Verified Solution • 2025–26 Session moderate

QJEEM23Ma5f06: JEE Main 2023 Mathematics - Class 12 JEEM23Ma5f06 Solution: For the system of linear equations , which of the following is…

For the system of linear equations

2x−y+3z=52x - y + 3z = 5

3x+2y−z=73x + 2y - z = 7

4x+5y+αz=β4x + 5y + \alpha z = \beta ,

which of the following is NOT correct?

Multiple Choice Options

  • A The system has infinitely many solutions for α=−6\alpha=-6 and β=9\beta=9
  • B The system has a unique solution for α\alpha ≠\ne −5-5 and β=8\beta=8
  • C The system is inconsistent for α=−5\alpha=-5 and β=8\beta=8
  • D The system has infinitely many solutions for α=−5\alpha=-5 and β=9\beta=9

Step-by-Step Verified Solution

Given system of linear equation is

2x−y+3z=53x+2y−z=74x+5y+αz=β Now, Δ=∣2amp;−1amp;33amp;2amp;−14amp;5amp;α∣=7(α+5)\begin{gathered}2 x-y+3 z=5 \\\\3 x+2 y-z=7 \\\\4 x+5 y+\alpha z=\beta \\\\\text { Now, } \Delta=\left|\begin{array}{ccc}2 & -1 & 3 \\3 & 2 & -1 \\4 & 5 & \alpha\end{array}\right|=7(\alpha+5)\end{gathered}

So, this system of equation has unique solution, if α≠−5\alpha \neq-5

and Δ1=∣5amp;−1amp;37amp;2amp;−1βamp;5amp;α∣=17α−5β+30\Delta_1=\left|\begin{array}{ccc}5 & -1 & 3 \\ 7 & 2 & -1 \\ \beta & 5 & \alpha\end{array}\right|=17 \alpha-5 \beta+30

and Δ2=∣2amp;5amp;33amp;7amp;−14amp;βamp;α∣=−11β+α+104\Delta_2=\left|\begin{array}{ccc}2 & 5 & 3 \\ 3 & 7 & -1 \\ 4 & \beta & \alpha\end{array}\right|=-11 \beta+\alpha+104

and Δ3=∣2amp;−1amp;53amp;2amp;74amp;5amp;β∣=7(β−9)\Delta_3=\left|\begin{array}{ccc}2 & -1 & 5 \\ 3 & 2 & 7 \\ 4 & 5 & \beta\end{array}\right|=7(\beta-9)

For infinitely many solutions,

Δ=Δ1=Δ2=Δ3=0\Delta=\Delta_1=\Delta_2=\Delta_3=0

amp; So, Δ=0amp;⇒7(α+5)=0amp;⇒α=−5\begin{aligned}& \text { So, } \Delta=0 \\\\& \Rightarrow 7(\alpha+5)=0 \\\\& \Rightarrow \alpha=-5\end{aligned}

amp;Δ3=0amp;⇒7(β−9)=0amp;⇒β=9\begin{aligned}& \Delta_3 =0 \\\\& \Rightarrow 7(\beta-9) =0 \\\\& \Rightarrow \beta =9\end{aligned}

If α=−5\alpha=-5 and β=8\beta=8 , then Δ\Delta equals zero but Δ3\Delta_3 does not, which would imply the system is inconsistent for α=−5\alpha=-5 and β=8\beta=8 .

Therefore, the option "The system is inconsistent for α=−5\alpha=-5 and β=8\beta=8 " is correct.

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