JEE MAIN - Mathematics (2023 - 11th April Evening Shift - No. 3)
If the system of linear equations
$$ \begin{aligned} & 7 x+11 y+\alpha z=13 \\\\ & 5 x+4 y+7 z=\beta \\\\ & 175 x+194 y+57 z=361 \end{aligned} $$
has infinitely many solutions, then $$\alpha+\beta+2$$ is equal to :
6
4
5
3
Explanation
Given,
$$ \begin{aligned} & 7 x+11 y+\alpha z=13 \\\\ & 5 x+4 y+7 z=\beta \\\\ & 175 x+194 y+57 z=361 \end{aligned} $$
$$ \text { For infinite solution, }\left|\begin{array}{ccc} 7 & 11 & \alpha \\ 5 & 4 & 7 \\ 175 & 194 & 57 \end{array}\right|=0 $$
$$ \Rightarrow\left|\begin{array}{ccc} 7 & 11 & \alpha \\ 5 & 4 & 7 \\ 0 & -81 & 57-25 \alpha \end{array}\right|=0 $$
$$ \begin{aligned} & \Rightarrow 81(49-5 \alpha)+(57-25 \alpha)(-27)=0 \\\\ & \Rightarrow 270 \alpha=-2430 \Rightarrow \alpha=-9 \end{aligned} $$
And $\Delta_1=0$
$$ \begin{aligned} & \left|\begin{array}{ccc} 13 & 11 & -9 \\ \beta & 4 & 7 \\ 361 & 194 & 57 \end{array}\right|=0 \\\\ & \Rightarrow \beta=11 \end{aligned} $$
$$ \therefore \alpha+\beta+2=4 $$
$$ \begin{aligned} & 7 x+11 y+\alpha z=13 \\\\ & 5 x+4 y+7 z=\beta \\\\ & 175 x+194 y+57 z=361 \end{aligned} $$
$$ \text { For infinite solution, }\left|\begin{array}{ccc} 7 & 11 & \alpha \\ 5 & 4 & 7 \\ 175 & 194 & 57 \end{array}\right|=0 $$
$$ \Rightarrow\left|\begin{array}{ccc} 7 & 11 & \alpha \\ 5 & 4 & 7 \\ 0 & -81 & 57-25 \alpha \end{array}\right|=0 $$
$$ \begin{aligned} & \Rightarrow 81(49-5 \alpha)+(57-25 \alpha)(-27)=0 \\\\ & \Rightarrow 270 \alpha=-2430 \Rightarrow \alpha=-9 \end{aligned} $$
And $\Delta_1=0$
$$ \begin{aligned} & \left|\begin{array}{ccc} 13 & 11 & -9 \\ \beta & 4 & 7 \\ 361 & 194 & 57 \end{array}\right|=0 \\\\ & \Rightarrow \beta=11 \end{aligned} $$
$$ \therefore \alpha+\beta+2=4 $$
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