(Đề 2018 câu 37, Mã 101)

Đáp án:

- Chọn hệ trục \( Ixyz \) như hình vẽ, xem cạnh hình vuông \( a = 1 \).  
- \( M(0, 0, \frac{1}{6}), D'(\frac{\sqrt{2}}{2}, 0, 0), C'(0, \frac{\sqrt{2}}{2}, 0) \).  

- Pt mp \((MC'D')\): \( \frac{x}{\frac{\sqrt{2}}{2}} + \frac{y}{\frac{\sqrt{2}}{2}} + \frac{z}{\frac{1}{6}} = 1 \Rightarrow \sqrt{2}x + \sqrt{2}y + 6z - 1 = 0. \)

\( A(0, -\frac{\sqrt{2}}{2}, 1) \Rightarrow \overrightarrow{AM} = (0, \frac{\sqrt{2}}{2}, -\frac{5}{6}) \)  
\( B(-\frac{\sqrt{2}}{2}, 0, 1) \Rightarrow \overrightarrow{BM} = (\frac{\sqrt{2}}{2}, 0, -\frac{5}{6}) \)  

\( \vec{n}_{(MAB)} = [\overrightarrow{AM}, \overrightarrow{BM}] = \left( \frac{-5\sqrt{2}}{12}, \frac{-5\sqrt{2}}{12}, -\frac{1}{12} \right),  \vec{n}_{(MCD)} = (\sqrt{2}, \sqrt{2}, 6) \)  

\( \cos \psi = \frac{|\vec{n}_{(MAB)} \cdot \vec{n}_{(MCD)}|}{|\vec{n}_{(MAB)}| |\vec{n}_{(MCD)}|} = \frac{\left| \frac{-10}{12} - \frac{10}{12} - 3 \right|}{\sqrt{\frac{136}{12}} \sqrt{40}} = \frac{56}{\sqrt{136 \times 40}} \)  

\( \cos \varphi = \frac{7}{\sqrt{85}} = \frac{7\sqrt{85}}{85} \Rightarrow \boxed{B}. \)