Verified Solution • 2025–26 Session moderate

QJEEM25Ma91d0: JEE Main 2025 Mathematics - Class 12 JEEM25Ma91d0 Solution: If the system of equations has infinitely many solutions, then…

If the system of equations

amp;2x−y+z=4amp;5x+λy+3z=12amp;100x−47y+μz=212\begin{aligned}& 2 x-y+z=4 \\& 5 x+\lambda y+3 z=12 \\& 100 x-47 y+\mu z=212\end{aligned}

has infinitely many solutions, then μ−2λ\mu-2 \lambda is equal to

Multiple Choice Options

  • A 56
  • B 59
  • C 57
  • D 55

Step-by-Step Verified Solution

amp;Δ=0⇒∣2amp;−1amp;15amp;λamp;3100amp;−47amp;μ∣=0amp;2(λμ+141)+(5μ−300)−235−100λ=0…(1)amp;Δ3=0⇒∣2amp;−1amp;45amp;λamp;12100amp;−47amp;212∣=0amp;6λ=−12⇒λ=−2amp; Put λ=2 in (1)amp;2(−2μ+141)+5μ−300−235+200=0amp;μ=53amp;∴57\begin{aligned}& \Delta=0 \Rightarrow\left|\begin{array}{ccc}2 & -1 & 1 \\5 & \lambda & 3 \\100 & -47 & \mu\end{array}\right|=0 \\& 2(\lambda \mu+141)+(5 \mu-300)-235-100 \lambda=0 \ldots (1)\\& \Delta_3=0 \Rightarrow\left|\begin{array}{ccc}2 & -1 & 4 \\5 & \lambda & 12 \\100 & -47 & 212\end{array}\right|=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 57\end{aligned}

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