First slide

System of linear equations

Question

If the system of linear equations $x + 2 a y + a z = 0, x + 3 b y + b z = 0, x + 4 c y + c z = 0$ has a non-zero solution, then a,b,c

Moderate

A

are in G.P
B

are in H.P
C

$satisfy a + 2 b + 3 c = 0$
D

are in A.P

Solution

$Given system of equations has a non-zero solution |\begin{array}{lll} 1 2 a a \\ 1 3 b b \\ 1 4 c c \end{array}|$
$\begin{aligned} C_{2} - 2 C_{1} \\ \Rightarrow |\begin{array}{ccc} 1 & 0 & a \\ 1 & b & b \\ 1 & 2 c & c \end{array}| = 0 \end{aligned}$
$\Rightarrow 2 a c = a b + b c \Rightarrow b = \frac{2 a c}{a + c} \Rightarrow a, b, c are in H . P$

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