System of linear equations 3 x-2 y-k z=10 ; 2 x-4 y-2 z=6 ; x+2 y-z=5, m is inconsistent if

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3 and 4 .Determinants and Matrices

medium

The system of linear equations $3 x-2 y-k z=10$; $2 x-4 y-2 z=6$ ; $x+2 y-z=5\, m$ is inconsistent if

$k =3, m =\frac{4}{5}$

$k \neq 3, m \in R$

$k \neq 3, m \neq \frac{4}{5}$

$k =3, m \neq \frac{4}{5}$

(JEE MAIN-2021)

Solution

$\Delta=\left|\begin{array}{ccc}3 & -2 & -k \\ 2 & -4 & -2 \\ 1 & 2 & -1\end{array}\right|=0$

$\Rightarrow \quad 24-2(0)- k (8)=0 \Rightarrow k =3$

$\Delta_{ x }=\left|\begin{array}{ccc}10 & -2 & -3 \\ 6 & -4 & -2 \\ 5 m & 2 & -1\end{array}\right|$

$=10(8)-2(-10 m+6)-3(12+20 m)$

$=8(4-5 m )$

$\Delta_{y}=\left|\begin{array}{ccc}3 & 10 & -3 \\ 2 & 6 & -2 \\ 1 & 5 m & -1\end{array}\right|$

$=3(-6+10 m)+10(0)-3(10 m-6)$

$=0$

$\Delta_{z}=\left|\begin{array}{ccc}3 & -2 & 10 \\ 2 & -4 & 6 \\ 1 & 2 & 5 m\end{array}\right|$

$=3(-20 m-12)-2(6-10 m)+10(8)$

$=40 m-32=8(5 m-4)$

for inconsistent

$k=3$ and $m \neq \frac{4}{5}$

Standard 12

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hard

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The system of linear equations $3 x-2 y-k z=10$; $2 x-4 y-2 z=6$ ; $x+2 y-z=5\, m$ is inconsistent if

Solution

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2&b&1 \\
b&{{b^2} + 1}&b \\
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\end{array}} \right]$ where $b > 0$. Then the minimum value of $\frac{{\det \left( A \right)}}{b}$ is