grammar - “Although in poor health, she continued...” vs “No matter how poor her health, she continued…”

The following is grammar question from an English as a second language exam

My girlfriend was showing me a copy of the exam she took several years ago after finishing high school. We were looking at the questions for fun, and one question was to pick the phrase that completed the sentence correctly.

a. Although in poor health, she continued to carry out her duties.

b. No matter how poor her health, she continued to carry out her duties.

The italicized portions are the choices for how to correctly complete the sentence. Of the four options, two were easily eliminated. The last two are above. I cannot decide which is correct.

I have a suspicion the first is correct. However, the prepositional phrase "in poor health" doesn't seem correct without a noun before it. It seems better to say "Although she was in poor health, she continued to carry out her duties."

If I were to say the second aloud, it feels more natural to add was, saying "No matter how poor her health was, she continued to carry out her duties."

Is it possible both are correct? This question seems pretty difficult for a high school student learning English as a second language.

EDIT: According to this exam's answer key, a is the correct answer.

Answer

You say that "the prepositional phrase 'in poor health' doesn't seem correct without a noun before it. It seems better to say 'Although she was in poor health, she continued to carry out her duties'."

There is support for your contention in The Right Word at the Right Time: A Guide to the English Language and How to Use It (p38). In its entry on although, though it states:

...(although) is still preferred at the beginning of a sentence -
provided that it introduces a full factual clause: 'Although I was
angry, I kept silent. ...if the verb is omitted from the clause, then
though is preferable: Though angry I kept silent.

This is the first time I encountered this particular differentiation of though/although. And I can find no corroboration in other descriptive grammars or style guides. In fact, the differentiation is contradicted in The Cambridge Grammar Of The English Language (p736):

Although and though are alternants, the latter slightly more informal. Their complement may be a full content clause or it may be
reduced to a participial and verbless clause.

• Although elected to the Council he can't take up his seat.

I would agree that this is a very unfair question for an English language learner. And what is more, the two sentences have different meanings:

Although in poor health, she continued to carry out her duties = She
was in poor health; nevertheless, she continued to carry out her
duties.

No matter how poor her health, she continued to carry out her duties =
It didn't matter whether her health was poor, very poor or terrible;
she still continued to carry out her duties.