$$\begin{gathered} \mathrm{Let} \mathrm{PQ} \mathrm{be} \mathrm{a} \mathrm{diameter} \mathrm{of} \mathrm{the} \mathrm{circle} {\mathrm{x}}^2 + {\mathrm{y}}^2 = 9. \mathrm{If} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{are} \mathrm{the} \mathrm{lengths} \mathrm{of} \mathrm{the} \mathrm{perpendiculars} \mathrm{from} \\ \mathrm{P} \mathrm{and} \mathrm{Q} \mathrm{on} \mathrm{the} \mathrm{straight} \mathrm{line}, \mathrm{x} + \mathrm{y} = 2 \mathrm{respectively}, \mathrm{then} \mathrm{the} \mathrm{maximum} \mathrm{value} \mathrm{of} \mathrm{\alpha \beta } \mathrm{is} ..... \end{gathered}$$

If the variance of the following frequency distribution :

Class :      10-20 20-30 30-40

Frequency : 2        x        2

is 50, then x is equal to ....

23.

A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is ......

$$\begin{gathered} \mathrm{Let} \left\{\mathrm{x}\right\} \mathrm{and} \left[\mathrm{x}\right] \mathrm{denote} \mathrm{the} \mathrm{fractional} \mathrm{part} \mathrm{of} \mathrm{x} \mathrm{and} \mathrm{the} \mathrm{greatest} \mathrm{integer} \mathrm{r} \le \mathrm{x} \mathrm{respectively} \mathrm{of} \mathrm{a} \mathrm{real} \mathrm{number} \mathrm{x}. \\ \mathrm{if} {\int }_0^{\mathrm{n}}\left\{\mathrm{x}\right\}d\mathrm{x} \mathrm{and} 10({\mathrm{n}}^2 – \mathrm{n}), (\mathrm{n} \in \mathrm{N}, \mathrm{n} > 1) \mathrm{are} \mathrm{three} \mathrm{consecutive} \mathrm{terms} \mathrm{of} \mathrm{a} \mathrm{G}.\mathrm{P}. \mathrm{then} \mathrm{n} \mathrm{is} \mathrm{equal} \mathrm{to} ...... \end{gathered}$$

$\mathrm{If} \overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} {\left|\hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right)\right|}^2 + {\left|\hat{\mathrm{j}}\left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}\right)\right|}^2 + {\left|\hat{\mathrm{k}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right)\right|}^2 = ?$