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Q.
If the equations $ax + 4y + z = 0$, $bx + 3y + z=0$, $cx + 2y + z = 0$ have non-trivial solution, then $a, b, c$ are in
Determinants
Solution:
$\left|\begin{matrix}a&4&1\\ b&3&1\\ c&2&1\end{matrix}\right|=0 \Rightarrow a\left(3-2\right)-4 \left(b-c\right)+1\left(2b-3c\right)=0$
$\Rightarrow \quad a-2b+c=0\quad\Rightarrow \quad2b=a+c$
Hence, a, b, c are in A.P.