Q102 / 160 · WBJEE 2012 · Mathematics

The system of linear equations $\lambda x+y+z=3$ $x-y-2 z=6$ $-x+y+z=\mu$ has

  • A
    infinite number of solutions for $\lambda \neq-1$ and all $\mu$
  • Correct answer
    B
    infinite number of solutions for $\lambda=-1$ and $\mu=3$
  • C
    no solution for $\lambda, x-1$
  • D
    unique solution for $\lambda=-1$ and $\mu=3$

Solution

Augmented matrix [A:B] = $\left[\begin{array}{rrrr}1 & -1 & -2 & 6 \\ -1 & 1 & 1 & \mu \\ \lambda & 1 & 1 & 3\end{array}\right]$ $\sim\left[\begin{array}{cccc}1 & -1 & -2 & 6 \\ 0 & 0 & -1 & \mu+6 \\ \lambda+1 & 0 & -1 & 9\end{array}\right]$ Applying $$ -\left[\begin{array}{cccc}1 & -1 & -2 & R_{2}+R_{1}, R_{3} \rightarrow R_{3}+R_{1} \\ 0 & 0 & -1 & \mu+6 \\ \lambda+1 & 0 & 0 & 3-\mu\end{array}\right] $$ Infinite number of solutions for $\lambda=-1$ and $\mu=3$

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

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