Question

Suppose$p,q,r\ne 0$ and the system of equations
$\begin{array}{ccc}(p+a)x+by+cz=0& & \\ ax+(q+b)y+cz=0& & \\ ax+by+(r+c)z=0& & \end{array}$
has a non-trivial solution,then the value of $\frac{p}{a}+\frac{q}{b}+\frac{r}{c},$is

• A. -1
• B. 0
• C. 1
• D. 2

Answer 1

Answer: (A)

-1

For non-trivial solution,
$\left|\begin{array}{ccc}p+a& b& c\\ a& q+b& c\\ a& b& r+c\end{array}\right|=0$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$\left|\begin{array}{ccc}p& -q& 0\\ 0& q& -r\\ a& b& r+c\end{array}\right|=0$
$\Rightarrow pqr+pqc+pbr+aqr=0$
$\Rightarrow \frac{c}{r}+\frac{b}{q}+\frac{a}{p}=-1$