Q62 / 110 · VITEEE 2022 · Mathematics

If the system of equations \(x+\lambda y+2=0\), \(\lambda x+y-2=0, \lambda x+\lambda y+3=0\) is consistent, then

  • Correct answer
    A
    \(\lambda= \pm 1\)
  • B
    \(\lambda= \pm 2\)
  • C
    \(\lambda=1,-2\)
  • D
    \(\lambda=-1,2\)

Solution

The system of equations will be consistent if \(\Delta=\left|\begin{array}{ccc} 1 & \lambda & 2 \\ \lambda & 1 & -2 \\ \lambda & \lambda & 3 \end{array}\right|=0\) To evaluate \(\Delta\) we use \(R_1 \rightarrow R_1+R_2\) followed by \(\mathrm{C}_2 \rightarrow \mathrm{C}_2-\mathrm{C}_1\) to obtain \(\begin{aligned} \Delta & =\left|\begin{array}{ccc} \lambda+1 & \lambda+1 & 0 \\ \lambda & 1 & -2 \\ \lambda & \lambda & 3 \end{array}\right|=\left|\begin{array}{ccc} \lambda+1 & 0 & 0 \\ \lambda & 1-\lambda & -2 \\ \lambda & 0 & 3 \end{array}\right| \\ & =3(\lambda+1)(1-\lambda)=3\left(1-\lambda^2\right) \end{aligned}\) For the system to be consistent, we must have \(1-\lambda^2=0\) or \(\lambda= \pm 1\).

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

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