Given:
\( 2pq - 20 = 52 - 2pq \)
\( \Rightarrow 4pq = 72 \)
\( \Rightarrow pq = 18 \) ...... (1)
Consider:
\( p^2 + q^2 - 29 = 2pq - 20 \)
\( \Rightarrow p^2 + q^2 - 2pq = 9 \)
\( \Rightarrow (p - q)^2 = 9 \)
\( \Rightarrow p - q = \pm 3 \)
From (1), \( pq = 18 \).
Using the identity \( p^2 + q^2 = 2pq + 9 \):
\( \Rightarrow p^2 + q^2 = 2(18) + 9 = 36 + 9 = 45 \).
Using the identity \( p^3 - q^3 = (p - q)(p^2 + pq + q^2) \):
\( p^2 + q^2 = 45 \) and \( pq = 18 \), so \( p^2 + pq + q^2 = 45 + 18 = 63 \).
Case 1: \( p - q = 3 \)
\( p^3 - q^3 = 3 \cdot 63 = 189 \).
Case 2: \( p - q = -3 \)
\( p^3 - q^3 = (-3) \cdot 63 = -189 \).
Difference between the two values:
\( 189 - (-189) = 189 + 189 = \mathbf{378} \).
Final Answer: (B) 378