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

(D) $(-\infty,-9) \cup[-8, \infty)$

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Matrices

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, 3 x + 2 y + 5 z = 3, 9 x + 4 y + (28 + [λ]) z = [λ]

has a solution is:

Answer: (C)

Write

n = [λ]

. The determinant of the coefficient matrix is

139124 15 28 + n = - (n + 9) .

For

n \neq = - 9

the determinant is non-zero, so the system has a unique solution. For

n = - 9

the third equation becomes

9 x + 4 y + 19 z = - 9

; the operations

R_{2} \to R_{2} - 3 R_{1}

and

R_{3} \to R_{3} - 9 R_{1}

give

- y + 2 z = - 9

and

- 5 y + 10 z = - 45

, the second being five times the first, so the system remains consistent. A solution therefore exists for every real value of

λ

JEE Main · 2021 · 27 Aug 2021 · Shift 2 · Single correct option