A) \[a=8,b\] can be any real number
B) \[b=15,\]a cab be any real number
C) \[a=R-\{8\}\] and \[\operatorname{b}\in \operatorname{R}-[15]\]
D) \[\operatorname{a}=8,b=15\]
Correct Answer: D
Solution :
Given system of equations can be written in matrix form as AX = B where \[A=\left( \begin{matrix} 1 & 2 & 3 \\ 1 & 3 & 5 \\ 2 & 5 & a \\ \end{matrix} \right)\]and \[B=\left( \begin{matrix} 6 \\ 9 \\ b \\ \end{matrix} \right)\] Since, system is consistent and has infinitely many solutions \[\therefore \] (adj. A) B = 0 \[\Rightarrow \] \[\left( \begin{matrix} 3a-25 & 15-2a & 1 \\ 10-a & a-6 & -2 \\ -1 & -1 & 1 \\ \end{matrix} \right)\left( \begin{matrix} 6 \\ 9 \\ b \\ \end{matrix} \right)=\left( \begin{matrix} 0 \\ 0 \\ 0 \\ \end{matrix} \right)\] \[\Rightarrow \] \[-6-9+b=0\Rightarrow b=15\] and \[6(10-a)+9(a-6)-2(b)=0\] \[\Rightarrow \] \[60-60a+9a-54-30=0\] \[\Rightarrow \] \[3a=24\Rightarrow a=8\] Hence, \[a=8,b=15.\]You need to login to perform this action.
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