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Question

# Let [λ] be the greatest integer less than or equal to λ. The set of all values of λ for which the system of linear equations x+y+z=4, 3x+2y+5z=3, 9x+4y+(28+[λ])z=[λ] has a solution is

A
(,9)[8,)
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B
[9,8)
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C
R
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D
(,9)(9,)
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Solution

## The correct option is C R x+y+z=4 3x+2y+5z=3 9x+4y+(28+[λ])z=[λ] For unique solution Δ≠0 ∣∣ ∣∣1113259428+[λ]∣∣ ∣∣≠0 ⇒(56+2[λ]−20)−1(84+3[λ]−45)+1(−6)≠0 ⇒36+2[λ]−39−3[λ]−6≠0 ⇒[λ]≠−9 ⇒λ∈(−∞,−9)∪[−8,∞) and if [λ]=−9, Δx=Δy=Δz=0 gives infinite solution. ∴ For λ∈R set of equations have solution.

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