mutinous would mean that someone has gone on a rebellion usually against an AUTHORITY. So, indelicate is a better option.

Just to add one point, the first blank OA is "agog", and from the available option, it can be the only option that could work. But, I wouldn't use the word in this context. Agog has a somewhat positive connotation. It means you are very excited or eager to watch a movie or go on vacation, something of that nature. Agog here has been used negatively or incorrectly. I have not seen the usage of this word in the manner this question has used.

So in this context agog means shocked or astonished ?

Analysis of the Sentence
1. Grandparents' Value: The grandparents "valued seemliness above all else."
- Seemliness means conformity to accepted standards of good taste and propriety; proper behavior.
2. The Granddaughter's Behavior: The behavior is "incorrigibly ________ (ii)."
- Since the grandparents valued propriety, the granddaughter's behavior must be the opposite: improper, unseemly, or inappropriate.
3. Grandparents' Reaction: They were " ______ (i)" at the behavior.
- Since the behavior violates their core value (seemliness), their reaction must be one of disapproval, astonishment, or distress.

Evaluating the Blanks
Blank (ii) - The Behavior (Must mean inappropriate/unseemly):
- (D) mutinous: Refusing to obey the orders of a person in authority. (Relates to rebellion, not specifically lack of propriety.)
- (E) indelicate: Lacking sensitivity, refinement, or propriety; coarse or unseemly. (Excellent fit. This is the direct opposite of seemliness.)
- (F) tortuous: Full of twists and turns; excessively long and complex. (Irrelevant.)
(E) indelicate is the best fit for Blank (ii).

Blank (i) - The Reaction (Must mean disapproval/astonishment, given their strong values):
- (A) loquacious: Talkative. (Irrelevant to their emotional reaction.)
- (B) agog: Highly excited or intensely eager. (Suggests pleasure or anticipation, which contradicts their disapproval of the behavior.)
- (C) nonchalant: Feeling or appearing casually calm and relaxed. (Contradicts the idea that the behavior violates their most valued principle.)

Since the behavior is incorrigibly (unable to be corrected) indelicate, and this violates what they value "above all else," they must be intensely or strongly affected. None of the available options directly mean disapproving or distressed. However, we must choose the best pairing:

Let's re-examine (B) agog. While it usually means excited in a positive sense, it can sometimes be used to mean astonished or overwhelmed with shock. If they valued seemliness above all else, they would certainly be astonished or overwhelmed by a lack of it.

If we look at the standard meaning, none of the options for (i) fit perfectly in a negative sense. However, let's confirm the likely intended pairing:
- Try (C) nonchalant + (E) indelicate: They valued seemliness, yet were nonchalant (calm) at unseemly behavior. (Contradictory to the strong value established.)
- Try (B) agog + (E) indelicate: They valued seemliness, and were intensely astonished/excited (in a negative sense, as in scandalized) at unseemly behavior. (Plausible in context.)

In the context of GRE vocabulary, often when the obvious negative reaction (like "distressed" or "horrified") is missing, and the behavior is extreme ("incorrigibly"), an intense state of shock or amazement, like agog, is intended, implying they were stunned by the extreme unseemliness.

The best and most logical pairing is (B) agog and (E) indelicate.
The completed sentence reads: "Her grandparents valued seemliness above all else, and were agog at her incorrigibly indelicate behavior."