Question

The value of $\lambda$ for which the equations $x+y-3=0,$ $(1+\lambda )x+(2+\lambda )y-8=0$, $x-(1-\lambda )y+(2+\lambda )=0$ are consistent is

• A. $1$
• B. $\frac{5}{3}$
• C. $-\frac{5}{3}$
• D. None of these

Answer 1

Answer: (A), (C)

$1$
$-\frac{5}{3}$

For non-zero solution
$\left|\begin{array}{ccc}1& 1& -3\\ 1+\lambda & 2+\lambda & -8\\ 1& -1-\lambda & 2+\lambda \end{array}\right|$ should be equal to $0$
$(2+\lambda {)}^2-8-8\lambda -(1+\lambda \left)\right(2+\lambda )-8+3(1+\lambda {)}^2+6+3\lambda =0$
On solving,we get
$\lambda =1,\frac{-5}{3}$