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### Q6: The maximum area \( A \) of the gate is (in m²)

This is a multiple-choice question with four options:
- \( A = 75 \)
- \( B = 75 \sqrt{3} \)
- \( C = 75 \sqrt{3}/2 \)
- \( D = 75/2 \)

Without more context or information about the gate, it's hard to calculate the exact maximum area. If there is additional information, please provide it for a detailed solution.

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### Q7: If the total number of non-empty relations from set \( A \) to set \( B \) are \( (2^{a} - 2^b) \), then what is the value of \( a + b \)?

From the information in the question:
- Let \( A = \{P, G\} \), the set of players, so the size of set \( A \) is \( |A| = 2 \).
- Let \( B = \{1, 2, 3, 4, 5, 6\} \), the set of possible outcomes, so the size of set \( B \) is \( |B| = 6 \).

The number of non-empty relations from set \( A \) to set \( B \) is given as:
\[
2^{a} - 2^{b}
\]
Where \( a \) is the size of set \( A \) and \( b \) is the size of set \( B \). So, \( a = 2 \) and \( b = 6 \).

Substituting into the equation:
\[
2^{a} - 2^{b} = 2^2 - 2^6 = 4 - 64 = -60
\]
Thus, the value of \( a + b = 2 + 6 = 8 \).

However, the problem asks for the value corresponding to \( a + b \), and the options do not match. Double-check the wording or steps provided in the image for any correction or context.

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### Q8: How many functions are possible from set \( B \) to set \( A \)?

To calculate the number of functions from set \( B \) to set \( A \), remember that for each element of set \( B \), we can assign any element from set \( A \).

The number of elements in set \( A \) is 2, and the number of elements in set \( B \) is 6. Therefore, the number of possible functions is:
\[
2^6 = 64
\]
Thus, the answer is \( 64 \), which corresponds to option \( C \).

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Let me know if you need any further clarification or additional details on any of these questions. 

Here are 5 related questions that could expand your understanding:
1. How is the number of relations between two sets generally calculated?
2. What is the difference between a function and a relation in set theory?
3. How can the area of a geometrical shape like a gate be determined if given different parameters like height and width?
4. Why do powers of two play a significant role in set theory problems?
5. Can you explain how non-empty subsets differ from normal subsets?

**Tip**: For set-theory questions, always pay attention to the size of the sets and how functions or relations between them are defined.

Q6: The maximum area A of the gate is (in m²). Q7: If the total number of non-empty relations from set A to set B are (2^a - 2^b), then what is the value of (a + b)? Q8: How many functions are possible from set B to set A?

This solution explores set theory concepts by calculating the number of relations and functions between two sets A and B, alongside a geometry question regarding the maximum area of a gate. This problem is suitable for students in Grades 10-12.

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