To solve this problem, we need to understand the relationship between beat frequency, tension, and frequency of the strings.$\\$Given:$\\$• Frequency of string A, $f_A = 530$ Hz$\\$• Beat frequency when strings A and B are played together is $6$ Hz$\\$• Beat frequency increases to $7$ Hz when tension in B is decreased.$\\$Let's denote:$\\$• Original frequency of string B as $f_B \\$• New frequency of string B after decreasing tension as $f_B' \\$The beat frequency is the absolute difference between the frequencies of the two strings:$\\ |f_A - f_B| = 6$ Hz$\\$This gives us two possibilities:$\\ f_B = f_A + 6 \quad$or$\quad f_B = f_A - 6 \\$Substituting the given frequency of A:$\\ f_B = 530 + 6 = 536$ Hz$\quad$or$\quad f_B = 530 - 6 = 524$ Hz$\\$When the tension in B is decreased, the frequency of B decreases, and the beat frequency becomes $7$ Hz.$\\$If $f_B = 536$ Hz, then $f_B' < 536$ Hz, and the new beat frequency is:$\\ |530 - f_B'| = 7$ Hz$\\$This would imply $f_B' = 537$ Hz or $f_B' = 523$ Hz, which is not possible since $f_B' < 536$ Hz.$\\$Therefore, the original frequency of B cannot be 536 Hz.$\\$If $f_B = 524$ Hz, then $f_B' < 524$ Hz, and the new beat frequency is:$\\ |530 - f_B'| = 7$ Hz$\\$This implies $f_B' = 523$ Hz or $f_B' = 537$ Hz. Since $f_B' < 524$ Hz, $f_B' = 523$ Hz.$\\$Thus, the original frequency of B is $524$ Hz.$\\$Therefore, the correct option is Option 1: 524 Hz.