Question 315 of 472MediumMCQ
The system of equations
\[αx+y+z=α−1x+αy+z=α−1x+y+αz=α−1 \]
has no solutions, if \[\alpha \] is : Solutions · 1
Official solution
\[ax + y + z = \alpha - 1\]
\[x + \alpha \,y + z = \alpha - 1;\]
\[x + y + z\alpha = \alpha - 1\]
\[\Delta = \left| {α111α111α } \right|\]
\[ = \alpha \left( {{\alpha ^2} - 1} \right) - 1\left( {\alpha - 1} \right) + 1\left( {1 - \alpha } \right)\]
\[ = \alpha \left( {\alpha - 1} \right)\left( {\alpha + 1} \right) - 1\left( {\alpha - 1} \right) - 1\left( {\alpha - 1} \right)\]
For infinite solutions, \[\Delta = 0\]
\[ \Rightarrow \left( {\alpha - 1} \right)\left[ {{\alpha ^2} + \alpha - 1 - 1} \right] = 0\]
\[ \Rightarrow \left( {\alpha - 1} \right)\left[ {{\alpha ^2} + \alpha - 2} \right] = 0\]
\[ \Rightarrow \alpha = - 2,1;\]
But \[\alpha \ne 1.\,\,\,\] \[\therefore\] \[\,\,\alpha = - 2\] Answer the question to unlock