81.

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{R} \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{x} + 4 \mathrm{for} \mathrm{x} < - 4 \\ 3\mathrm{x} + 2 \mathrm{for} - 4 \le \mathrm{x} < 4 \\ \mathrm{x} - 4 \mathrm{for} \mathrm{x} \ge 4\right\} \end{gathered}$$

then the correct matching of List-I from List-II is :

        List-I                      list-II

(A)  f(- 5) + f(- 4)     (i)    14

(B)  $\mathrm{f}\left(\left|\mathrm{f}\left(- 8\right)\right|\right)$           (ii)     4

(C)  $\mathrm{f}\left(\mathrm{f}\left(- 7\right) + \mathrm{f}\left(3\right)\right)$    (iii)    - 11

(D)  f(f(f(f(0)))) + 1 (iv)    1

                              (v)     0

A. A B C D	(i) (iii) (vi) (ii) (v)
B. A B C D	(ii) (iii) (iv) (ii) (v)
C. A B C D	(iii) (iv) (iii) (ii) (i)
D. A B C D	(iv) (iii) (vi) (v) (ii)

If $\mathrm{x} = \sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}}$, then x²(x - 4)² is equal to :

• 7

• 4

• 2

• 1

$\sqrt{12 - \sqrt{68 + 48\sqrt{2}}}$ 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