Question

Select the correct option from the given alternatives:

The determinant D = `|("a", "b", "a" + "b"),("b", "c", "b" + "c"),("a" + "b", "b" + "c", 0)|` = 0 if

Options

• a, b, c are in A.P.

• a, b, c are in G.P

• a, b, c are in H.P.

• α is root of ax² + 2bx + c = 0

Answer 1

a, b, c are in G.P.

Explanation:

Applying R₃ → R₃ – (R₁ + R₂), we get

`|("a", "b", "a" + "b"),("b", "c", "b" + "c"),(0, 0, -("a" + 2"b" + "c"))|` = 0

∴ a[–c(a + 2b + c) – 0] – b[–b(a + 2b + c) – 0] + (a + b) (0  – 0) = 0

∴ (–ac + b²) (a + 2b + c) = 0

∴ –ac + b² = 0 or a + 2b + c = 0

∴ b² = ac 

∴ a, b, c are in G.P.