Question

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 $x+y+z=4,3 x+2 y+5 z=3$, $9 x+4 y+(28+[\lambda]) z=[\lambda]$ has a solution is:

• A. $R$
• B. $(-\infty,-9) \cup(-9, \infty)$
• C. $[-9,-8)$
• D. $(-\infty,-9) \cup[-8, \infty)$

Answer 1

Answer: (A)

$D =\begin{vmatrix}1 & 1 & 1 \\3 & 2 & 5 \\9 & 4 & 28+[\lambda]\end{vmatrix}=-24-[\lambda]+15=-[\lambda]-9$
if $[\lambda]+9 \neq 0$ then unique solution
if $[\lambda]+9=0$ then $D _{1}= D _{2}= D _{3}=0$
so infinite solutions
Hence $\lambda$ can be any red number.