The values of $m, n$, for which the system of equations $\begin{aligned} & x+y+z=4, \\ & 2 x+5 y+5 z=17, \\…

The values of $m, n$, for which the system of equations $\begin{aligned} & x+y+z=4, \\ & 2 x+5 y+5 z=17, \\ & x+2 y+\mathrm{m} z=\mathrm{n} \end{aligned}$ has infinitely many solutions, satisfy the equation:
  1. $m^2+n^2-m n=39$
  2. $m^2+n^2-m-n=46$
  3. $m^2+n^2+m+n=64$
  4. $m^2+n^2+m n=68$

Solution

$\begin{aligned} & \mathrm{D}=\left|\begin{array}{lll}1 & 1 & 1 \\ 2 & 5 & 5 \\ 1 & 2 & \mathrm{~m}\end{array}\right|=0 \Rightarrow \mathrm{m}=2 \\ & \mathrm{D}_3=\left|\begin{array}{ccc}1 & 1 & 4 \\ 2 & 5 & 17 \\ 1 & 2 & \mathrm{n}\end{array}\right|=0 \Rightarrow \mathrm{n}=7\end{aligned}$

Asked in: JEE Main 2024 (05 Apr Shift 2)

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