**Rule:** The adjective 'alarmed' is followed by 'at' or 'by'. 'Alarmed at' is used for a situation or an event. The company is alarmed at the sudden drop in sales, so 'at' is the correct preposition. 'Alarmed by' is used for the cause of the alarm. Both are possible, but 'at' is more common in this context. Options (c) and (d) are incorrect.

**Rule:** The adjective 'careful' is followed by the preposition 'about' when it means to be cautious or to pay attention to something. He is careful about his health, so 'about' is the correct preposition. Option (b) 'of' can also be used, but 'about' is more common in this context. Option (c) 'with' is used when talking about how something is handled. Option (d) 'for' is grammatically incorrect.

**Rule:** The verb 'enveloped' is followed by the preposition 'in'. 'Enveloped in' means to be covered or surrounded by something. The company is enveloped in a fog of mystery, so 'in' is the correct preposition. Options (b), (c), and (d) are incorrect.

**Rule:** The adjective 'absorbed' is followed by the preposition 'in'. 'Absorbed in' means to be very interested in something and give it all your attention. He was absorbed in his work, so 'in' is the correct preposition. Option (b) 'with' is incorrect. Option (c) 'by' is incorrect. Option (d) 'at' is grammatically incorrect.

**Rule:** The adjective 'anxious' is followed by 'about' when it means to be worried or nervous. She is not anxious about the exam, so 'about' is the correct preposition. 'Anxious for' means to be eager for something. Options (c) and (d) are incorrect.

**Rule:** The verb 'operate' is followed by the preposition 'on'. 'Operate on' is used when a doctor cuts a body for medical reasons. However, in this context, it means to use or control a machine. The company has to operate on the new machine, so 'on' is the correct preposition. Options (b), (c), and (d) are incorrect.

**Rule:** The verb 'matched' is followed by 'by', 'with', or 'to'. 'Matched by' is used to show that something is equal to something else in a particular quality. The company is matched by its rival, so 'by' is the correct preposition. 'Matched with' is used to show that two things go well together. 'Matched to' is used to find something that is suitable for a particular person or purpose. Option (d) is incorrect.

**Rule:** The adjective 'firm' is followed by the preposition 'in'. 'Firm in' one's decision or belief means to be strong and not easily changed. She is firm in her decision, so 'in' is the correct preposition. Options (b), (c), and (d) are incorrect.

**Rule:** The adjective 'replete' is followed by the preposition 'with'. 'Replete with' means to be full of something. The company is replete with talent, so 'with' is the correct preposition. Options (b), (c), and (d) are incorrect.

**Rule:** The adjective 'thankful' is followed by 'to' and 'for'. 'Thankful to' is used for the person you are thankful to. She is thankful to her parents, so 'to' is the correct preposition. 'Thankful for' is used for the reason for being thankful. Options (c) and (d) are incorrect.