Question

Give that $q^2-pr<0,p>0,$ the value of $\left|\begin{array}{ccc}p& q& px+qy\\ q& r& qx+ry\\ px+qy& qx+ry& 0\end{array}\right|$is

• A. 0
• B. > 0
• C. < 0
• D. $q^2+pr$

Answer 1

Answer: (C)

Let $\Delta =\left|\begin{array}{ccc}p& q& px+qy\\ q& r& qx+ry\\ px+qy& qx+ry& 0\end{array}\right|$
Operate $R_3-x{R}_1-y{R}_2$, we get
$\therefore$ $\Delta =\left|\begin{array}{ccc}p& q& px+qy\\ q& r& qx+ry\\ 0& 0& -(p{x}^2+2qxy+r{y}^2)\end{array}\right|$
= $(q^2-pr)(p{x}^2+2qxy+r{y}^2)$
Since $q^2-pr<0,p>0$
$\therefore$ $p{x}^2+2qxy+r{y}^2>0$ and hence $\Delta$ is -ve