Question

If the three planes px + 4y + z = 0, 2y + 3z – 1 = 0 and 3x – qz + 2 = 0 have a common line then the value of p + q is___

Answer 1

Since the three plane meet at one line, the given system of equation have infinite no. of solutions
$\Delta =\left|\begin{array}{ccc}p& 4& 1\\ 0& 2& 3\\ 3& 0& -q\end{array}\right|=0\Rightarrow pq=15$
${\Delta }_1=\left|\begin{array}{ccc}0& 4& 1\\ 1& 2& 3\\ -2& 0& -q\end{array}\right|=0\Rightarrow q=5\Rightarrow p=3$