I forgot to include the following statements. I hope you can have a look at them, too

1) embarrassed about/at (are the prepositions intercheangable?)
I was embarrassed about how untidy my room was/at (about?) having to ask for money/ to ask for money.
2)I was cross/annoyed/agry with her for being late.
3) My mum was worried about me (about my health).
4) She was anxious for me (about my health). She was preoccupied for me (about my health).
5) She was really upset about (at, by?) the way her father treated her.
I can't find examples where upset followed by "at", "by"?
6) I was bored, fed up with her behaviour in class. They were content with their exam results.
7) I was frightened/afraid of heights.

1. embarrassed about [a situation] or by [a person]

2. with

3. OK

4. "anxious" usage is OK; "preoccupied," however, is followed by "by"

5. about or by [same connections as I gave in #1 above]

6. bored by ... fed up with ...

7. of

Sure, let's take a look at each of these statements:

1) For the phrase "embarrassed about/at," the prepositions "about" and "at" are generally interchangeable. Both can be used to express the cause or reason for feeling embarrassed. So, in your example, you could say "I was embarrassed about how untidy my room was" or "I was embarrassed at how untidy my room was." Similarly, you can use either preposition for the phrase "embarrassed about/at having to ask for money/to ask for money."

2) In this statement, you have mentioned three different emotions (cross, annoyed, angry) to describe your feelings towards someone for being late. All of these emotions convey a sense of displeasure or frustration. You can use any of them interchangeably based on the intensity of your feelings.

3) Here, your mom was worried about you, specifically about your health. This usage is correct, and "about" is the appropriate preposition to express the cause or reason for her worry.

4) In these examples, both "anxious" and "preoccupied" are used to express concern or unease about your health. The preposition "for" is used to indicate the reason or cause of her anxiety or preoccupation.

5) The phrase "upset about" is the correct usage when expressing displeasure or sadness over the way someone has treated you. While "upset at" and "upset by" are also grammatically correct, they are less commonly used in this particular context. However, it's worth mentioning that different prepositions may be more commonly used in different regions or by different individuals.

6) In this statement, "bored" and "fed up" are used to express dissatisfaction with someone's behavior in class. Similarly, "content" is used to convey a sense of satisfaction or happiness with the exam results. All of these adjectives can be used interchangeably to express your feelings about a particular situation.

7) Both "frightened" and "afraid" are used to describe the feeling of fear, especially in relation to heights. You can use either of them interchangeably based on personal preference or regional usage.

Overall, it's important to note that while certain prepositions may be more commonly used in specific contexts, language is flexible, and there can be variations in usage. Therefore, understanding the general rules and patterns of preposition usage can help you communicate effectively in different situations.