JEE Main 2020 (Online) 5th September Morning ShiftQuestion 1
Let $\lambda\in\mathbf{R}$. The system of linear equations $2x_1-4x_2+\lambda x_3=1$, $x_1-6x_2+x_3=2$, $\lambda x_1-10x_2+4x_3=3$ is inconsistent for :
Determinant
Solution
Coefficient determinant $=2(\lambda-3)(3\lambda+2)$, zero for $\lambda=3,-\tfrac23$. $\lambda=3$: rank of coefficient matrix $=2=$ rank of augmented matrix, so consistent. $\lambda=-\tfrac23$: coefficient rank $2$, augmented rank $3$, so inconsistent. Exactly one negative value of $\lambda$. Option (C).