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If the system of linear equations $x_{1} + 2 x_{2} + 3 x_{3} = 6; x_{1} + 3 x_{2} + 5 x_{3} = 9; 2 x_{1} + 5 x_{2} + {ax}_{3} = b$ is consistent and has infinite number of solutions, then

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a = 8, b can be any real number

b = 15, a can be any real number

$a \in R - [8] and b \in R - [15]$

$a = 8, b = 15$

answer is D.

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Detailed Solution

Make $Δ = 0$
$|\begin{array}{lll} 1 2 3 \\ 1 3 5 \\ 2 5 9 \end{array}| = 0 \Rightarrow a = 8$
Make $Δ_{3} = 0$
$|\begin{array}{lll} 1 2 6 \\ 1 3 9 \\ 2 5 6 \end{array}| = 0 \Rightarrow b = 15$

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