The system of equations aligned & x+y+z=6 & x+2 y+5 z=9, & x+5 y+λ z=μ, aligned has no solution if

Correct answer: C. λ=17, μ ≠ 18 aligned & D=| array lll 1 & 1 & 1 1 & 2 & 5 1 & 5 & λ array |=0 & λ=17 & D_z=| array lll 1 & 1 & 6 1 & 2 & 9 1 & 5 & μ array | ≠ 0 & μ ≠ 18 aligned

JEE Main 202523 JanuaryEvening Shift

Exam: JEE Main
Year: 2025
Date: 23 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Determinants

JEE Main 2025 · 23 January · Evening Shift · Mathematics · Determinants

The system of equations aligned & x+y+z=6 & x+2 y+5 z=9, & x+5 y+λ...

The system of equations

\begin{aligned} & x+y+z=6 \\ & x+2 y+5 z=9, \\ & x+5 y+\lambda z=\mu, \end{aligned}

has no solution if

Options

A $\lambda=15, \mu \neq 17$
B $\lambda \neq 17, \mu \neq 18$
C $\lambda=17, \mu \neq 18$
D $\lambda=17, \mu=18$

Correct answer

C. $\lambda=17, \mu \neq 18$

Step-by-step solution

\begin{aligned} & \mathrm{D}=\left|\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 5 \\ 1 & 5 & \lambda\end{array}\right|=0 \\ & \lambda=17 \\ & D_z=\left|\begin{array}{lll}1 & 1 & 6 \\ 1 & 2 & 9 \\ 1 & 5 & \mu\end{array}\right| \neq 0 \\ & \mu \neq 18\end{aligned}

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