If system of equations mathrm{x}+4 mathrm{y}-mathrm{z}=λ , 7 x+9 y+μ z=-3,5 x+y+2 z=-1 has infinit

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

medium

If the system of equations $\mathrm{x}+4 \mathrm{y}-\mathrm{z}=\lambda$, $7 x+9 y+\mu z=-3,5 x+y+2 z=-1$ has infinitely many solutions, then $(2 \mu .+3 \lambda)$ is equal to :

A

$2$

B

$-3$

C

$3$

D

$-2$

(JEE MAIN-2024)

Solution

$\Delta=\left|\begin{array}{ccc}1 & 4 & -1 \\ 7 & 9 & \mu \\ 5 & 1 & 2\end{array}\right|=0$

$\Rightarrow(18-\mu)-4(14-5 \mu)-(7-45)=0 \Rightarrow \mu=0$

$\Delta=\Delta_{\mathrm{x}}=\Delta_{\mathrm{y}}=\Delta_{\mathrm{z}}=0$ (For infinite solution)

$\Delta_x=\left|\begin{array}{ccc}\lambda & 4 & -1 \\ -3 & 9 & \mu \\ -1 & 1 & 2\end{array}\right|=0$

$ \lambda(18-\mu)-4(-6+\mu)-1(-3+9)=0 $

$ 18 \lambda+24-6=0 \Rightarrow \lambda=-1$

Standard 12

Mathematics

Share

Similar Questions

If $A = \left| {\,\begin{array}{*{20}{c}}{\sin (\theta + \alpha )}&{\cos (\theta + \alpha )}&1\\{\sin (\theta + \beta )}&{\cos (\theta + \beta )}&1\\{\sin (\theta + \gamma )}&{\cos (\theta + \gamma )}&1\end{array}\,} \right|$ ,then

medium

If $\left| {\,\begin{array}{*{20}{c}}{x + 1}&1&1\\2&{x + 2}&2\\3&3&{x + 3}\end{array}\,} \right| = 0,$ then $x$ is

easy

The value of the determinant given below $\left| {{\rm{ }}\begin{array}{*{20}{c}}1&2&3\\3&5&7\\8&{14}&{20}\end{array}} \right|$ is

normal

Let $\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in R$. If $x(\alpha, 1,2)+y(1, \beta, 2)$ $+z(2,3, \gamma)=(0,0,0)$ for some $x, y, z \in R, x y z \neq$ 0 , then $6 \alpha+4 \beta+\gamma$ is equal to…………..

hard

(JEE MAIN-2024)

If $\left| {\begin{array}{*{20}{c}}
{a – b – c}&{2a}&{2a}\\
{2b}&{b – c – a}&{2b}\\
{2c}&{2c}&{c – a – b}
\end{array}} \right|$ $ = \left( {a + b + c} \right)\,{\left( {x + a + b + c} \right)^2}$ , $x \ne 0$ and $a + b + c \ne 0$, then $x$ is equal to

hard

(JEE MAIN-2019)