Relations and Functions

A and B are two points on the Argand plane such that the segment AB is bisected at the point (0, 0). If the point A, which is in the third quadrant has principal amplitude 0, then the principal amplitude of the point B is

• $- \mathrm{\theta }$

• $\mathrm{\pi } - \mathrm{\theta }$

• $\mathrm{\theta } - \mathrm{\pi }$

• $\mathrm{\pi } + \mathrm{\theta }$

The relation on the set A = {x : $\left|\mathrm{x}\right| < 3, \mathrm{x} \in \mathrm{Z}$} is defined by R = {(x, y) : y = $\left|\mathrm{x}\right|, \mathrm{x} \ne - 1$} Then, the number of elements in the power set of R is

• 8

• 16

• 32

• 64

f(x) = (20 - x⁴)^1/4 for 0 < x < $\sqrt{5}$, then $\mathrm{f}\left(\mathrm{f}\left(\frac{1}{2}\right)\right)$ is equal to

• 2^{- 4}

• 2^{- 3}

• 2^{- 2}

• 2^{- 1}

$\frac{\sqrt{8 + \sqrt{28}} + \sqrt{8 - \sqrt{28}}}{\sqrt{8 + \sqrt{28}} - \sqrt{8 - \sqrt{28}}}$ is equal to

• 2

• 7

• $\sqrt{7}$

• $\sqrt{2}$

x = log_0.1(0.001), y = log₉(81), then $\sqrt{\mathrm{x} - 2\sqrt{\mathrm{y}}}$ is equal to

• $3 - \sqrt{2}$

• $\sqrt{3} - 2$

• $\sqrt{2} - 1$

• $\sqrt{2} - 2$

$1 + {\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{a}\right) + \frac{{\mathrm{x}}^2}{2!}{\left({\log }_{\mathrm{e}}\left(\mathrm{a}\right)\right)}^2 + \frac{{\mathrm{x}}^3}{3!}{\left({\log }_{\mathrm{e}}\left(\mathrm{a}\right)\right)}^3 + ...\left(\mathrm{a} > 0, \mathrm{x} \in \mathrm{R}\right)$, is equal to

• a^x

• a^x

• ${\mathrm{a}}^{{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}$

• x

If the sum of two of the roots of x³ + px² - qx + r = 0 is zero, then pq is equal to

• - r

• r

• 2r

• - 2r

For any integer n > 1, the number of positive divisors of n is denoted by d(n). Then, for a prime P, d (d (d(P)⁷)) is equal to

• 1

• 2

• 3

• p

$\sum _{\mathrm{k} = 1}^5\frac{1^3 + 2^3 + .... + {\mathrm{k}}^3}{1 + 3 + 5 + ... + \left(2\mathrm{k} - 1\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 22.5

• 24.5

• 28.5

• 32.5

$$\begin{gathered} \mathrm{The} \mathrm{function} \mathrm{f}: \mathrm{C} \to \mathrm{C} \mathrm{defined}, \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{ax} + \mathrm{d}}{\mathrm{cx} + \mathrm{d}} \mathrm{for} \mathrm{x} \in \mathrm{C} \mathrm{where} \\ \mathrm{bd} \ne 0 \mathrm{reduces} \mathrm{to} \mathrm{a} \mathrm{constant} \mathrm{function}, \mathrm{if} \end{gathered}$$

• a = c

• b = d

• ad = bc

• ab = cd