The value of the expression 1 . (2 - w)(2 - w²) + 2 . (3 - w)(3 - w²) + ... + (n - 1)(n - w²), where w is an imaginary cube root of unity is

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

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

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

• None of these

Inverse of function f(x) = $\frac{{10}^{\mathrm{x}} - {10}^{- \mathrm{x}}}{{10}^{\mathrm{x}} + {10}^{- \mathrm{x}}}$ is

• log₁₀(2 - x)

• $\frac{1}{2}{\log }_{10}\left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right)$

• $\frac{1}{2}{\log }_{10}\left(2\mathrm{x} - 1\right)$

• $\frac{1}{4}{\log }_{10}\left(\frac{2\mathrm{x}}{2 - \mathrm{x}}\right)$

Let f : R - {x} $\to$ R be a function defined by f(x) =  $\frac{\mathrm{x} - \mathrm{m}}{\mathrm{x} - \mathrm{n}}$, where m $\ne$ n. Then

• f is one-one onto

• f is one-one into

• f is many one onto

• f is many one into

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

276.

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