Let λ ∈ R. the system of linear equations
Question:

Let $\lambda \in \mathrm{R}$. The system of linear equations

$2 x_{1}-4 x_{2}+\lambda x_{3}=1$

$x_{1}-6 x_{2}+x_{3}=2$

$\lambda x_{1}-10 x_{2}+4 x_{3}=3$

is inconsistent for :

1. exactly one negative value of $\lambda$

2. exactly one positive value of $\lambda$.

3. every value of $\lambda$.

4. exactly two values of $\lambda$.

Correct Option: 1

Solution:

$D=\left|\begin{array}{ccc}2 & -4 & \lambda \\ 1 & -6 & 1 \\ \lambda & -10 & 4\end{array}\right|$

$=2(3 \lambda+2)(\lambda-3)$

$\mathrm{D}_{1}=-2(\lambda-3)$

$\mathrm{D}_{2}=-2(\lambda+1)(\lambda-3)$

$\mathrm{D}_{3}=-2(\lambda-3)$

When $\lambda=3$, then

$\mathrm{D}=\mathrm{D}_{1}=\mathrm{D}_{2}=\mathrm{D}_{3}=0$

$\Rightarrow$ Infinite many solution

when $\lambda=-\frac{2}{3}$ then $\mathrm{D}_{1}, \mathrm{D}_{2}, \mathrm{D}_{3}$ none of them

is zero so equations are inconsistant

$\therefore \lambda=-\frac{2}{3}$