QJEEM22Ma4c4d: JEE Main 2022 Mathematics - Class 12 JEEM22Ma4c4d Solution: If the system of linear equations where, R, has no solution, then

Question

Verified Solution • 2025–26 Session moderate

If the system of linear equations

$2x + 3y - z = - 2$

$x + y + z = 4$

$x - y + |\lambda |z = 4\lambda - 4$

where, $\lambda$ $\in$ R, has no solution, then

Accepted Answer

$Δ = | \begin{matrix} 2 & 3 & - 1 \\ 1 & 1 & 1 \\ 1 & - 1 & | λ | \end{matrix} | = 0 \Rightarrow | λ | = 7$

But at $\lambda = 7,\,{D_x} = {D_y} = {D_z} = 0$

${P_1}:2x + 3y - z = - 2$

${P_2}:x + y + z = 4$

${P_3}:x - y + |\lambda |z = 4\lambda - 4$

So clearly $5{P_2} - 2{P_1} = {P_3}$ , so at $\lambda = 7$ , system of equation is having infinite solutions.

So $\lambda = - 7$ is correct answer.

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