Exams Joint Entrance Examination Mathematics Matrices and Determinants Question 412 Question 412 of 472 Medium MCQ
Let the system of linear equations
x + 2 y + z = 2 x + 2y + z = 2 x + 2 y + z = 2 ,
\[\alpha x + 3y - z = \alpha \],
\[ - \alpha x + y + 2z = - \alpha \]
be inconsistent. Then \[\alpha\] is equal to :
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x + 2 y + z = 2 x + 2y + z = 2 x + 2 y + z = 2
\[\alpha x + 3y - z = \alpha \]
\[ - \alpha x + y + 2z = - \alpha \]
\[\Delta = \left| { ( 1 2 1 α 3 − 1 − α 1 2 ) \begin{pmatrix} 1 & 2 & 1 \\
\alpha & 3 & { - 1} \\
{ - \alpha } & 1 & 2 \\
\end{pmatrix} 1 α − α 2 3 1 1 − 1 2 } \right| = 1(6 + 1) - 2(2\alpha - \alpha ) + 1(\alpha + 3\alpha )\]
\[ = 7 + 2\alpha \]
\[\Delta = 0 \Rightarrow \alpha = - {7 \over 2}\]
\[{\Delta _1} = \left| { ( 2 2 1 α 3 − 1 − α 1 2 ) \begin{pmatrix} 2 & 2 & 1 \\
\alpha & 3 & { - 1} \\
{ - \alpha } & 1 & 2 \\
\end{pmatrix} 2 α − α 2 3 1 1 − 1 2 } \right| = 14 + 2\alpha \ne 0\] for \[\alpha = - {7 \over 2}\]
\[\therefore\] For no solution \[\alpha = - {7 \over 2}\] Answer the question to unlock