21.

• 0

• 1

• $- \frac{1}{2}$

$\mathrm{If} \mathrm{f}\left(0\right) = 0, \mathrm{f}\left(1\right) = 1, \mathrm{f}\hspace{0.17em}\left(2\right) = 2 \mathrm{and} \mathrm{f}\left(\mathrm{x}\right) = \mathrm{f}\left(\mathrm{x} - 2\right) + \mathrm{f}\left(\mathrm{x} - 3\right) \mathrm{for} \mathrm{x} = 3, 4, 5, . . . , \mathrm{then} \mathrm{f}\left(9\right) = ?$

• 12

• 13

• 14

• 10

The numbers a_n = 6ⁿ - 5n for n = 1, 2, 3, . . . when divided by 25 leave the remainder

• 9

• 7

• 3

• 1

Let n = 1! + 4! + 7! + . . . + 400!. Then ten's digit of n is

• 1

• 6

• 2

• 7

Let a = $\frac{{10}^{\mathrm{n}}}{\mathrm{n}!}$ for n = 1, 2, 3 . . . then the greatest  value of n for which a_n  is the greatest is

• 11

• 20

• 10

• 8

A polygon has 54 diagonals. Then, the number of its sides is

• 7

• 9

• 10

• 12

$\mathrm{If} (1 + 2\mathrm{x} + 3{\mathrm{x}}^2{)}^{10} = {\mathrm{a}}_0 + {\mathrm{a}}_1\mathrm{x} + {\mathrm{a}}_2{\mathrm{x}}^2 + . . . + {\mathrm{a}}_{20}{\mathrm{x}}^{20}, \mathrm{then} \frac{{\mathrm{a}}_2}{{\mathrm{a}}_1} = ?$

• 10.5

• 21

• 10

• 5.5

$\mathrm{For} \left|\mathrm{x}\right| < \frac{1}{5}, \mathrm{the} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^3 \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \frac{1}{\left(1 - 5\mathrm{x}\right)\left(1 - 4\mathrm{x}\right)} \mathrm{is}$

• 369

• 370

• 371

• 372

${\log }_4\left(2\right) - {\log }_8\left(2\right) + {\log }_{16}\left(2\right) - . . = ?$

• e²

• ${\log }_{\mathrm{e}}\left(2\right)$

• $1 + {\log }_{\mathrm{e}}\left(3\right)$

• $1 - {\log }_{\mathrm{e}}\left(2\right)$

$\mathrm{For} \mathrm{x} \in \mathrm{R}, \mathrm{the} \mathrm{least} \mathrm{value} \mathrm{of} \frac{{\mathrm{x}}^2 - 6\mathrm{x} + 5}{{\mathrm{x}}^2 + 2\mathrm{x} + 1} \mathrm{is}$

• - 1

• $- \frac{1}{2}$

• $- \frac{1}{4}$

• $- \frac{1}{3}$