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 –digits 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. x+2y+z

Answer 1

Answer: (C)

0

Answer 2

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

$= \frac{1}{1000}\left| \begin{matrix} 50 & 40 & 30 \\ 100x + 50 + 1 & 100y + 40 + 1 & 100z + 30 + 1 \\ 100x & 100y & 100z \end{matrix} \right|$Applying $R_{2} \rightarrow R_{2} - \left( R_{1} + R_{3} \right)$ & then

$C_{2} \rightarrow C_{2} - C_{1}C_{3} \rightarrow C_{3} - C_{2}$

= $- \frac{1}{1000}\left\{ - 1000(z - y) + 1000(y - x) \right\}$

= $(z + x - 2y)$ = 0 $(\therefore x,y,zareinA.P.)$