The correct answer is - B. While Padmi looks frantically for her pen, her sister calmly writes with it.
Clause 1, Padmi looks frantically for her pen, is dependent here because it is a subordinate clause which cannot exist on its own in this context. Clause 2 is independent here, meaning that it doesn't depend on Clause 1.
A and D are incorrect because both Clauses are independent here, C is incorrect because Clause 1 is independent, and Clause 2 is dependent, which is not what you need here.