| Let \[x,\,\,\,y,\,\,\,z\] are three integers lying between \[1\] and \[9\] such that \[x\,51,\,\,\,y\,41\] and \[z\,31\] are three digit numbers. |
| Statement-1: The value of the determinant\[\left| \begin{matrix} 5 & 4 & 3 \\ x\,51 & y\,41 & z\,31 \\ x & y & z \\ \end{matrix} \right|is\,\,zero\]. |
| Statement-2: The value of a determinant is zero if the entries in any two rows (or columns) of the determinant are correspondingly proportional. |
A) Statement-1 is false, Statement-2 is true.
B) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
C) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
D) Statement-1 is true, Statement-2 is false.
Correct Answer: D
Solution :
\[\Delta =\left| \begin{matrix} 5 & 4 & 3 \\ x\,51 & y\,41 & z\,31 \\ x & y & z \\ \end{matrix} \right|\] \[=\left| \begin{matrix} 5 & 4 & 3 \\ 100x+51 & 100y+41 & 100z+31 \\ x & y & z \\ \end{matrix} \right|\] \[=\left| \begin{matrix} 5 & 4 & 3 \\ 1 & 1 & 1 \\ x & y & z \\ \end{matrix} \right|\] \[[{{R}_{2}}\to {{R}_{2}}-100{{R}_{3}}-10{{R}_{1}}]\] which is zero provided\[x,\,\,y,\,\,z\]are in\[A.P.\]
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