Let [ λ ] be the greatest integer less than or equal to λ . The set...

Given, the system of linear equations,

$x + y + z = 4 . . . . . . . . . . . . . . . . . . . . . . . . \left(i\right)$

$3x + 2y + 5z = 3 . . . . . . . . . . . . . . . . \left(ii\right)$

$9x + 4y + ( 28 + [ \lambda ] ] z = [ \lambda ] . . . . . . . . . . . . . . . . . . . . . \left(iii\right)$

D = \left|\begin{cases}1 & 1 & 1 \\ 3 & 2 & 5 \\ 9 & 4 & 28 + [ \lambda ]\end{cases}\right|

$= 1\left\{56 + 2\left[\lambda\right] - 20\right\} - 1\left\{84 + 3\left[\lambda\right] - 45\right\} + 1\left(12 - 18\right)$

$D = - ( [ \lambda ] + 9 )$

If $D = - \left([ \lambda ] + 9\right) \ne 0$ then unique solution
If $D = - \left([ \lambda ] + 9\right) = 0$ then ${D}_{1} = {D}_{2} = {D}_{3} = 0$ infinite solutions

Hence, $\lambda \in R$