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

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

The correct matching of List-I from List-II is :

         List- I                               List-II

(A)   ${\left(1 - \mathrm{x}\right)}^{- \mathrm{n}}$                     (i)      $\frac{\mathrm{x}}{\mathrm{x} + 1}$

(B)    ${\left(1 + \mathrm{x}\right)}^{- \mathrm{n}}$                   (ii)      $1 - \mathrm{nx} + \frac{\mathrm{n}\left(\mathrm{n} + 1\right)}{2!}{\mathrm{x}}^2 - ... \mathrm{if} \left|\mathrm{x}\right| < 1$

(C)    if x > 1, then             (iii)     $1 + \mathrm{nx} + \frac{\mathrm{n}\left(\mathrm{n} + 1\right)}{2!}{\mathrm{x}}^2 + ... \mathrm{if} \left|\mathrm{x}\right| < 1$

$1 + \frac{1}{\mathrm{x}} + \frac{1}{{\mathrm{x}}^2} + ... \mathrm{is}$

(D) $$\begin{gathered} \mathrm{If} \left|\mathrm{x}\right| > 1, \mathrm{then} \\ 1 - \frac{2}{{\mathrm{x}}^2} + \frac{3}{{\mathrm{x}}^4} - \frac{4}{{\mathrm{x}}^6} + \end{gathered}$$    (iv)     $\frac{\mathrm{x}}{\mathrm{x} - 1}$

                                        (v)     $\frac{{\mathrm{x}}^4}{{\left({\mathrm{x}}^2 + 1\right)}^2}$

                                        (vi)     $\frac{{\mathrm{x}}^4}{{\left({\mathrm{x}}^2 - 1\right)}^2}$    

If f : R $\to$ R is defined by f(x)=x - [ x] - $\frac{1}{2}$ for x $\in$ R, where [x] is the greatest mteger not exceeding x, then $\left\{\mathrm{x} \in \mathrm{R} : \mathrm{f}\left(\mathrm{x}\right) = \frac{1}{2}\right\}$ is equal to :

• Z, the set of all integers

• N, the set of all natural numbers

• $\mathrm{\varphi }$, the empty set

• R

If f : R $\to$ R is defined by f(x) = [2x] - 2[x] for x $\in$ R, where [x] is the greatest integer not exceeding x, then the range of f is :

• $\left\{\mathrm{x} \in \mathrm{R} : 0 \le \mathrm{x} \le 1\right\}$

• {0, 1}

• $\left\{\mathrm{x} \in \mathrm{R} : \mathrm{x} > 0\right\}$

• $\left\{\mathrm{x} \in \mathrm{R} : \mathrm{x} \le 0\right\}$

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

Eight different letters of an alphabet are given. Words of four letters from these are formed.The number of such words with at least one letter repeated is :

• $\left(\begin{array}{c}8\\ 4\end{array}\right) - {}^8\mathrm{P}_4$

• $8^4 + \left(\begin{array}{c}8\\ 4\end{array}\right)$

• $8^4 - {}^8\mathrm{P}_4$

• $8^4 - \left(\begin{array}{c}8\\ 4\end{array}\right)$

The number of natural numbers less than 1000, in which no two digits are repeated, is :

• 738

• 792

• 837

• 720

$1 + \frac{2}{4} + \frac{2}{4} \frac{5}{8} + \frac{2}{4} \frac{5}{8} \frac{8}{12} + \frac{2}{4} \frac{5}{8} \frac{8}{12} \frac{11}{16} + . . . . \mathrm{is} \mathrm{equal} \mathrm{to}:$

• $4^{ - \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• $\sqrt[3]{16}$

• $\sqrt[3]{4}$

• $4^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$