JEE MAIN - Mathematics (2022 - 28th June Morning Shift - No. 2)

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

$$\lambda$$ = 7

$$\lambda$$ = $$-$$7

$$\lambda$$ = 8

$$\lambda$$² = 1

$$\Delta = \left| {\matrix{ 2 & 3 & { - 1} \cr 1 & 1 & 1 \cr 1 & { - 1} & {|\lambda |} \cr } } \right| = 0 \Rightarrow |\lambda | = 7$$

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.