$\mathrm{If} {\mathrm{a}}^{\mathrm{x}} = {\mathrm{b}}^{\mathrm{y}} = {\mathrm{c}}^{\mathrm{z}} = {\mathrm{d}}^{\mathrm{w}}, \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{x}\left(\frac{1}{\mathrm{y}} +\hspace{0.17em}\frac{1}{\mathrm{z}} + \frac{1}{\mathrm{w}}\right) \mathrm{is}$

• ${\log }_{\mathrm{a}}\left(\mathrm{abc}\right)$

• ${\log }_{\mathrm{a}}\left(\mathrm{bcd}\right)$

• ${\log }_{\mathrm{b}}\left(\mathrm{cda}\right)$

• ${\log }_{\mathrm{c}}\left(\mathrm{dab}\right)$

If $\frac{3\mathrm{x}}{\left(\mathrm{x} - \mathrm{a}\right)\left(\mathrm{x} - \mathrm{b}\right)} = \frac{2}{\mathrm{x} - \mathrm{a}} + \frac{1}{\mathrm{x} - \mathrm{b}}$, then a : b is equal to

• 1 : 2

• - 2 : 1

• 1 : 3

• 3 : 1

$$\begin{gathered} \mathrm{If} : \mathrm{R}\to \mathrm{R} \mathrm{and} \mathrm{g} : \mathrm{R}\to \mathrm{R} \mathrm{are} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x}\right| \mathrm{and} \\ \mathrm{g}\left(\mathrm{x}\right) = \left[\mathrm{x} - 3\right] \mathrm{for} \mathrm{x} \in \mathrm{R}, \mathrm{then} \\ \left\{\mathrm{g}\left(\mathrm{f}\left(\mathrm{x}\right)\right) : - \frac{8}{5} < \mathrm{x} < \frac{8}{5}\right\} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• {0, 1}

• {1, 2}

• {- 3, - 2}

• {2, 3}

$$\begin{gathered} \mathrm{f} : [-6, 6] \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^2 - 3 \\ \mathrm{for} \mathrm{x} \in \mathrm{R}, \mathrm{then} \\ \left(\mathrm{fofof}\right) (-1) + \left(\mathrm{fofof}\right) \left(0\right) + \left(\mathrm{fofof}\right)\left(1\right) \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\mathrm{f}\left(4\sqrt{2}\right)$

• $\mathrm{f}\left(3\sqrt{2}\right)$

• $\mathrm{f}\left(2\sqrt{2}\right)$

• $\mathrm{f}\left(\sqrt{2}\right)$

$$\begin{gathered} \mathrm{Given} \mathrm{that} \mathrm{a}, \mathrm{b} \in \left\{0, 1, 2, . . . , 9\right\} \mathrm{with} \\ \mathrm{a} +\hspace{0.17em}\mathrm{b} \ne 0 \mathrm{and} \mathrm{that} {\left(\mathrm{a} + \frac{\mathrm{b}}{10}\right)}^{\mathrm{x}} = {\left(\frac{\mathrm{a}}{\mathrm{b}} + \frac{\mathrm{b}}{100}\right)}^{\mathrm{y}} = 1000. \mathrm{Then}, \\ \frac{1}{\mathrm{x}} - \frac{1}{\mathrm{y}} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• 1

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$

$\mathrm{If} \mathrm{x} = \frac{1}{2}\left(\sqrt{7} + \frac{1}{\sqrt{7}}\right), \mathrm{then} \frac{\sqrt{{\mathrm{x}}^2 - 1}}{\mathrm{x} - \sqrt{{\mathrm{x}}^2 - 1}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 1

• 2

• 3

• 4

$\mathrm{If} \frac{{\mathrm{x}}^2 + \mathrm{x} + 1}{{\mathrm{x}}^2 + 2\mathrm{x} + 1} = \mathrm{A} +\hspace{0.17em}\frac{\mathrm{B}}{\mathrm{x} +\hspace{0.17em}1} + \frac{\mathrm{C}}{{\left(\mathrm{x} + 1\right)}^2}, \mathrm{then} \mathrm{A} - \mathrm{B} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 4C

• 4C + 1

• 3C

• 2C

$\sum _{\mathrm{k} = 1}^{\infty }\frac{1}{\mathrm{k}!}\left(\sum _{\mathrm{n} = 1}^{\mathrm{k}}{2}^{\mathrm{n} - 1}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• e

• e² + e

• e²

• e² - e

$$\begin{gathered} \mathrm{If} \mathrm{f} : \left[2, 3\right]\to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f} \left(\mathrm{x}\right) = {\mathrm{x}}^3 + 3\mathrm{x} - 2, \mathrm{then} \mathrm{the} \mathrm{range} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \\ \mathrm{contained} \mathrm{in} \mathrm{the} \mathrm{interval} \end{gathered}$$

• [1, 12]

• [12, 34]

• [35, 50]

• [- 12, 12]

$\left\{\mathrm{x} \in \mathrm{R} :\hspace{0.17em}\frac{2\mathrm{x} - 1}{{\mathrm{x}}^3 +\hspace{0.17em}4{\mathrm{x}}^2 +\hspace{0.17em}3\mathrm{x}} \in \mathrm{R}\right\} \mathrm{equals}$

• R - {0}

• R - {0, 1, 3}

• R - {0, - 1, - 3}

• R - $\left\{0, - 1, - 3, +\frac{1}{2}\right\}$