The domain of definition of function $\mathrm{f}\left(\mathrm{x}\right) = \frac{1 +\hspace{0.17em}2{\left(\mathrm{x} +\hspace{0.17em}4\right)}^{- 0.5}}{2 - {\left(\mathrm{x} +\hspace{0.17em}4\right)}^{- 0.5}} + {\left(\mathrm{x} + 4\right)}^{0.5} + 4{\left(\mathrm{x} + 4\right)}^{0.5}$ is

• R

• (- 4, 4)

• R⁺

• $\left(- 4, 0\right) \cup \left(0, \infty \right)$

If f(x) = $\frac{4^{\mathrm{x}}}{4^{\mathrm{x}} + 2}$, then $\mathrm{f}\left(\frac{1}{97}\right) + \mathrm{f}\left(\frac{2}{97}\right) + ... + \mathrm{f}\left(\frac{96}{97}\right)$ is equal to

• 1

• 48

• - 48

• - 1

Let a relation R be defined on set of all real numbers by a R b if and only if 1 + ab > 0. Then, R is

• reflexive, transitive but not symmetric

• reflexive, symmetric but not transitive

• symmetric, transitive but not reflexive

• an equivalence relation

If $\frac{\mathrm{xy}}{\mathrm{x} + \mathrm{y}} = \frac{2}{3}, \frac{{\displaystyle \mathrm{yz}}}{{\displaystyle \mathrm{y} + \mathrm{z}}} = \frac{{\displaystyle 6}}{{\displaystyle 5}}, \frac{{\displaystyle \mathrm{xz}}}{{\displaystyle \mathrm{x} + \mathrm{z}}} = \frac{{\displaystyle 3}}{{\displaystyle 4}}$, then (x, y, z) is equal to

• (1, 2, 3)

• (2, 1, 3)

• (3, 1, 2)

• (3, 2, 1)

Let f(x) = $\frac{1}{2} - \tan \left(\frac{\mathrm{\pi x}}{2}\right)$, - 1 < x < 1 and g(x) = $\sqrt{3 + 4\mathrm{x} - 4{\mathrm{x}}^2}$, then dom(f + g) is given by

• $\left[\frac{1}{2}, 1\right]$

• $[\frac{1}{2}, - 1)$

• $[ - \frac{1}{2}, 1)$

• $\left[- \frac{1}{2}, - 1\right]$

Let the functions f, g, h are defined from the set of real numbers R to R such that

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^2 - 1, \mathrm{g}\left(\mathrm{x}\right) = \sqrt{\left({\mathrm{x}}^2 + 1\right)} \mathrm{and} \\ \mathrm{h}\left(\mathrm{x}\right) = \left\{0, \mathrm{if} \mathrm{x} < 0 \\ \mathrm{x}, \mathrm{if} \mathrm{x} \ge 0\right\} \end{gathered}$$

then ho(fog)(x) is defined by

• x

• x²

• 0

• None of these

If f(x) = cos(log(x)), then f(x)f(y) $- \frac{1}{2}\left[\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) + \mathrm{f}\left(\mathrm{xy}\right)\right]$ has the value :

• - 1

• $\frac{1}{2}$

• - 2

• zero

If y = 3^{x - 1} + 3^{-x - 1} (x real), then the least value ofy is

• 2

• 6

• 2/3

• None of these

If f(x) = (a - xⁿ)^1/n where a > 0 and n $\in$ N, then fof(x) is equal to :

• a

• x

• xⁿ

• aⁿ

If R denotes the set of all real numbers, then the function f : $\mathrm{R} \to \mathrm{R}$ defined f(x) = $\left|\mathrm{x}\right|$ is :

• one-one only

• onto only

• both one-one and onto

• neither one-one nor onto