The sentence tjat effectively combines the clauses to make clause one dependent of clause two is the third option: While Padmi looks frantically for her pen, her sister calmly writes with it.
The clause "while Padmi looks frantically for her pen" could not stand alone as it cannot convey a total thought. I would not make sense, hence it depends of the second clause which conveys the total meaning of the statement. The word while, which is a conjuction, provides as the clue to spot the dependent clause.
Dependent clause :a clause that cannot form a separate sentence but can form a sentence when joined with a main clause