$\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} :$

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

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

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

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

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

• {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

$\left\{\mathrm{n}\left(\mathrm{n} + 1\right)\left(2\mathrm{n} + 1\right) : \mathrm{n} \in \mathrm{Z}\right\} \subset$

• $\left\{6\mathrm{k} : \mathrm{k} \in \mathrm{Z}\right\}$

• $\left\{12\mathrm{k} : \mathrm{k} \in \mathrm{Z}\right\}$

• $\left\{18\mathrm{k} : \mathrm{k} \in \mathrm{Z}\right\}$

• $\left\{24\mathrm{k} : \mathrm{k} \in \mathrm{Z}\right\}$

If $\frac{{\mathrm{x}}^2}{\left(2\mathrm{x} - 1\right)\left(\mathrm{x} + 2\right)\left(\mathrm{x} - 3\right)} = \mathrm{A} + \frac{\mathrm{B}}{2\mathrm{x} - 1} + \frac{\mathrm{C}}{\mathrm{x} + 2} + \frac{\mathrm{D}}{\mathrm{x} - 3}$, then A is equal to

• $\frac{1}{2}$

• $- \frac{1}{50}$

• $- \frac{8}{25}$

• $\frac{27}{25}$

$\mathrm{If} \mathrm{m}\left[- 3, 4\right] + \mathrm{n}\left[4 - 3\right] = \left[10 - 11\right], \mathrm{then} 3\mathrm{m} + 7\mathrm{n} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 3

• 5

• 10

• 1

80.

If f : R $\to$ R is defined by f(x) = [2x] - 2[x] for x $\in$ R, where [x] is the greatest integer not exceeding x, then the range of f is :

• $\left\{\mathrm{x} \in \mathrm{R} : 0 \le \mathrm{x} \le 1\right\}$

• {0, 1}

• $\left\{\mathrm{x} \in \mathrm{R} : \mathrm{x} > 0\right\}$

• $\left\{\mathrm{x} \in \mathrm{R} : \mathrm{x} \le 0\right\}$