Consider the curves

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \mathrm{x}\left|\mathrm{x}\right| - 1 \mathrm{and} \mathrm{g}\left(\mathrm{x}\right) = \left\{\frac{3\mathrm{x}}{2}, \mathrm{x} > 0 \\ 2\mathrm{x}, \mathrm{x} \le 0\right\} \end{gathered}$$

Where do the curves intersect ?

• At (2, 3) only

• At ( - 1, - 2) only

• At (2, 3) and ( - 1, 2)

• Neither at (2, 3) nor at ( - 1,  - 2)

Consider the function $\mathrm{f}\left(\mathrm{x}\right) = \frac{27\left({\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} - \mathrm{x}\right)}{4}$

How may solutions does the function f(x) = 1 have ?

• one

• Two

• Three

• Four

Consider the function $\mathrm{f}\left(\mathrm{x}\right) = \frac{27\left({\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} - \mathrm{x}\right)}{4}$

How may solutions does the function f(x) = - 1 have ?

• one

• two

• three

• four

Suppose there is a relation * between the positive numbers x and y given by x * y if and only if $\mathrm{x} \le {\mathrm{y}}^2$. Then which one of the following is correct ?

• * is reflexive hut not 'transitive and symmetric

• * is transitive but not reflexive and symmetric

• * is symmetric and relexive but not transitive

• * is symmetric but not reflexive and transitive

$$\begin{gathered} \mathrm{If} \mathrm{f}\left({\mathrm{x}}_1\right) - \mathrm{f}\left({\mathrm{x}}_2\right) = \mathrm{f}\left(\frac{{\mathrm{x}}_1 - {\mathrm{x}}_2}{1 - {\mathrm{x}}_1{\mathrm{x}}_2}\right) \\ \mathrm{for} {\mathrm{x}}_1, {\mathrm{x}}_2 \in \left( - 1, 1\right), \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{f}\left(\mathrm{x}\right) = ? \end{gathered}$$

• $\ln \left(\frac{1 - \mathrm{x}}{1 +\hspace{0.17em}\mathrm{x}}\right)$

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

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

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

Consider the function

$\mathrm{f}\left(\mathrm{x}\right) = \left|{\mathrm{x}}^2 - 5\mathrm{x} + 6\right|$

What is f'(4) = ?

•  - 4

•  - 3

• 3

• 2

Consider the function

$\mathrm{f}\left(\mathrm{x}\right) = \left|{\mathrm{x}}^2 - 5\mathrm{x} + 6\right|$

What is f''(2.5) = ?

•  - 3

•  - 2

• 0

• 2

Let, f(x) be the greatest integer function and g(x) bethe modulus function.

What is $\left(\mathrm{g} \mathrm{o} \mathrm{f}\right)\left( - \frac{5}{3}\right) - \left(\mathrm{f} \mathrm{o} \mathrm{f}\right)\left( - \frac{5}{3}\right) = ?$

•  - 1

• 0

• 1

• 2

Let, f(x) be the greatest integer function and g(x) bethe modulus function.

What is $\left(\mathrm{f} \mathrm{o} \mathrm{f}\right)\left( - \frac{9}{5}\right) + \left(\mathrm{g} \mathrm{o} \mathrm{g}\right)\left( - 2\right) = ?$

•  - 1

• 0

• 1

• 2

What is the binary equivalent of the decimal number 0.3125 ?

• 0.0111

• 0.1010

• 0.0101

• 0.1101