QJEEM23Mab140: JEE Main 2023 Mathematics - Class 12 JEEM23Mab140 Solution: If the system of linear equations has infinitely many…

Question

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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 :

Accepted Answer

Given,

\begin{aligned}&amp; 7 x+11 y+\alpha z=13 \\\\&amp; 5 x+4 y+7 z=\beta \\\\&amp; 175 x+194 y+57 z=361\end{aligned}

\text { For infinite solution, }\left|\begin{array}{ccc}7 &amp; 11 &amp; \alpha \\5 &amp; 4 &amp; 7 \\175 &amp; 194 &amp; 57\end{array}\right|=0

\Rightarrow\left|\begin{array}{ccc}7 &amp; 11 &amp; \alpha \\5 &amp; 4 &amp; 7 \\0 &amp; -81 &amp; 57-25 \alpha\end{array}\right|=0

\begin{aligned}&amp; \Rightarrow 81(49-5 \alpha)+(57-25 \alpha)(-27)=0 \\\\&amp; \Rightarrow 270 \alpha=-2430 \Rightarrow \alpha=-9\end{aligned}

And

\Delta_1=0

\begin{aligned}&amp; \left|\begin{array}{ccc}13 &amp; 11 &amp; -9 \\\beta &amp; 4 &amp; 7 \\361 &amp; 194 &amp; 57\end{array}\right|=0 \\\\&amp; \Rightarrow \beta=11\end{aligned}

\therefore \alpha+\beta+2=4

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