QJEEM24Mab599: JEE Main 2024 Mathematics - Class 12 JEEM24Mab599 Solution: Let the system of equations have infinite number of solutions.…

Question

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Let the system of equations

x+2 y+3 z=5,2 x+3 y+z=9,4 x+3 y+\lambda z=\mu

have infinite number of solutions. Then

\lambda+2 \mu

is equal to :

Accepted Answer

\begin{aligned}&amp; x+2 y+3 z=5 \\\\&amp; 2 x+3 y+z=9 \\\\&amp; 4 x+3 y+\lambda z=\mu\end{aligned}

For infinite following

\Delta=\Delta_1=\Delta_2=\Delta_3=0

\begin{aligned} &amp; \Delta=\left|\begin{array}{lll}1 &amp; 2 &amp; 3 \\ 2 &amp; 3 &amp; 1 \\ 4 &amp; 3 &amp; \lambda\end{array}\right|=0 \Rightarrow \lambda=-13 \\\\ &amp; \Delta_1=\left|\begin{array}{llc}5 &amp; 2 &amp; 3 \\ 9 &amp; 3 &amp; 1 \\ \mu &amp; 3 &amp; -13\end{array}\right|=0 \Rightarrow \mu=15 \\\\ &amp; \Delta_2=\left|\begin{array}{ccc}1 &amp; 5 &amp; 3 \\ 2 &amp; 9 &amp; 1 \\ 4 &amp; 15 &amp; -13\end{array}\right|=0\end{aligned}

\Delta_3=\left|\begin{array}{ccc}1 &amp; 2 &amp; 5 \\2 &amp; 3 &amp; 9 \\4 &amp; 3 &amp; 15\end{array}\right|=0

For

\lambda=-13, \mu=15

system of equation has infinite solution hence

\lambda+2 \mu=17

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