If the system of equations |
x + y + z = 5 |
x + 2y + 3z = 9 |
\[x+3y+az=\beta \] |
has infinitely many solutions, then p - a equals: |
A) 21
B) 8
C) 18
D) 5
Correct Answer: B
Solution :
\[x+y-z=5\] |
\[x+2y+3z=9.\] |
\[x+3y+az=\beta \] |
\[D=\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 0 \\ \end{matrix} \right|=0\] |
\[\Rightarrow \] \[(2\alpha -9)+(3-\alpha )+(3-2)=0\] |
\[\Rightarrow \] \[\alpha =5\] |
Now, \[{{D}_{3}}=\left| \begin{matrix} 1 & 1 & 5 \\ 1 & 2 & 9 \\ 1 & 2 & \beta \\ \end{matrix} \right|=0\] |
\[\Rightarrow \] \[2\beta -27+9\beta -5(3-2)=0\] |
\[\Rightarrow \] \[\beta =13\] |
\[\Rightarrow \] at \[\alpha =5,~\beta =13\]above 3 planes from common line. |
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