If the system of equations $2 x + 3 y - z = 0, x + k y - 2 z = 0$ and $2 x - y + z = 0$ has a non trivial solution. Then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k =$

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$\frac{1}{2}$

$\frac{3}{2}$

$\frac{5}{2}$

$\frac{7}{2}$

answer is A.

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Detailed Solution

$|\begin{matrix} 2 & 3 & - 1 \\ 1 & k & - 2 \\ 2 & - 1 & 1 \end{matrix}| = 0 \Rightarrow K = \frac{9}{2}$

Given equations are $2 x + 3 y - z = 0 \to (1)$

$x + \frac{9}{2} y - 2 z = 0 \to (2), 2 x - y + z = 0 \to (3)$

$(1) - (3) \Rightarrow 4 y - 2 z = 0 \Rightarrow \frac{z}{y} = 2$

$(1) + (3) \Rightarrow \frac{x}{y} = \frac{- 1}{2} \to (5)$

$(4) \times (5) \Rightarrow \frac{y}{z} \times \frac{x}{y} = \frac{- 1}{4} \Rightarrow \frac{z}{x} = - 4$

$\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k = - \frac{1}{2} + \frac{1}{2} - 4 + \frac{9}{2} = \frac{1}{2}$

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