Question

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

• A. $\frac{5}{3}$
• B. $-\frac{5}{3}$
• C. $1$
• D. $-1$

Answer 1

Answer: (B), (C)

$1$

Given system of equations is consistent.
$\therefore \Delta =0$
$\Rightarrow \left|\begin{array}{ccc}1& 1& -3\\ 1+\lambda & 2+\lambda & -8\\ 1& -1-\lambda & 2+\lambda \end{array}\right|=0$
$\text{Applying}{C}_2\to C_2-C_1,C_3\to C_3+3C_1$, we get
$\Rightarrow \left|\begin{array}{ccc}1& 0& 0\\ 1+\lambda & 1& -5+3\lambda \\ 1& -2-\lambda & 5+\lambda \end{array}\right|=0\Rightarrow (5+\lambda )+(2+\lambda )(-5+3\lambda )=0=3{\lambda }^2+2\lambda -5=0\Rightarrow (\lambda -1)(3\lambda +5)=0\Rightarrow \lambda =1,-\frac{5}{3}$