Relations and Functions

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

• (- 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

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

• - 1

• 1

• 0

• 2

$\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

Let Q be the set of all rational numbers in [0, 1]and f: [0, 1] $\to$ [0,1] be defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{x}, \mathrm{for} \mathrm{x} \in \mathrm{Q} \\ 1 - \mathrm{x} \mathrm{for} \mathrm{x}\notin \mathrm{Q}\right\} \\ \mathrm{Then}, \mathrm{the} \mathrm{set} \mathrm{S} = \left\{\mathrm{x} \in \left[0, 21\right] : \left(\mathrm{fof}\right)\left(\mathrm{x}\right)\right\} = ? \end{gathered}$$

• $[0, 1]$

• - Q

• $[0, 1]\; -\; Q$

• (0, 1)

$\mathrm{If} \mathrm{f} : \mathrm{R} \to \mathrm{R}, \mathrm{g} : \mathrm{R} \to \mathrm{R} \mathrm{are} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = 5\mathrm{x} - 3, \mathrm{g}\left(\mathrm{x}\right) = {\mathrm{x}}^2 + 3, \mathrm{then} {\mathrm{gof}}^{ - 1}\left(3\right) = ?$

• $\frac{25}{3}$

• $\frac{111}{25}$

• $\frac{9}{25}$

• $\frac{25}{111}$

$\mathrm{If} \mathrm{A} = \left\{\mathrm{x} \in \mathrm{Rl}\frac{\mathrm{\pi }}{4} \le \mathrm{x} \le \frac{\mathrm{\pi }}{3}\right\} \mathrm{and} \mathrm{f}\left(\mathrm{x}\right) = \sin \left(\mathrm{x}\right) - \mathrm{x}, \mathrm{then} \mathrm{f}\left(\mathrm{A}\right) =?$

• $\left[\frac{\sqrt{3}}{2} - \frac{\mathrm{\pi }}{3}, \frac{1}{\sqrt{2}} - \frac{\mathrm{\pi }}{4}\right]$

• $\left[\frac{- 1}{\sqrt{2}} - \frac{\mathrm{\pi }}{4}, \frac{\sqrt{3}}{2} - \frac{\mathrm{\pi }}{3}\right]$

• $\left(- \frac{\mathrm{\pi }}{3}, - \frac{\mathrm{\pi }}{4}\right)$

• $\left[\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{3}\right]$

$\mathrm{In} \mathrm{a} ∆\mathrm{ABC}, \frac{\mathrm{a}}{\tan \left(\mathrm{A}\right)} \hspace{0.17em}+ \frac{\mathrm{b}}{\tan \left(\mathrm{B}\right)} + \frac{\mathrm{c}}{\tan \left(\mathrm{C}\right)} = ?$

• 2r

• r +2R

• 2r +R

• 2(r + R)

If f is defined in [1, 3] by f(x) = x³ + bx² + ax,such that f(1) - f(3) = 0 and f'(c) = 0, where c = 2 + $\frac{1}{\sqrt{3}}$, then (a, b) is equal to

• ( - 6, 11)

• $\left(2 - \frac{1}{\sqrt{3}}, 2 + \frac{1}{\sqrt{3}}\right)$

• (11, - 6)

• (6, 11)

The domain of the function f(x) = $\sqrt{{\log }_{0.5}\left(\mathrm{x}!\right)} \mathrm{is}$

• $\left\{0, 1, 2, 3, ...\right\}$

• $\left\{0, 1, 2, 3, ...\right\}$

• $\left(0, \infty \right)$

• $\left\{0, 1\right\}$