Relations and Functions

Let A = {1, 2, 3} and B = {2, 3, 4}, then which of the following relations is a function from A to B ?

• {(1, 2), (2, 3), (3, 4), (2, 2)}

• {(1, 2),(2, 3), (1, 3)}

• {(1, 3), (2, 3), (3, 3)}

• {(1, 1), (2, 3), (3, 4)}

If $\mathrm{y} = 2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\log }_{\mathrm{x}}\left(8\right)$}\right.}$, then x is equal to

• y

• y²

• y³

• None of these

The domain of the function

$\mathrm{f}\left(\mathrm{x}\right) = {\log }_{10}\left(\sqrt{\mathrm{x} - 4} + \sqrt{6 - \mathrm{x}}\right)$ is

• [4, 6]

• $\left(- \infty , 6\right)$

• (2, 3)

• None of these

The inverse of the function f(x) = $\mathrm{f}\left(\mathrm{x}\right) = \frac{{10}^{\mathrm{x}} - {10}^{- \mathrm{x}}}{{10}^{\mathrm{x}} + {10}^{- \mathrm{x}}}$ is

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

• $\frac{1}{2}{\log }_{10}\left(\frac{1 +\hspace{0.17em}\mathrm{x}}{1 - \mathrm{x}}\right)$

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

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

The number of solutions of ${\log }_2\left(\mathrm{x} - 1\right) = 2{\log }_2\left(\mathrm{x} - 3\right)$

• 2

• 1

• 6

• 7

$\mathrm{If} \mathrm{}$ \mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{x} - 1}{\mathrm{x} + 1}$\frac{\mathrm{}}{}$, then f(2x) is

• $\frac{\mathrm{f}\left(\mathrm{x}\right) + 1}{\mathrm{f}\left(\mathrm{x}\right) + 3}$

• $\frac{3\mathrm{f}\left(\mathrm{x}\right) + 1}{\mathrm{f}\left(\mathrm{x}\right) + 3}$

• $\frac{\mathrm{f}\left(\mathrm{x}\right) + 3}{\mathrm{f}\left(\mathrm{x}\right) + 1}$

• $\frac{\mathrm{f}\left(\mathrm{x}\right) + 3}{3\mathrm{f}\left(\mathrm{x}\right) + 1}$

∼ (P ∨ q) v ( ∼ p ∧ q) is logically equivalent to

• - p

• p

• q

Given, $\mathrm{f}\left(\mathrm{x}\right) = \log \left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right)$ and $\mathrm{g}\left(\mathrm{x}\right) = \frac{3\mathrm{x} + {\mathrm{x}}^3}{1 +\hspace{0.17em}3{\mathrm{x}}^2}$, then fog(x) equals

• [f(x)]³

• - f(x)

• 3f(x)

• None of these

Which of the following proposition is a tautology?

• $\left(~ \mathrm{P} \vee ~ \mathrm{q}\right) \wedge \left(\mathrm{p} \vee ~ \mathrm{q}\right)$

• $~ \mathrm{q} \wedge \left(\mathrm{p} \vee ~ \mathrm{q}\right)$

• $~ \mathrm{P} \wedge \left(\mathrm{p} \vee ~ \mathrm{q}\right)$

• $\left(~ \mathrm{P} \vee ~ \mathrm{q}\right) \vee \left(\mathrm{p} \vee ~ \mathrm{q}\right)$

If N denote the set of all natural numbers and R be the relation on N $\times$ N defined by (a, b) R (c, d), if ad(b + c) = bc(a + d), then R is

• symmetric only

• reflexive only

• transitive only

• an equivalence relation