**Rule:** The verb 'commenting' is followed by 'on' or 'about'. 'Commenting on' means to express an opinion about something. She is commenting on the latest political developments, so 'on' is the correct preposition. 'Commenting about' is also possible. Both are correct. Options (c) and (d) are incorrect.

**Rule:** The adjective 'conversant' is followed by the preposition 'with'. 'Conversant with' means to be familiar with something. The company is conversant with the latest technology, so 'with' is the correct preposition. Options (b), (c), and (d) are incorrect.

**Rule:** The verb 'counted' is followed by the preposition 'as'. 'Counted as' means to be considered as something. He is counted as one of the greatest scientists, so 'as' is the correct preposition. 'Counted among' is also possible. Both are correct. Options (c) and (d) are incorrect.

**Rule:** The adjective 'displeased' is followed by the preposition 'with'. 'Displeased with' means to be annoyed or unhappy with something. He is displeased with his performance, so 'with' is the correct preposition. Options (b), (c), and (d) are incorrect.

**Rule:** The adjective 'acquainted' is followed by the preposition 'with'. 'Acquainted with' means to be familiar with something. He is acquainted with the latest developments, so 'with' is the correct preposition. Options (b), (c), and (d) are incorrect.

**Rule:** The verb 'know' is followed by 'about' or 'of'. 'Know about' means to have knowledge or information about something. The company has to know about the latest market trends, so 'about' is the correct preposition. 'Know of' means to be aware of the existence of something. Option (b) is less appropriate here. Options (c) and (d) are incorrect.

**Rule:** The adjective 'anxious' is followed by the preposition 'about' when it means to be worried or nervous. She is anxious about the results, so 'about' is the correct preposition. Option (b) 'for' would mean to be eager for something. Option (c) 'with' is incorrect. Option (d) 'of' is grammatically incorrect.

**Rule:** The adjective 'alive' is followed by the preposition 'to'. 'Alive to' means to be aware of and interested in something. The company is alive to the latest market trends, so 'to' is the correct preposition. Options (b), (c), and (d) are incorrect.

**Rule:** The verb 'armed' is followed by the preposition 'with'. 'Armed with' means to be equipped with something. The company is armed with the latest technology, so 'with' is the correct preposition. Options (b), (c), and (d) are incorrect.

**Rule:** The adjective 'aware' is followed by the preposition 'of'. 'Aware of' means to know that something exists, or to have knowledge or experience of a particular thing. He is aware of the latest developments, so 'of' is the correct preposition. Options (b), (c), and (d) are incorrect.