Step-by-step Solution:

Determine the characteristics of the relation ( aRb ) if \( a^2 = b^2 \): \[\] Analyzing the given relation:\[\] The relation ( aRb ) is defined by \( a^2 = b^2 \). We will check if it satisfies properties like symmetric, reflexive, transitive, etc.\[\] \[\] 1. Symmetric: The relation is symmetric if \( aRb \implies\) bRa . Here, \( a^2 = b^2 \) \(\implies\) \( b^2 = a^2 \), which is true. Thus, the relation is symmetric.\[\] \[\] 2. Reflexive: The relation is reflexive if \( aRa \) holds for every element ( a ). Substituting \( b = a \), \( a^2 = a^2 \) is always true. Hence, the relation is reflexive.\[\] \[\] 3. Transitive: The relation is transitive if \( aRb \) and \( bRc \) \(\implies\) \( aRc \). If \( a^2 = b^2 \) and \( b^2 = c^2 \), then \( a^2 = c^2 \) follows. Hence, the relation is transitive.\[\] \[\] Final Answer:\[\] The relation is symmetric, reflexive, and transitive.\[\] \[\] Correct Option: (d)