Let λ ∈ R . system of linear equations 2 x_{1}-4 x_{2}+λ x_{3}=1 x_{1}-6 x_{2}+x_{3}=2 λ x_{1}-

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3 and 4 .Determinants and Matrices

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Let $\lambda \in R .$ 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

A

exactly one negative value of $\lambda$.

B

exactly one positive value of $\lambda$.

C

every value of $\lambda$.

D

exactly two values of $\lambda$.

(JEE MAIN-2020)

Solution

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

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

$D_{1}=-2(\lambda-3)$

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

$D_{3}=-2(\lambda-3)$

When $\overline{\lambda=3}$, then

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

$\Rightarrow$ Infinite many solution

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

is zero so equations are inconsistant

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

Standard 12

Mathematics

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