If the system of equations 2 x - y + z =4 5 x+λ y+3 z=12 100 x-47 y+μ z=212 , has infinitely many solutions, then μ-2 λ is equal to

Mathematics · Matrices

JEE Main 2025 — 24 January, Morning Shift — Question 20

If the system of equations

$2 \mathrm{x}-\mathrm{y}+\mathrm{z}=4$

$5 x+\lambda y+3 z=12$

$100 x-47 y+\mu z=212$ ,

has infinitely many solutions, then $\mu-2 \lambda$ is equal to

\Delta = 0 \Rightarrow \begin{vmatrix} 2 & -1 & 1 \\ 5 & \lambda & 3 \\ 100 & -47 & \mu \end{vmatrix} = 0

2(\lambda \mu + 141) + (5 \mu - 300) - 235 - 100 \lambda = 0 \dots (1)

\Delta_3 = 0 \Rightarrow \begin{vmatrix} 2 & -1 & 4 \\ 5 & \lambda & 12 \\ 100 & -47 & 212 \end{vmatrix} = 0

6 \lambda = -12 \Rightarrow \lambda = -2

\text{Put } \lambda = -2 \text{ in } (1)

2(-2 \mu + 141) + 5 \mu - 300 - 235 + 200 = 0

\mu = 53

\therefore \mu - 2\lambda = 53 + 4 = 57

Answer key and solution verified before publishing.

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