Analysis of the Statements

The goal is to find a single, unique value for x² + y². Let's evaluate the sufficiency of each statement.

• 
  Statement 1 alone ("xy = 5"): This is not sufficient. For example, if x=1 and y=5, then x² + y² = 1 + 25 = 26. But if x=2 and y=2.5, then x² + y² = 4 + 6.25 = 10.25. Since we can get different values, this statement alone is not enough.
  


• 
  Statement 2 alone ("x + y = 10"): This is also not sufficient. For example, if x=1 and y=9, then x² + y² = 1 + 81 = 82. But if x=2 and y=8, then x² + y² = 4 + 64 = 68. Since we can get different values, this statement alone is not enough.
  


• 
  Both Statements Together: This is sufficient. We can use the well-known algebraic identity:
  
  (x + y)² = x² + 2xy + y²
  
  We can rearrange this formula to solve for our target expression:
  
  x² + y² = (x + y)² - 2xy
  

  Now, we can substitute the known values from both statements into this new equation:

  
  x² + y² = (10)² - 2(5)

x² + y² = 100 - 10

x² + y² = 90
  

  Since we can find one unique value, the statements together are sufficient.

  
  

Conclusion: Neither statement is sufficient on its own, but they are sufficient when used together.

Final Answer: The correct option is B.