Correct Answer: 83

Solution : a(a + b + c) = 45 ------------------------- (1)
b(a + b + c) = 75 ------------------------- (2)
c(a + b + c) = 105 ------------------------ (3)
Adding (1), (2), and (3), we get (a + b + c) ² = 225
⇒ (a + b + c) = 15 or (a + b + c) = –15
Case 1: (a + b + c) = 15 -------------------(4)
Dividing (4) by (1), we get a = 3
Dividing (4) by (2), we get b = 5
Dividing (4) by (1), we get c = 7
(a ² + b ² + c ² ) = 3 ² + 5 ² + 7 ² = 83
Case 2: (a + b+ c) = –15 -------------(5)
Dividing (5) by (1), we get a = –3
Dividing (5) by (2), we get b = –5
Dividing (5) by (1), we get c = –7
(a ² + b ² + c ² ) = (–3) ² + (–5) ² + (–7) ² = 83
Hence, the correct answer is 83.