Question

If the system of linear equations

x + 2ay + az = 0

x + 3by + bz = 0

x + 4cy + cz = 0

has a non-zero solutions, then a, b, c are in-

• A. A.P.
• B. G.P.
• C. H.P.
• D. Satisfy a + 2b + 3c = 0

Answer 1

Answer: (C)

H.P.

Answer 2

The system of linear equation has a non-zero

solution ∆ = $\left| \begin{matrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4c & c \end{matrix} \right|$ = 0

applying R₂ → R₂ – R₁, R₃ → R₃ – R₁

$\left| \begin{matrix} 1 & 2a & a \\ 0 & 3b - 2a & b - a \\ 0 & 4c - 2a & c - a \end{matrix} \right|$ = 0

Solving 2ac = bc + ab

$\frac{2}{b}$= $\frac{1}{a}$+ $\frac{1}{c}$⇒ a, b, c are in H.P.