Q35 / 75 · JEE Main 2025 (23 Jan Shift 2) · Mathematics

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

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

Solution

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

Hence, the correct answer is option: 'C'.

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