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

• 12

• 13

• 14

• 10

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

• 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

• $- \frac{1}{2}$

• $- \frac{1}{4}$

• $- \frac{1}{3}$

$\left\{\mathrm{x} \in \mathrm{R} : \frac{14\mathrm{x}}{\mathrm{x} + 1} - \frac{9\mathrm{x} - 30}{\mathrm{x} - 4} < 0 = ?\right\}$

• (- 1, 4)

• $\left(1, 4\right) \cup \left(\left(5, 7\right)\right)$

• (1, 7)

• $\left(- 1, 1\right) \cup \left(4, 6\right)$

If a, b and n are natural numbers, then a^{2n - 1}  + b^{2n - 1} is divisible by

• a + b

• a - b

• a³ + b³

• a² + b²

106.

$$\begin{gathered} \frac{{\mathrm{x}}^2 + \mathrm{x} + 1}{\left(\mathrm{x} - 1\right)\left(\mathrm{x} - 2\right)\left(\mathrm{x} - 3\right)} = \frac{\mathrm{A}}{\mathrm{x} - 1} + \frac{\mathrm{B}}{\mathrm{x} - 2} + \frac{\mathrm{C}}{\mathrm{x} - 3} \\ \Rightarrow \mathrm{A} + \mathrm{C} = \end{gathered}$$

• 4

• 5

• 6

• 8

$$\begin{gathered} \mathrm{Let} \mathrm{f} :\hspace{0.17em}\mathrm{R} \to \mathrm{R} \mathrm{be} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{\alpha } + \frac{\sin \left(\mathrm{x}\right)}{\mathrm{x}}, \mathrm{if} \mathrm{x}> 0 \\ 2, \mathrm{if} \mathrm{x} = 0 \\ \mathrm{\beta } + \left[\frac{\sin \left(\mathrm{x}\right) - \mathrm{x}}{{\mathrm{x}}^3}\right], \mathrm{if} \mathrm{x} < 0\right\} \\ \mathrm{where}, \left[\mathrm{x}\right] \mathrm{denotes} \mathrm{the} \mathrm{integral} \mathrm{part} \mathrm{of} \mathrm{x}. \\ \mathrm{If} \mathrm{f} \mathrm{continuous} \mathrm{at} \mathrm{x} = 0, \mathrm{then} \mathrm{\beta } - \mathrm{\alpha } \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• - 1

• 1

• 0

• 2

If a, b, c and d ∈ R such that a² + b² = 4 and ${\mathrm{c}}^2 + {\mathrm{d}}^2 = 4$and if (a + ib) = (c + id)² (x + iy), then x² + y² is equal to

• 4

• 3

• 2

• 1

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = {\left(\mathrm{p} - {\mathrm{x}}^{\mathrm{n}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{n}$}\right.}, \mathrm{p} > 0 \mathrm{and} \mathrm{n} \mathrm{is} \mathrm{a} \mathrm{positive} \mathrm{integer}, \mathrm{then} \mathrm{f}\left[\mathrm{f}\left(\mathrm{x}\right)\right] = ?$

• x

• xⁿ

• p^1/n

• p - xⁿ

If R is the set of all real numbers and $\mathrm{f} : \mathrm{R} - \left\{2\right\} \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \frac{2 +\hspace{0.17em}\mathrm{x}}{2 - \mathrm{x}} \mathrm{for} \mathrm{x} \in \mathrm{R} - \left\{2\right\}$

• R - {- 2}

• R

• R - {1}

• R - {- 1}