Relations and Functions

If : R $\to$ R and g : R $\to$ R are defined by f(x) = x - [x] and g(x) = [x] for x $\in$ R, where[x] is the greatest integer not exceeding x, then for every x $\in$ R, f(g(x)) is equal to

• x

• 0

• f(x)

• g(x)

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

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

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

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

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

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

• {0, 1}

• {1, 2}

• {- 3, - 2}

• {2, 3}

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

• 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

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

• e

• e² + e

• e²

• e² - e

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

• R - {0}

• R - {0, 1, 3}

• R - {0, - 1, - 3}

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

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

• 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

• $- \frac{1}{2}$

• $- \frac{1}{4}$

• $- \frac{1}{3}$