If the four points, whose position vectors are $3\widehat{\mathrm{i}}-4\widehat{\mathrm{j}}+2\widehat{\mathrm{k}},\widehat{\mathrm{i}}+2\widehat{\mathrm{j}}-\widehat{\mathrm{k}},-2\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}\text{ and }5\widehat{\mathrm{i}}-2\alpha \widehat{\mathrm{j}}+4\widehat{\mathrm{k}}$ are coplanar then $\alpha$is equal to

$\overrightarrow{\mathrm{OA}}=3\widehat{\mathrm{i}}-4\widehat{\mathrm{j}}+2\widehat{\mathrm{k}};\overrightarrow{\mathrm{OB}}=\widehat{\mathrm{i}}+2\widehat{\mathrm{j}}-\widehat{\mathrm{k}};\overrightarrow{\mathrm{OC}}=-2\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+3\widehat{\mathrm{k}};\overrightarrow{\mathrm{OD}}=5\widehat{\mathrm{i}}-2\alpha \widehat{\mathrm{j}}+4\widehat{\mathrm{k}}$

$\overrightarrow{\mathrm{AB}}=\widehat{\mathrm{i}}+2\widehat{\mathrm{j}}-\widehat{\mathrm{k}}-3\widehat{\mathrm{i}}+4\widehat{\mathrm{j}}-2\widehat{\mathrm{k}}=-2\widehat{\mathrm{i}}+6\widehat{\mathrm{j}}-3\widehat{\mathrm{k}}$

$\overrightarrow{\mathrm{AC}}=-2\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}-3\widehat{\mathrm{i}}+4\widehat{\mathrm{j}}-2\widehat{\mathrm{k}}=-5\widehat{\mathrm{i}}+3\widehat{\mathrm{j}}-\widehat{\mathrm{k}}$

$\overrightarrow{\mathrm{AD}}=5\widehat{\mathrm{i}}-2\alpha \widehat{\mathrm{j}}+4\widehat{\mathrm{k}}-3\widehat{\mathrm{i}}+4\widehat{\mathrm{j}}-2\widehat{\mathrm{k}}=-2\widehat{\mathrm{i}}+\left(-2\alpha +4\right)+2\widehat{\mathrm{k}}$

$\mathrm{ABCD}\text{ are coplanr }\left[\begin{array}{ccc}\overrightarrow{\mathrm{AB}}& \overrightarrow{\mathrm{AC}}& \overrightarrow{\mathrm{AD}}\end{array}\right]=0$

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

$-2(6+2\alpha -4)-6(-10-2)-3(10\alpha -20-6)=0$

$-2(2+2\alpha )-6(-12)-3(10\alpha -26)=0$

$=4-4\alpha +72-30\alpha +78=0$

$-34\alpha +146=0$

$-34\alpha =-146$

$\alpha =\frac{146}{34}=\frac{73}{17}$