Match the following columns.

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

• $\frac{25}{3}$

• $\frac{111}{25}$

• $\frac{9}{25}$

• $\frac{25}{111}$

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

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

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

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

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

The value of the sum 1 · 2 . 3 + 2 . 3 . 4 + 3 . 4 . 5 +  ... upto n terms is equal to

• $\frac{1}{6}{\mathrm{n}}^2\left(2{\mathrm{n}}^2 + 1\right)$

• $\frac{1}{6}\left({\mathrm{n}}^2 - 1\right)\left(2\mathrm{n} - 1\right)\left(2\mathrm{n} + 3\right)$

• $\frac{1}{8}\left({\mathrm{n}}^2 + 1\right)\left({\mathrm{n}}^2 +\hspace{0.17em}5\right)$

• $\frac{1}{4}\mathrm{n}\left(\mathrm{n} +\hspace{0.17em}1\right)\left(\mathrm{n} + 2\right)\left(\mathrm{n} + 3\right)$

$\sum _{\mathrm{k} = 1}^6 \left[\sin \left(\frac{{\displaystyle 2\mathrm{k\pi }}}{{\displaystyle 7}} - \mathrm{icos}\left(\frac{{\displaystyle 2\mathrm{k\pi }}}{{\displaystyle 7}}\right)\right)\right] = ?$

•  - 1

• 0

•  - i

• i

$$\begin{gathered} \mathrm{If} \mathrm{\omega } \mathrm{is} \mathrm{a} \mathrm{complex} \mathrm{cube} \mathrm{root} \mathrm{of} \mathrm{unity}, \mathrm{then} \\ {\mathrm{\omega }}^{\left(\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \infty \right)} + \mathrm{\omega }\left(\frac{1}{2} + \frac{3}{8} + \frac{9}{32} + \infty \right) = ? \end{gathered}$$

• 1

• - 1

• $\mathrm{\omega }$

• i

The common roots of the equations ${\mathrm{z}}^3 +\hspace{0.17em}2{\mathrm{z}}^2 + 2\mathrm{z} + 1 = 0, {\mathrm{z}}^{2014} + {\mathrm{z}}^{2015} + 1 = 0 \mathrm{are}$

• $\mathrm{\omega }, {\mathrm{\omega }}^2$

• $1, \mathrm{\omega }, {\mathrm{\omega }}^2$

• $- 1, \mathrm{\omega }, {\mathrm{\omega }}^2$

• $- \mathrm{\omega }, - {\mathrm{\omega }}^2$

If a, b, c are distinct and the roots of (b - c)x² + (c - a)x + (a - b) = 0 are equal, then a, b and c are in

• anthmet1c progression

• geometnc progression

• harmonic progression

• arithmetico-geometnc progression

If the roots of x³ - kx² + 14x - 8 = 0 are in geometric progression, then k is equal to

• - 3

• 7

• 4

• 0

If the harmonic mean of the roots of $\sqrt{2}{\mathrm{x}}^2 - \mathrm{bx} + \left(8 - 2\sqrt{\mathrm{d}}\right) = 0$ is 4, then the value of b is

• 2

• 3

• $4 - \sqrt{5}$

• $4 + \sqrt{5}$