Q.

If $x, y, z$ are integers in A.P lying between 1 and 9 and $x 51, y 41$ and $z 31$ are three digit number then the value of

$|\begin{matrix} 5 & 4 & 3 \\ x 51 & y 41 & z 31 \\ x & y & z \end{matrix}|$ is

Moderate

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$x + y + z$

$x - y - z$

$x + 2 y + z$

answer is C.

(Detailed Solution Below)

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Detailed Solution

$|\begin{matrix} 5 & 4 & 3 \\ x 51 & y 41 & z 31 \\ x & y & z \end{matrix}| R_{2} \to R_{2} - 100 R_{3} - 10 R_{1} {|\begin{matrix} 5 & 4 & 3 \\ 1 & 1 & 1 \\ x & y & z \end{matrix}|}_{C_{1} \to C_{1} + C_{3} - 2 C_{2}} \Rightarrow |\begin{matrix} 0 & 4 & 3 \\ 0 & 1 & 1 \\ 0 & y & z \end{matrix}| = 0 (x + z = 2 y)$

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