$\mathrm{If} \mathrm{y} = {\tan }^{-1}\left(\frac{\sqrt{1 +\hspace{0.17em}{\mathrm{a}}^2{\mathrm{x}}^2} - 1}{\mathrm{ax}}\right), \mathrm{then} \left(1 + {\mathrm{a}}^2{\mathrm{x}}^2\right)\mathrm{y}\text{'}\text{'} +\hspace{0.17em}2{\mathrm{a}}^2\mathrm{y}\text{'} = ?$

• - 2a²

• a²

• 2a²

• 0

$\mathrm{If} {\mathrm{x}}^2 + {\mathrm{y}}^2 = \mathrm{t} +\hspace{0.17em}\frac{1}{\mathrm{t}} \mathrm{and} {\mathrm{x}}^4 + {\mathrm{y}}^4 = {\mathrm{t}}^2 +\hspace{0.17em}\frac{1}{{\mathrm{t}}^2}, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $-\frac{\mathrm{x}}{\mathrm{y}}$

• $-\frac{\mathrm{y}}{\mathrm{x}}$

• $\frac{{\mathrm{x}}^2}{{\mathrm{y}}^2}$

• $\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}$

$\mathrm{If} \mathrm{x} = {\mathrm{at}}^2 \mathrm{and} \mathrm{y} = 2\mathrm{at}, \mathrm{then} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} \mathrm{at} \mathrm{t} = \frac{1}{2} \mathrm{is}$

• $- \frac{2}{\mathrm{a}}$

• $\frac{4}{\mathrm{a}}$

• $\frac{8}{\mathrm{a}}$

• $- \frac{4}{\mathrm{a}}$

$$\begin{gathered} \mathrm{The} \mathrm{equations} \mathrm{x} - \mathrm{y} + 2\mathrm{z} = 4 \\ 3\mathrm{x} + \mathrm{y} + 4\mathrm{z} = 6 \\ \mathrm{x} + \mathrm{y} + \mathrm{z} = 1 \mathrm{have} \end{gathered}$$

• unique solution

• infinitely many solutions

• no solution

• two solutions

The value (s) of x for which the function

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) \left\{= 1 - \mathrm{x}, \mathrm{x} < 1 \\ =\left(1 - \mathrm{x}\right)\left(2 - \mathrm{x}\right), 1 \le \mathrm{x} \le 2 \\ 3 - \mathrm{x}, \mathrm{x} > 2\right\} \\ \mathrm{fails} \mathrm{to} \mathrm{be} \mathrm{continuous} \mathrm{is} \left(\mathrm{are}\right) \end{gathered}$$

• 1

• 2

• 3

• all real numbers

$\mathrm{If} \mathrm{y} = {\log }_2\left({\log }_2\left(\mathrm{x}\right)\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $\frac{{\log }_{\mathrm{e}}\left(2\right)}{{\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{x}\right)}$

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

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

• $\frac{1}{\mathrm{x}{\left({\log }_2\left(\mathrm{x}\right)\right)}^2}$

The angle of intersection between the curves y² + x² = a²$\sqrt{2}$ and x² - y² = a² is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{12}$

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{k}}_1{\left(\mathrm{x} - \mathrm{\pi }\right)}^2, \mathrm{x} \le \mathrm{\pi } \\ {\mathrm{k}}_2\cos \left(\mathrm{x}\right), \mathrm{x} > \mathrm{\pi }\right\} \mathrm{is} \mathrm{twice} \mathrm{differentiable}, \\ \mathrm{then} \mathrm{the} \mathrm{ordered} \mathrm{pair} ({\mathrm{k}}_1, {\mathrm{k}}_2) \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\left(\frac{1}{2}, - 1\right)$

• $\left(\frac{1}{2}, 1\right)$

• (1, 0)

• (1, 1)

$\mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \mathrm{x}, \left[\frac{\mathrm{x}}{2}\right] \mathrm{for} –10 < \mathrm{x} < 10, \mathrm{where} \left[\mathrm{t}\right] \mathrm{denotes} \mathrm{the} \mathrm{greatest} \mathrm{integer} \mathrm{function}. \mathrm{Then} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{points} \mathrm{of} \mathrm{discontinuity} \mathrm{of} \mathrm{f} \mathrm{is} \mathrm{equal} \mathrm{to} \mathrm{\_\_\_\_}$

$$\begin{gathered} \mathrm{Let} \mathrm{f} : \mathrm{R} \to \mathrm{R} \mathrm{be} \mathrm{defined} \mathrm{as} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{x}}^5\sin \left(\frac{1}{\mathrm{x}}\right) + 5{\mathrm{x}}^2, \mathrm{x} < 0 \\ 0, x = 0 \\ {\mathrm{x}}^5\cos \left(\frac{1}{\mathrm{x}}\right) + {\mathrm{\lambda x}}^2, \mathrm{x} > 0\right\} \\ \mathrm{The} \mathrm{value} \mathrm{of} \mathrm{\lambda } \mathrm{for} \mathrm{which} \mathrm{f}\text{'}\text{'}\left(0\right) \mathrm{exists}, \mathrm{is} \end{gathered}$$