Let a unit vector $\widehat{\mathrm{OP}}$  make angle $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$  with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where $\mathrm{\beta }\in \left(0,\frac{{\displaystyle \mathrm{\pi }}}{{\displaystyle 2}}\right)$ . If  $\widehat{\mathrm{OP}}$ is perpendicular to the plane though points $(1,2,3), (2,3,4) and (1,5,7),$then which one of the following is true?

Dc’s of OP:  $\mathrm{cos\alpha },\mathrm{cos\beta },\mathrm{cos\gamma }(\mathrm{cos\beta }>0)$
OP is parallel to the normal of the plane containing $$\begin{gathered} \text{ Equation of plane passing through the points} \\ (1,2,3),(2,3,4),(1,5,7) \mathrm{is} \end{gathered}$$ i

$\left|\begin{array}{ccc}x-1& y-2& z-3\\ 1& 1& 1\\ 0& 3& 4\end{array}\right|=0$

$\Rightarrow x-4y+3z=2$

Dr’s of above normal: 1,-4,3 
$\begin{array}{l}\frac{\mathrm{cos\alpha }}{-1}=\frac{\mathrm{cos\beta }}{4}=\frac{\mathrm{cos\gamma }}{-3}\\ \Rightarrow \mathrm{cos\alpha }<0,\mathrm{cos\gamma }<0\end{array}$

$\frac{\pi }{2}<\alpha <\pi ,\frac{\pi }{2}<\gamma <\pi$