Let $\lambda \varepsilon \mathrm{R}$. The system of linear equations. $2 x_{1}-4 x_{2}+\lambda x_{3}=1$ $x_{1}-6 x_{2}+x_{3}=2$ $\lambda \mathrm{x}_{1}-10 \mathrm{x}_{2}+4 \mathrm{x}_{3}=3$ Is inconsistent for:

(B) exactly one negative value of $\lambda$

(A) exactly one positive value of $\lambda$

(D) exactly two values of $\lambda$

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Matrices

Let

λ ε R

. The 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} - 10 x_{2} + 4 x_{3} = 3

Is inconsistent for:

Answer: (B)

The system fails to have a solution when its coefficient determinant vanishes but the equations do not agree. Expanding,

D = 21 λ - 4 - 6 - 10 λ 14 = (λ - 3) (3 λ + 2)

, which is zero at

λ = 3

and

λ = - \frac{2}{3}

. At

λ = 3

the equations stay consistent, whereas at

λ = - \frac{2}{3}

one of the replaced determinants is non-zero, so the system is inconsistent there. That is exactly one negative value of

λ

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Single correct option