If the set of equations $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+a z=b$ has unique solution, then

If the set of equations $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+a z=b$ has unique solution, then
  1. $a=8, b=15$
  2. $a \neq 8, b \in \mathrm{R}$
  3. $a=8, b \neq 15$
  4. $a \neq 15, b=8$

Solution

For unique solution $\left|\begin{array}{lll}1 & 2 & 3 \\ 1 & 3 & 5 \\ 2 & 5 & a\end{array}\right| \neq 0$ $\begin{aligned} & \Rightarrow 3 a-25-2(a-10)+3(5-6) \neq 0 \\ & \Rightarrow a-8 \neq 0 \Rightarrow a \neq 8 \text { and } b \in R . \end{aligned}$

Asked in: AP EAMCET 2024 (22 May Shift 1)

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