Matrices & Determinants
Homogeneous system; non-trivial solution condition
Grade Class 12
Question:
If the system of equations $x+2ay+az=0$, $x+3by+bz=0$, $x+4cy+cz=0$ has a non-zero solution, then $a,b,c$
are in A.P.
are in G.P.
are in H.P.
satisfy $a+2b+3c=0$
Step-by-Step Solution
Key Concept: Non-trivial solution ⟺ $\det=0$. Expand: $1(3bc-4bc)-2a(b-c)+a(4c-3b)=0$ ⟹ $-bc+2a(c-b)+a(4c-3b)=0$... simplify to $2/b=1/a+1/c$ ⟹ HP.
$a,b,c$ are in H.P.
Correct Answer: 3