Verified Solution • 2025–26 Session moderate

QJEEM23Maaea6: JEE Main 2023 Mathematics - Class 12 JEEM23Maaea6 Solution: Consider the following system of equations for some . Then…

Consider the following system of equations

αx+2y+z=1\alpha x+2y+z=1

2αx+3y+z=12\alpha x+3y+z=1

3x+αy+2z=β3x+\alpha y+2z=\beta

for some α,β∈R\alpha,\beta\in \mathbb{R} . Then which of the following is NOT correct.

Multiple Choice Options

  • A It has a solution for all α≠−1\alpha\ne-1 and β=2\beta=2
  • B It has no solution if α=−1\alpha=-1 and β≠2\beta\ne2
  • C It has no solution for α=−1\alpha=-1 and for all β∈R\beta \in \mathbb{R}
  • D It has no solution for α=3\alpha=3 and for all β≠2\beta\ne2

Step-by-Step Verified Solution

D=∣αamp;2amp;12αamp;3amp;13amp;αamp;2∣=0⇒α=−1,3D=\left|\begin{array}{ccc}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & 2\end{array}\right|=0 \Rightarrow \alpha=-1,3

Dx=∣2amp;1amp;13amp;1amp;1αamp;2amp;β∣=0⇒β=2D_{x}=\left|\begin{array}{ccc}2 & 1 & 1 \\ 3 & 1 & 1 \\ \alpha & 2 & \beta\end{array}\right|=0 \Rightarrow \beta=2

Dy=∣αamp;1amp;12αamp;1amp;13amp;2amp;β∣=0D_{y}=\left|\begin{array}{ccc}\alpha & 1 & 1 \\ 2 \alpha & 1 & 1 \\ 3 & 2 & \beta\end{array}\right|=0

Dz=∣αamp;2amp;12αamp;3amp;13amp;αamp;β∣=0D_{z}=\left|\begin{array}{ccc}\alpha & 2 & 1 \\ 2 \alpha & 3 & 1 \\ 3 & \alpha & \beta\end{array}\right|=0

β=2,α=−1\beta=2, \alpha=-1

α=−1,β=2\alpha=-1, \beta=2 Infinite solution

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