The two appropriate uses of email in a professional setting from the options provided are:
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Sending detailed project plans - This is relevant and necessary for collaboration and communication about work-related tasks.
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Communicating with external clients - Email is a standard tool for professional communication with clients, ensuring clear and formal exchanges.
The other options are generally less appropriate for a professional email context. Sharing confidential information and opinions on management can lead to security and professionalism issues, casual conversations are better suited for other platforms like chat applications, and scheduling personal appointments doesn’t typically belong in a professional email context.