Solve the Following Questions
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:

1. $\mathbf{R}$

2. $(-\infty,-9) \cup(-9, \infty)$

3. $[-9,-8)$

4. $(-\infty,-9) \cup[-8, \infty)$

Correct Option: 1

Solution:

$\mathrm{D}=\left|\begin{array}{llc}1 & 1 & 1 \\ 3 & 2 & 5 \\ 9 & 4 & 28+[\lambda]\end{array}\right|=-24-[\lambda]+15=-[\lambda]-9$

if $[\lambda]+9 \neq 0$ then unique solution

if $[\lambda]+9=0$ then $\mathrm{D}_{1}=\mathrm{D}_{2}=\mathrm{D}_{3}=0$

so infinite solutions

Hence $\lambda$ can be any red number.