If a ≠ b, b, c satisfy |(1,2b,c),(3,b,c),(4,a,b)| = 0, then abc = … (a) a + b + c

Question

If a ≠ b, b, c satisfy \(\begin{vmatrix} a &2b&2c \\[0.3em]3&b&c\\4&a&b\end{vmatrix}\) = 0, then abc = …

(a) a + b + c 

(b) 0 

(c) b³ 

(d) ab + bc

Answer 1

(c) b³

Expanding along R₁

a(b² – ac) – 2b (3b – 4c) + 2c (3a – 4b) = 0

(b² – ac) (a – b) = 0

b² = ac (or) a = b 

⇒ abc = b(b²) = b³