Question

If $p$ and $q$ are real so that the system of equations $pz+4y+z=0,2y+3z=1$ and $3x–qz=–2$ has infinite solutions then $\sqrt{q^2-p^2}$ is equal to

Answer 1

$D=\left|\begin{array}{ccc}p& 4& 1\\ 0& 2& 3\\ 3& 0& -q\end{array}\right|=0\Rightarrow pq=15$

$D_1=\left|\begin{array}{ccc}0& 4& 1\\ 1& 2& 3\\ -2& 0& -q\end{array}\right|=0\Rightarrow q=5$

$\therefore p=3$

$\therefore \sqrt{q^2-p^2}=\sqrt{25-9}=4$