboard.(edited to fix formatting only)
(b) She was happy (on) that day.
(c) She writes/ sends letters to her pen pal by emails.
(d) They /Their both hobby is collecting stamps.
Vincent Teo(a) He answered the questions on / at the blackboard.
"At the blackboard" refers to his location.
"On the blackboard" could refer to either the location of the questions or the location of his answers.
(b) She was happy (on) that day.