In the triangle with vertices at A(6, 3), B(- 6, 3) and C(- 6, - 3), the median through A meets BC at P, the line AC meets the x-axis at Q, while R and S respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is

      List-I                        List-II

(i)    P                      (A)    (0, 0)

(ii)   Q                      (B)    (6, 0)

(iii)   R                     (C)    (- 2, 1)

(iv)   S                     (D)    (- 6, 0)

                               (E)    (- 6, - 3)

                               (F)    (- 6, 3)

If a = $\frac{1 - \mathrm{i}\sqrt{3}}{2}$, then the correct matching of List-I from List-II is

         List-I                       List-II

(i)      $\mathrm{a}\overline{\mathrm{a}}$                            $- \frac{\mathrm{\pi }}{3}$

(ii)    $\arg \left(\frac{1}{\overline{\mathrm{a}}}\right)$                       $- \mathrm{i}\sqrt{3}$

(iii)    $\mathrm{a} - \overline{\mathrm{a}}$                        $\raisebox{1ex}{$2\mathrm{i}$}\!\left/ \!\raisebox{-1ex}{$\sqrt{3}$}\right.$ 

(iv)     $\mathrm{Im}\left(\frac{4}{3\mathrm{a}}\right)$                    1

                                        $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

                                        $\frac{2}{\sqrt{3}}$

correct match is 

$$\begin{gathered} \mathrm{If} \mathrm{Q} \mathrm{denotes} \mathrm{the} \mathrm{set} \mathrm{of} \mathrm{all} \mathrm{rational} \mathrm{numbers} \mathrm{and} \mathrm{f}\left(\frac{\mathrm{p}}{\mathrm{q}}\right) = \sqrt{{\mathrm{p}}^2 - {\mathrm{q}}^2} \mathrm{for} \\ \mathrm{any} \frac{\mathrm{p}}{\mathrm{q}} \in \mathrm{Q}, \mathrm{then} \mathrm{observe} \mathrm{the} \mathrm{following} \mathrm{statements}. \\ \mathrm{I}. \mathrm{f}\left(\frac{\mathrm{p}}{\mathrm{q}}\right) \mathrm{is} \mathrm{real} \mathrm{for} \mathrm{each} \frac{\mathrm{p}}{\mathrm{q}} \in \mathrm{Q} \\ \mathrm{II}. \mathrm{f}\left(\frac{\mathrm{p}}{\mathrm{q}}\right) \mathrm{is} \mathrm{a} \mathrm{complex} \mathrm{number} \mathrm{for} \mathrm{each} \frac{\mathrm{p}}{\mathrm{q}} \in \mathrm{Q}. \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ? \end{gathered}$$

• Both I and II are true

• I is true, II is false

• I is false, II is true

• Both I and II are false

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

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

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

• (1, 2)

• $\left[1, 2\right]$

5.

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

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

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

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

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

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

$\mathrm{If} {\mathrm{S}}_{\mathrm{n}} = 1^3 +\hspace{0.17em}{2}^3 + ... +\hspace{0.17em}{\mathrm{n}}^3 \mathrm{and} {\mathrm{T}}_{\mathrm{n}} = 1 +\hspace{0.17em}2 + ... + \mathrm{n}, \mathrm{then}$

• ${\mathrm{S}}_{\mathrm{n}} = {\mathrm{T}}_{{\mathrm{n}}^3}$

• ${\mathrm{S}}_{\mathrm{n}} = {\mathrm{T}}_{{\mathrm{n}}^2}$

• ${\mathrm{S}}_{\mathrm{n}} = {{\mathrm{T}}_{\mathrm{n}}}^2$

• ${\mathrm{S}}_{\mathrm{n}} = {{\mathrm{T}}_{\mathrm{n}}}^3$

$\mathrm{If} {\mathrm{a}}_{\mathrm{k}} \mathrm{is} \mathrm{the} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^{\mathrm{k}} \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} {\left(1 + \mathrm{x} + {\mathrm{x}}^2\right)}^{\mathrm{n}} \mathrm{for} \mathrm{k} = 0, 1, 2, . . , 2\mathrm{n}, \mathrm{then}$

• - a₀

• 3ⁿ

• n 3^{n + 1}

• n 3ⁿ

The sum of the series $\frac{3}{4 . 8} - \frac{3 . 5}{4 . 8 . 12} + \frac{3 . 5 . 7}{4 . 8 . 12 . 16} - ...$

• $\sqrt{\frac{3}{2}} - \frac{3}{4}$

• $\sqrt{\frac{2}{3}} - \frac{3}{4}$

• $\sqrt{\frac{3}{2}} - \frac{1}{4}$

• $\sqrt{\frac{2}{3}} - \frac{1}{4}$