If the four points, whose position vectors are 3 i ^ - 4 j ^ + 2 k ^ , i ^ + 2 j ^ - k ^ , - 2 i ^ - j ^ + 3…

Let the four points be $A: (3,-4,2)$, $B: (1,2,-1)$, $C: (-2,-1,3)$ and $D: (5,-2\alpha , 4)$

$A, B, C, D$ are coplanar points, then

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

$\Rightarrow \left|\begin{array}{ccc}-2& 6& -3\\ -5& 3& 1\\ 2& -2\alpha +4& 2\end{array}\right|=0$

$\Rightarrow -3\alpha +146=0$

$\Rightarrow \alpha =\frac{73}{17}$