Question

The value of $\lambda$ for which the system of equations $2x-y-z=12,x-2y+z=-4,x+y+\lambda z=4$ has no solution is

• A. 3
• B. -3
• C. 2
• D. -2

Answer 1

The correct option is D -2

Here, $D=\left|\begin{array}{ccc}2& -1& -1\\ 1& -2& 1\\ 1& 1& \lambda \end{array}\right|$

$=2(-2\lambda -1)+1(\lambda -1)-1(1+2)$

$=-3\lambda -6$
For no solution $D=0$

$\Rightarrow \lambda =-2$
Now, $D_1=\left|\begin{array}{ccc}2& -1& 12\\ 1& -2& -4\\ 1& 1& 4\end{array}\right|$

$=2(-8+4)+1(4+4)+12(1+2)$
$\Rightarrow D_1=36\ne 0$

So, $D=0$ and at least one of $D_1\ne 0$

Hence, there is no solution for $\lambda =-2$