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Fundamental Principle of Counting

If x,y,z are ...

Question

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 numbers, then the value of
$∣ ∣ ∣ ∣ \begin{matrix} 5 & 4 & 3 x 51 & y 41 & z 31 x & y & z \end{matrix} ∣ ∣ ∣ ∣$ is

x + y + z

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x - y + z

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0

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x + 2 y + z

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Solution

The correct option is C $0$
$∣ ∣ ∣ ∣ \begin{matrix} 5 & 4 & 3 x 51 & y 41 & z 31 x & y & z \end{matrix} ∣ ∣ ∣ ∣ Applying R_{2} \to R_{2} - 100 R_{3} - 10 R_{1} ∣ ∣ ∣ ∣ \begin{matrix} 5 & 4 & 3 1 & 1 & 1 x & y & z \end{matrix} ∣ ∣ ∣ ∣ Applying C_{1} \to C_{1} + C_{3} - 2 C_{2} = ∣ ∣ ∣ ∣ \begin{matrix} 0 & 4 & 3 0 & 1 & 1 0 & y & z \end{matrix} ∣ ∣ ∣ ∣ = 0 (∵ x + z = 2 y)$

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Fundamental Principle of Counting

Standard XII Mathematics

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If x,y,z are integers in A.P. lying between 1 and 9 and x51,y41 and z31 are three digit numbers, then the value of ∣∣ ∣∣543x51y41z31xyz∣∣ ∣∣ is

The correct option is C 0∣∣ ∣∣543x51y41z31xyz∣∣ ∣∣Applying R2→R2−100R3−10R1∣∣ ∣∣543111xyz∣∣ ∣∣Applying C1→C1+C3−2C2=∣∣ ∣∣0430110yz∣∣ ∣∣=0(∵x+z=2y)

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 numbers, then the value of
$∣ ∣ ∣ ∣ \begin{matrix} 5 & 4 & 3 x 51 & y 41 & z 31 x & y & z \end{matrix} ∣ ∣ ∣ ∣$ is