$\sqrt{12 - \sqrt{68 + 48\sqrt{2}}}$ is equal to

• $\sqrt{2} - 3$

• $2 + \sqrt{2}$

• $2 - \sqrt{2}$

• $6 - 2\sqrt{8}$

For all integers n > 1, which of the following is divisible by 9 ?

• 8ⁿ + 1

• 4ⁿ - 3n - 1

• 3²ⁿ + 3n + 1

• 10ⁿ + 1

$$\begin{gathered} \mathrm{If} \frac{3\mathrm{x} + 2}{\left(\mathrm{x} + 1\right)\left(2{\mathrm{x}}^2 + 3\right)} = \frac{\mathrm{A}}{\mathrm{x} + 1} + \frac{\mathrm{Bx} + \mathrm{C}\hspace{0.17em}}{2{\mathrm{x}}^2 +\hspace{0.17em}3}, \mathrm{then} \\ \mathrm{A} + \mathrm{C} - \mathrm{B} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• 0

• 2

• 3

• 5

• $\mathrm{x} \in \left(- \frac{3}{2}, \frac{1}{4}\right)$

• $\mathrm{x} \in \left(- \frac{3}{4}, \frac{1}{4}\right)$

• $\mathrm{x} \in \left(- \frac{3}{4}, \frac{1}{2}\right)$

• $\mathrm{x} < \frac{1}{4}$

${\mathrm{e}}^{\log \left(\cos (\mathrm{h}{)}^{- 1}\left(2\right)\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\log \left(2 - \sqrt{3} \right)$

• $\log \left(\sqrt{3} - 2\right)$

• $\log \left(2 + \sqrt{3}\right)$

• $\log \left(2 + \sqrt{5}\right)$

$$\begin{gathered} \mathrm{If} \mathrm{R}\to \mathrm{R} \mathrm{and} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \frac{1}{2 - \cos \left(3\mathrm{x}\right)} \\ \mathrm{fpr} \mathrm{each} \mathrm{x} \in \mathrm{R}, \mathrm{then} \mathrm{the} \mathrm{range} \mathrm{of} \mathrm{f} \mathrm{is} \end{gathered}$$

• $\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right., - 1\right)$

• $\left[\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right., 1\right]$

• (1, 2)

• $\left[1, 2\right]$

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)

$\sqrt{2+ \sqrt{5} - \sqrt{6 - 3\sqrt{5} + \sqrt{14 - 6\sqrt{5}}}} = ?$

• 1

• 2

• 3

• 4

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

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

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

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

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

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