Question

Find the value of $\lambda$, if the following equations are consistent
$x+y-3=0$
$(1+\lambda )x+(2+\lambda )y-8=0$
$x-(1+\lambda )y+(2+\lambda )=0$

• A. $1$
• B. $-5/3$
• C. $5/3$
• D. $-1$

Answer 1

Answer: (A), (B)

The correct options are
A $1$
B $-5/3$

The given equations in two unknowns are consistent, then $\Delta =0$
i.e.,
$\left|\begin{array}{ccc}1& 1& -3\\ 1+\lambda & 2+\lambda & -8\\ 1& -(1+\lambda )& 2+\lambda \end{array}\right|=0$
Applying $C_2\to C_2-C_1$ and $C_3\to C_3+3C_1$
$\therefore \left|\begin{array}{ccc}1& 0& 0\\ 1+\lambda & 1& 3\lambda -5\\ 1& -2-\lambda & 5+\lambda \end{array}\right|=0$
$\Rightarrow 1\cdot \left|\begin{array}{cc}1& 3\lambda -5\\ -2-\lambda & 5+\lambda \end{array}\right|=0$
$\Rightarrow (5+\lambda )-(3\lambda -5)(-2-\lambda )=0$
$\Rightarrow 5+\lambda +6\lambda +3{\lambda }^2-10-5\lambda =0$
$\Rightarrow 3{\lambda }^2+2\lambda -5=0$
$\therefore \lambda =1,-5/3$