| \[x+y+z=5\] \[x+2y+3z=9\] \[x+3y+\alpha z=\beta \] |
| has infinitely many solutions, then \[\beta -\alpha \] equals- |
A) 8
B) 21
C) 18
D) 5
Correct Answer: A
Solution :
\[\Delta =\left| \begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix}\,\,\,\,\,\,\begin{matrix} 1 \\ 2 \\ 3 \\ \end{matrix}\,\,\,\,\,\,\begin{matrix} 1 \\ 3 \\ \alpha \\ \end{matrix} \right|=2\alpha +3+3-2-9-\alpha =0\] \[\Rightarrow \,\,\,\alpha =5\] Here \[{{P}_{1}}+{{P}_{3}}=2{{P}_{2}}\] For \[\infty \] solution \[\frac{5+\beta }{2}=9\,\,\Rightarrow \,\beta =13\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\therefore \,\,\,\beta -\alpha =8\]
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