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

Mathematics

$S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x + y + z =1$ ; $x +\lambda y + z =1$ ; $x + y +\lambda z =1$ is inconsistent, then $\sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)$ is equal to

hard

(JEE MAIN-2023)

If the system of linear equations : $x+y+2 z=6$, $2 x+3 y+a z=a+1$, $-x-3 y+b z=2 b$ where $a, b \in R$, has infinitely many solutions, then $7 a+3 b$ is equal to :

medium

(JEE MAIN-2025)

The existence of unique solution of the system of equations, $x+y+z=\beta $ , $5x-y+\alpha z=10$ , $2x+3y-z=6$ depends on

normal

Find area of the triangle with vertices at the point given in each of the following: $(1,0),(6,0),(4,3)$

easy

For which of the following ordered pairs $(\mu, \delta)$ the system of linear equations $x+2 y+3 z=1$ ; $3 x+4 y+5 z=\mu$ ; $4 x+4 y+4 z=\delta$ is inconsistent?