Values of mathrm{m}, mathrm{n} , for which system of equations x+y+z=4 2 x+5 y+5 z=17 x+2 y+m z=n ha

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3 and 4 .Determinants and Matrices

hard

The values of $\mathrm{m}, \mathrm{n}$, for which the system of equations

$ x+y+z=4 $

$ 2 x+5 y+5 z=17 $

$ x+2 y+m z=n$

has infinitely many solutions, satisfy the equation :

A

$m^2+n^2-m-n=46$

B

$m^2+n^2+m+n=64$

C

$\mathrm{m}^2+\mathrm{n}^2+\mathrm{mn}=68$

D

$m^2+n^2-m n=39$

(JEE MAIN-2024)

Solution

$\mathrm{D}=\left|\begin{array}{ccc}1 & 1 & 1 \\ 2 & 5 & 5 \\ 1 & 2 & \mathrm{~m}\end{array}\right|=0 \Rightarrow \mathrm{m}=2$

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

Standard 12

Mathematics

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