Question

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 $\left| \begin{matrix} 5 & 4 & 3 \\ x51 & y41 & z31 \\ x & y & z \end{matrix} \right|$ is

• A. $x + y + z$
• B. $x - y + z$
• C. 0
• D. None of these

Answer 1

Answer: (C)

0

Answer 2

$\because x51 = 100x + 50 + 1$,

$y41 = 100y + 40 + 1$

$z31 = 100z + 30 + 1$

$\therefore\Delta = \left| \begin{matrix} 5 & 4 & 3 \\ 100x + 50 + 1 & 100y + 40 + 1 & 100z + 30 + 1 \\ x & y & z \end{matrix} \right|$Applying $R_{2} \rightarrow R_{2} - 100R_{3} - 10R_{1}$

5 & 4 & 3 \\ 1 & 1 & 1 \\ x & y & z \end{matrix} \right| = x - 2y + z$$ $\because$ x, y, z are in A.P. , $\therefore$ $x - 2y + z = 0$, $\therefore$ $\Delta = 0$