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

• A. $x+y+z$
• B. $x-y+z$
• C. $-(2y-x-z)=0$
• D. $x+2y+z$

Answer 1

Answer: (C)

$-(2y-x-z)=0$

$\left|\begin{array}{ccc}5& 4& 3\\ 100x+50+1& 100y+40+1& 100z+30+1\\ x& y& z\end{array}\right|$

$R_2\to R_2-100R_3-10R_2$

$\left|\begin{array}{ccc}5& 4& 3\\ 1& 1& 1\\ x& y& z\end{array}\right|$

$C_1\to C_1-C_2$

$C_2\to C_2-C_3$

$\left|\begin{array}{ccc}1& 1& 3\\ 0& 0& 1\\ x-y& y-z& z\end{array}\right|$

$-1(y-z-x+y)$

$-(2y-x-z)=0$