Writing clear and concise answers in Mathematics Optional for UPSC requires a systematic approach to ensure that the examiner understands the solution without any ambiguity. Here are some tips to achieve this:

1. **Understand the question**: Carefully read and comprehend the given question to identify its requirements and constraints. Break it down into smaller parts if necessary.

2. **Organize your answer**: Structure your answer in a logical and coherent manner. Use headings and subheadings to clearly identify different sections of your response.

3. **Start with a concise introduction**: Begin your answer with a brief introduction that provides context and outlines the approach you will take to solve the problem.

4. **State the key formulas and concepts**: Clearly mention the relevant formulas, theorems, and concepts that are necessary to solve the problem. This shows your understanding of the subject matter and provides a foundation for your solution.

5. **Provide a step-by-step solution**: Break down the problem into smaller steps and explain each step in a clear and concise manner. Use bullet points to present each step, making it easier for the examiner to follow your thought process.

6. **Use appropriate mathematical notation**: Utilize mathematical symbols, equations, and notation to express your solution accurately. This helps in reducing unnecessary explanations and makes the answer more concise.

7. **Include necessary justifications**: Provide reasoning and explanations for each step of your solution. This demonstrates your understanding of the underlying concepts and ensures that your answer is well-supported.

8. **Avoid unnecessary elaboration**: While it is important to provide explanations, avoid unnecessary elaboration that can make your answer lengthy. Stick to the relevant points and focus on clarity and precision.

9. **Review and revise**: Once you have completed your answer, review it to ensure that it is clear, concise, and free from errors. Check for any missing steps, incorrect calculations, or unclear explanations.

10. **Practice and time management**: Regular practice is key to developing the skill of writing clear and concise answers. Additionally, during the examination, manage your time effectively to allocate sufficient time for each question and avoid rushing through your answers.

In conclusion, writing clear and concise answers in Mathematics Optional for UPSC requires a systematic approach, effective organization, and a focus on clarity and precision. By following these tips, you can effectively communicate your solutions and increase your chances of scoring well in the examination.