The values of m , 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: [2024]

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Q.
The values of $m, n$ 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: [2024]

1 $m^{2} + n^{2} + m n = 68$

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

3 $m^{2} + n^{2} - m n = 39$

4 $m^{2} + n^{2} + m + n = 64$

Ans.
(3)

Since the system of equations has infinitely many solutions

$∴$ $Δ = 0 and Δ_{3} = 0$

$\Rightarrow$ $| \begin{matrix} 1 & 1 & 1 \\ 2 & 5 & 5 \\ 1 & 2 & m \end{matrix} |$ $= 0$ and $| \begin{matrix} 1 & 1 & 4 \\ 2 & 5 & 17 \\ 1 & 2 & n \end{matrix} |$ $= 0$

$\Rightarrow (5 m - 10) - (2 m - 5) + (- 1) = 0$ and $(5 n - 34) - (2 n - 17) + 4 (- 1) = 0$

$\Rightarrow 3 m - 6 = 0$ and $3 n - 21 = 0$

$\Rightarrow m = 2$ and $n = 7$

Clearly $m^{2} + n^{2} - m n = 4 + 49 - 14 = 39$

Solution

Q.
The values of $m, n$ 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: [2024]

1 $m^{2} + n^{2} + m n = 68$

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

3 $m^{2} + n^{2} - m n = 39$

4 $m^{2} + n^{2} + m + n = 64$

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Solution

Q. The values of m,n for which the system of equationsx+y+z=4, 2x+5y+5z=17, x+2y+mz=n has infinitely many solutions, satisfy the equation: [2024]

1 m2+n2+mn=68

2 m2+n2-m-n=46

3 m2+n2-mn=39

4 m2+n2+m+n=64

Ans. (3) Since the system of equations has infinitely many solutions ∴ Δ=0 and Δ3=0 ⇒|11125512m|=0 and |114251712n|=0 ⇒(5m-10)-(2m-5)+(-1)=0 and (5n-34)-(2n-17)+4(-1)=0 ⇒3m-6=0 and 3n-21=0 ⇒m=2 and n=7 Clearly m2+n2-mn=4+49-14=39

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Q.
The values of $m, n$ 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: [2024]

1 $m^{2} + n^{2} + m n = 68$

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

3 $m^{2} + n^{2} - m n = 39$

4 $m^{2} + n^{2} + m + n = 64$