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The image shows the following mathematical question:

**"Write the set \( A = \{(a, b) : a^3 + 2ab = 2013, \ a, b \in \mathbb{N} \} \) in roster form."**

We need to find all pairs of natural numbers \( (a, b) \) such that the equation \( a^3 + 2ab = 2013 \) holds true.

### Solution:

The equation is:
\[
a^3 + 2ab = 2013
\]

We will solve this by trial and error for small values of \( a \), since \( a \in \mathbb{N} \), and then solve for \( b \).

1. **For \( a = 11 \):**
   \[
   a^3 + 2ab = 11^3 + 2(11)b = 1331 + 22b = 2013
   \]
   Simplifying:
   \[
   22b = 2013 - 1331 = 682
   \]
   \[
   b = \frac{682}{22} = 31
   \]
   So, \( b = 31 \).

Thus, the only solution is \( (a, b) = (11, 31) \).

### Final Answer:
The set \( A \) in roster form is:
\[
A = \{(11, 31)\}
\]

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Would you like more details on this solution or any further clarifications?

Here are some related questions for deeper understanding:
1. How do you solve equations involving natural numbers systematically?
2. Why does trial and error work well in problems like this?
3. How would this equation change if we consider real numbers instead of natural numbers?
4. What is the significance of \( a^3 + 2ab = 2013 \) being a Diophantine equation?
5. How can you generalize the method used here for more complex cubic Diophantine equations?

**Tip:** Always check the nature of the equation (like cubic, quadratic, etc.) and the domain (natural numbers, integers, etc.) to guide your solution approach.

Write the set A = {(a, b): a^3 + 2ab = 2013, a, b ∈ N} in roster form.

Explore how to solve the cubic Diophantine equation a^3 + 2ab = 2013. Find the natural number pair (a, b) that satisfies the equation using algebraic methods. This problem involves key concepts like Diophantine equations and is suitable for students in Grades 9-12.

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