Suppose X = {1, 2, 3, 4) and R is a relation on X. If R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}, then which one of the following is correct ?

• R is reflexive and symmetric, but not transitive

• R is symmetric and transitive, but not reflexive

• R is reflexive and transitive, but not symmetric

• R is neither reflexive nor transitive, but symmetric

A relation R is defined on the set N of natural numbers as xRy ⇒ x² - 4xy + 3y² = 0. Then which one of the following is correct ?

• R is reflexive and symmetric, but not transitive

• R is reflexive and transitive, but not symmetric

• R is reflexive, symmetric and transitive

• R is reflexive, but neither symmetric nor transitive

3.

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = {\log }_{10}\left(1 + \mathrm{x}\right), \mathrm{then} \mathrm{what} \mathrm{is} 4\mathrm{f}\left(4\right) = 5\mathrm{f}\left(1\right) - {\log }_{10}\left(2\right) = ?$

• 0

• x² + 1

• x = 1

• x²

$\mathrm{A} \mathrm{function} \mathrm{f} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \ln \left(\sqrt{{\mathrm{x}}^2 + 1} - \mathrm{x}\right) \mathrm{is} :$

• an even function

• an odd function

• Both even and odd function

• Neither even nor odd

The domain of the function f defined by $\mathrm{f}\left(\mathrm{x}\right) = {\log }_{\mathrm{x}}\left(10\right) \mathrm{is} :$

• $\mathrm{x} > 10$

• $\mathrm{x} > 0 \mathrm{excluding} \mathrm{x} = 10$

• $\mathrm{x} \ge 10$

• $\mathrm{x} > 0 \mathrm{excluding} \mathrm{x} = 1$

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = 3^{1 + \mathrm{x}}, \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right)\mathrm{f}\left(\mathrm{z}\right) = ?$

• f(x + y + z)

• f(x + y + z + 1)

• f(x + y + z + 2)

• f(x + y + z + 3)

The domain of the function f(x) = $\sqrt{\left(2 - \mathrm{x}\right)\left(\mathrm{x} - 3\right)} \mathrm{is} :$

• $\left(0, \infty \right)$

• $\left(0, \infty \right)$

• $\left[2, 3\right]$

• $\left(2, 3\right)$

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

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

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

• $\frac{\mathrm{x} + 2}{\mathrm{x} - 2}$

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

The binary number expression of the decimal number 31 is :

• 1111

• 10111

• 11011

• 11111

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{ℝ} \to \mathrm{S} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = 4\sin \left(\mathrm{x}\right) - 3\cos \left(\mathrm{x}\right) + 1 \mathrm{is} \mathrm{onto}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{S} \mathrm{equal} \mathrm{to} ? \end{gathered}$$

• $\left[ - 5, 5\right]$

• ( - 5, 5)

• ( - 4, 6)

• [ - 4, 6]