Explanation

The sentence is about the implications of the activists’ energetic work for some assertion about the woman suffrage and temperance movements. The second blank, however, obscures the nature of that assertion. But it is clear that the “energetic work” could either support an assertion that the two movements were similar, or undermine an assertion that the two movements were opposed. “Supports” is offered as a choice for the first blank (as is the somewhat similar “underscores”), but there is no corresponding term in the second blank, nothing along the lines of “similar” or “compatible.” “Undermines” and “inimical” make for the only meaningful statement.
Thus, the correct answer is undermines (Choice A) and inimical (Choice E).
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Let us look at all logical possibilities.

The assertion = the two movements were:

1. co-existing
2. against each other
3. unrelated

The activists’ energetic work in the service of both woman suffrage and the temperance movement:

1. supports the assertion
2. destroys the assertion
3. ignores the assertion


Given that the activists put energetic work in the service of both movements

it would support the assertion that the two movements were coexisting
OR
it would destroy the assertion that the two movements were against each other

There is no choice for the second blank to suggest that the two movements were coexisting. So the word choice for the first blank cannot be supports. Or underscores.

There is a word choice for the second blank - inimical - to mean that the two movements were against each other. And we do have a word choice for the first blank to convey the meaning of "destroy". And that is undermines.

The word choices predominant and diffuse do not fit it into the above logical possibilities. Neither are there any clear word choices for the first blank which would meaningfully complete the sentence using these two words for the second blank.

And no word choices exist to support the third logical possibility - that of ignoring the assertion that they are unrelated movements.
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Why can't the 'energetic work' of the activist support the assertion that the movements were predominant?
After all, a predominant movement ought to receive more attention and work.

Moreover, how does inimical relate to 'energetic work'? This question is ambivalent.

The activist has worked in two types of movement at a same time frame however we cannot conclude that at the time when the activist was engaged in those two movements that there were only those two predominant movement there could have been other more significant movements.
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I agree with james21. Also, there is no clues to realize that there should be some similarity between the two movements (as amorphous mentioned there are no reasons to support the fact that the two movements are predominant)

I am really confused with some of the GRE questions

answer is C and F

I completely disagree with this line of thought. ALWAYS the sentence has a clue even though our clue is feeble and super tiny

The activists’ energetic work in the service of both woman suffrage and the temperance movement in the late nineteenth century (i)__________ the assertion that the two movements were (ii)__________.

The words energetic work, which work as a specific indication how the work was AND provides a context of the quality of the support to the woman at the time, make our sentence or TOTALLY positive or TOTALLY negative.

I.E. the first and second blanks must go in the same direction. OR the two movements were akin OR were, in contrast, each other.

Now, in the FIRST blank clearly, C is out of scope. So everything boils down to A or B. However, we said that the two blanks must go in the same direction BUT B does not have a counterpart in the second blanks.

So the answer must be undermins and inimical.

ETS is very cunning and astute to craft similar question. If the students do not see it that does not mean the question is flawed.

Ask if somethinng is still unclear to you.

Regards
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Thank you for the explanation, however, I never implied the question was flawed. All that examinees need to learn is to break into the test maker's mind and think in the same way that they do.
Frankly speaking, I severely disagree with your resolution (POE) since eliminating "predominant", because it is not a proper counterpart for "support", is not acceptable( in my opinion ). Although, by struggling a few more hours with the text (and myself), I realized that the only clue that we can grasp to justify the answer is "both" in the highlighted text you mentioned which states that same activists were contributing to "both" movements which infers a sense of alignment between the movements

I am happy for the help Sir.

Regards
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If you describe something as diffuse, you mean that it is vague and difficult to understand or explain.
So answer A,D

Just to take "truffaldino from bergamo" and then we understand that energetic work on service is possible even on different hosts. So there are only reason related with understanding. So diffuse is correct in all sences.

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I cannot understand why these 2 movements cannot coexist if it took place in reality in the history of many countries? Why we should count that these 2 events are against each other - prohibition era and female rights? Based on history, in reality, both movements were highly supported by activists, were coexisted and were predominant.
So why?

Every question is different. While in some questions, the meaning is extremely important, in some others, elimination is more important. For instance, in this case, elimination is just as important as the sentence itself. The point is, if we choose option B or option C for Blank (i), none of the options in Blank (ii) create a pair that makes any logical sense. With 'support', we would need an option similar to 'symbiotic'.
So, the only possible option that fits in the 1st blank is option A. Now, looking at the 2nd blank, again, options D and F don't make logical sense with 'undermines'. So, based on elimination, it must be option E.
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Diffuse means lacking concentration, so since they were supported energetically, it undermined the assertion that both events were diffuse. (Not concentrated or focused).
Can someone clarify?

The person was energetic in his work, and not the events. You cannot compare apples with oranges.
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not an astonishing position. That's because GRE is a nonsense exam ) we are going to study PhD, and we get stock in such a rubbish exam

GRE apparently has no sense.
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What would you choose if prompt goes like this Carcass

The activists` prodigious activity on behalf of both woman suffrage and the temperance movement in the late nineteenth century belies the assertion that the two movements were __________.
A influential
B controversial
C inimical
D elitist
E transitory
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Using belie A,B, and C could fit the bill alike.

The questions without the second blank choices could be ambiguous
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