21.

$\mathrm{If} \underset{\mathrm{x}\to 1}{\lim }\frac{\mathrm{x} +{\mathrm{x}}^2 + {\mathrm{x}}^3 +\hspace{0.17em}... + {\mathrm{x}}^{\mathrm{n}} - \mathrm{n}}{\mathrm{x} - 1} = 820, \left(\mathrm{n} \in \mathrm{N}\right) \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{n} =?$

The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x² + y² – 2x – 4y + 4 = 0 at two distinct points is .... 

If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ...

$$\begin{gathered} \mathrm{Let} \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}} \mathrm{be} \mathrm{three} \mathrm{unit} \mathrm{vectors} \mathrm{such} \mathrm{that} {\left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{c}}\right|}^2 + {\left|\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{c}}\right|}^2 = 8. \\ \mathrm{Then} {\left|\overrightarrow{\mathrm{a}} + 2\overrightarrow{\mathrm{b}}\right|}^2 + {\left|\overrightarrow{\mathrm{a}} + 2\overrightarrow{\mathrm{c}}\right|}^2 = ? \end{gathered}$$

$\mathrm{The} \mathrm{integral} {\int }_0^2\left|\left|\mathrm{x} - 1\right| - \mathrm{x}\right|d\mathrm{x} = ?$