If f : $\mathrm{R} \to \mathrm{R} \mathrm{and} \mathrm{g} :\hspace{0.17em}\mathrm{R} \to \mathrm{R}$ are given by f(x) = $\left|\mathrm{x}\right|$ and g(x) = [x] for each x $\in$ R, then $\left\{\mathrm{x} \in \mathrm{R} : \mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right) \le \mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)\right\}$ is equal to

• $\mathrm{Z} \cup \left(- \infty , 0\right)$

• $\left(- \infty , 0\right)$

• Z

• R

If f(x) = $\frac{1}{\sqrt{\mathrm{x} + 2\sqrt{2\mathrm{x} - 4}}} + \frac{1}{\sqrt{\mathrm{x} - 2\sqrt{2\mathrm{x} - 4}}}$ for x > 2, then f(11) is equal to

• $\frac{7}{6}$

• $\frac{5}{6}$

• $\frac{6}{7}$

• $\frac{5}{7}$

If ${\mathrm{e}}^{\mathrm{f}\left(\mathrm{x}\right)} = \frac{10 + \mathrm{x}}{10 - \mathrm{x}}, \mathrm{x} \in \left(- 10, 10\right)$ and f(x) = $\mathrm{kf}\left(\frac{200\mathrm{x}}{100 + {\mathrm{x}}^2}\right)$, then k is equal to

• 0.5

• 0.6

• 0.7

• 0.8

Let a, b, and c be such that $\frac{1}{\left(1 - \mathrm{x}\right)\left(1 - 2\mathrm{x}\right)\left(1 - 3\mathrm{x}\right)} = \frac{\mathrm{a}}{1 - \mathrm{x}} + \frac{\mathrm{b}}{1 - 2\mathrm{x}} + \frac{\mathrm{c}}{1 - 3\mathrm{x}}$ then, $\frac{\mathrm{a}}{1} + \frac{\mathrm{b}}{3} + \frac{\mathrm{c}}{5}$ is equal to

• $\frac{1}{15}$

• $\frac{1}{6}$

• $\frac{1}{5}$

• $\frac{1}{3}$

$\mathrm{If} 0 < \mathrm{y} < 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} \mathrm{and} \mathrm{x}\left({\mathrm{y}}^3 - 1\right) = 1, \mathrm{then} \frac{2}{\mathrm{x}} + \frac{2}{3{\mathrm{x}}^3} + \frac{2}{5{\mathrm{x}}^5} + ...\mathrm{is} \mathrm{equal} \mathrm{to} :$

• $\log \left(\frac{{\mathrm{y}}^3}{2 - {\mathrm{y}}^3}\right)$

• $\log \left(\frac{{\mathrm{y}}^3}{1 - {\mathrm{y}}^3}\right)$

• $\log \left(\frac{2{\mathrm{y}}^3}{1 - {\mathrm{y}}^3}\right)$

• $\log \left(\frac{{\mathrm{y}}^3}{1 - 2{\mathrm{y}}^3}\right)$

66.

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{N} \to \mathrm{Z} \mathrm{is} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{n}\right) =\left\{ 2 \mathrm{if} \mathrm{n} = 3\mathrm{k}, \mathrm{k} \in \mathrm{Z} \\ 10 \mathrm{if} \mathrm{n} = 3\mathrm{k} + 1, \mathrm{k} \in \mathrm{Z} \\ 0 \mathrm{if} \mathrm{n} = 3\mathrm{k} + 2, \mathrm{k} \in \mathrm{Z}\right\} \\ \mathrm{Then}, \left\{\mathrm{n} \in \mathrm{N} : \mathrm{f}\left(\mathrm{n}\right) > 2\right\} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• {3, 6, 4}

• {1, 4, 7}

• {4, 7}

• {7}

The value of $\sqrt{42 + \sqrt{42 + \sqrt{42 + ...}}}$ is equal to

• 7

• - 6

• 5

• 4

$\mathrm{If} {\log }_{27}\left({\log }_3\left(\mathrm{x}\right)\right) = \frac{1}{3}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{x} \mathrm{is}$

• 3

• 6

• 9

• 27

$\mathrm{x} = \frac{1}{2}\left(\sqrt{3} + \frac{1}{\sqrt{3}}\right), \mathrm{then} \frac{\sqrt{{\mathrm{x}}^2 - 1}}{\mathrm{x} - \sqrt{{\mathrm{x}}^2 - 1}}$

• 1

• 2

• 3

• $\frac{1}{2}$

If a, b, c $\ne$ 0 and belong to the set to {0, 1, 2, 3, ..., 9}, then ${\log }_{10}\left(\frac{\mathrm{a} + 10\mathrm{b} + {10}^2\mathrm{c}}{{10}^{- 4}\mathrm{a} + {10}^{- 3}\mathrm{b} + {10}^{- 2}\mathrm{c}}\right)$ is equal to

• 1

• 2

• 3

• 4