Question

The greatest value of $c\in R$ for which the system of linear equations
$x-cy-cz=0cx-y+cz=0cx+cy-z=0$
has a non-trivial solution, is :

• A. $-1$
• B. $0$
• C. $\frac{1}{2}$
• D. $2$

Answer 1

The correct option is C $\frac{1}{2}$

For non-trivial solution $\Delta =0$
$\Rightarrow \left|\begin{array}{ccc}1& -c& -c\\ c& -1& c\\ c& c& -1\end{array}\right|=0\Rightarrow 1(1-c^2)+c(-c-c^2)-c(c^2+c)=0\Rightarrow 2c^3+3c^2-1=0\Rightarrow (c+1{)}^2(2c-1)=0\Rightarrow c=-1,\frac{1}{2}$

Hence, greatest value of $c=\frac{1}{2}$