JEE Main 2021 (Online) 27th August Evening ShiftQuestion 1
Let $[\lambda]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4$, $3x+2y+5z=3$, $9x+4y+(28+[\lambda])z=[\lambda]$ has a solution is :
Determinant
Solution
Coefficient determinant for $n=[\lambda]$: $\det\begin{pmatrix}1&1&1\\3&2&5\\9&4&28+n\end{pmatrix}=-(n+9)$. If $n\neq-9$ unique solution exists. If $n=-9$: $R_3=3R_1+\ldots$ i.e. $9x+4y+19z=-9$; rank of $A$ and of $[A|b]$ are both $2$ (checked), so it is consistent. So a solution exists for all $\lambda$: $\mathbf R$. Option (C).