Question

The equations $\lambda x - y = 2, 2x - 3y = - \lambda$ and $3x - 2y = -\lambda$ are consistent for

• A. $\lambda = - 4 $
• B. $\lambda = 1, 4 $
• C. $\lambda = 1, - 4 $
• D. $\lambda = - 1, 4 $
• E. $\lambda = - 1 $

Answer 1

Answer: (D)

In a consistent, the intersection point of two lines, satisfy the third line.
Consider $\lambda=-1$, then given equation become
$-x-y=2$
$2 x-3 y=1$
$\Rightarrow x=-1, y=-1$
Third equation is $3 x-2 y=-1$
Put $x=-1, y=-1$
$\therefore -3+2=-1$
$\Rightarrow -1=-1$, true
Consider $\lambda=4$, then given equation become
$4 x-y=2$
$2 x-3 y=-4$
$\Rightarrow y=2, x=1$
Third equation is $3 x-2 y=-1$
Put $y=2, x=1$
$\therefore 3-4=-1$
$\Rightarrow -1=-1$, true