Question

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

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

Answer 1

Answer: (A), (C)

The correct options are
A $1$
C $-\frac{5}{3}$

Here, the equations are in two variables $x$ and $y$. If they are consistent then the values of $x$ and $y$ ontained from first two equations should satisfy the third equation and hence $D=0$

$\Rightarrow \left|\begin{array}{ccc}1& 1& -3\\ 1+\lambda & 2+\lambda & -8\\ 1& -1-\lambda & 2+\lambda \end{array}\right|=0$

Applying $C_2\to C_2-C_1,C_3\to C_3+3C_1$, we get

$\Rightarrow \left|\begin{array}{ccc}1& 0& 0\\ 1+\lambda & 1& -5+3\lambda \\ 1& -2-\lambda & 5+\lambda \end{array}\right|=0$

$\Rightarrow (5+\lambda )+(2+\lambda )(-5+3\lambda )=3{\lambda }^2+2\lambda -5=0$

$\Rightarrow (\lambda -1)(3\lambda +5)=0\Rightarrow \lambda =1,-\frac{5}{3}$