21.

Let AD and BC be two vertical poles at A and B respectively on a horizontal ground. If AD = 8 m, BC = 11 m and AB = 10 m; then the distance (in meters) of a point M on AB from the point A such that MD² + MC² is minimums is ___

Set A has m elements and Set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m - n is _____

$$\begin{gathered} \mathrm{Let} \mathrm{f} : \mathrm{R} \to \mathrm{R} \mathrm{be} \mathrm{defined} \mathrm{as} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{x}}^5\sin \left(\frac{1}{\mathrm{x}}\right) + 5{\mathrm{x}}^2, \mathrm{x} < 0 \\ 0, x = 0 \\ {\mathrm{x}}^5\cos \left(\frac{1}{\mathrm{x}}\right) + {\mathrm{\lambda x}}^2, \mathrm{x} > 0\right\} \\ \mathrm{The} \mathrm{value} \mathrm{of} \mathrm{\lambda } \mathrm{for} \mathrm{which} \mathrm{f}\text{'}\text{'}\left(0\right) \mathrm{exists}, \mathrm{is} \end{gathered}$$

$\mathrm{If} \overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} \mathrm{are} \mathrm{unit} \mathrm{vectors}, \mathrm{then} \mathrm{the} \mathrm{greatest} \mathrm{value} \mathrm{of} \sqrt{3}\left|\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}\right| + \left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}}\right| = ?$

The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be 45°. After waling a distance of 80 meters towards the top, up a slope inclined at angle of 30° to the horizontal plane the angle of elevation of the top of the hill becomes 75°. Then the height of the hill (in meters) is _____.