Question 1

Both II and III must be true.

Since \( |x| = |y| \) and \( x = y \), it follows that both \( x \) and \( y \) must be equal in magnitude and sign. This leads to the conclusion that \( xy < 0 \) cannot be true, while \( x^2 = y^2 \) must hold.

A) III only

This option suggests that only \( x^2 = y^2 \) is true. While it is indeed true that \( x^2 = y^2 \) given \( |x| = |y| \) and \( x = y \), this option fails to consider that \( xy < 0 \) is not applicable under these conditions, making this option incomplete.

B) I and II only

This choice states that both \( x + y = 0 \) and \( xy < 0 \) are true. However, since \( x = y \), the equation \( x + y = 0 \) cannot be true unless both \( x \) and \( y \) are zero, which contradicts the requirement that \( xy < 0 \). Thus, this option is incorrect.

C) I and III only

This option claims that both \( x + y = 0 \) and \( x^2 = y^2 \) are true. While \( x^2 = y^2 \) is indeed true, \( x + y = 0 \) cannot be true if \( x = y \) unless both are zero, which is not a necessary condition. Therefore, this option is also incorrect.

D) II and III

This option correctly identifies that \( xy < 0 \) and \( x^2 = y^2 \) must be true. Given that \( x = y \), it follows that both conditions are satisfied, making this the only correct combination of statements.

Conclusion

The correct answer is D because both \( xy < 0 \) and \( x^2 = y^2 \) must be true provided the initial conditions. Options A, B, and C each fail to satisfy the requirements presented in the question, thus reinforcing the validity of option D as the only correct choice.