$$\begin{gathered} \mathrm{Let} \mathrm{S} \mathrm{be} \mathrm{the} \mathrm{set} \mathrm{of} \mathrm{all} \mathrm{integer} \mathrm{solutions}, (\mathrm{x}, \mathrm{y}, \mathrm{z}), \mathrm{of} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations} \\ \mathrm{x} – 2\mathrm{y} + 5\mathrm{z} = 0 \\ – 2\mathrm{x} + 4\mathrm{y} + \mathrm{z} = 0 \\ – 7\mathrm{x} + 14\mathrm{y} + 9\mathrm{z} = 0 \\ \mathrm{Such} \mathrm{that} 15 < {\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 < 150. \mathrm{Then}, \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{elements} \mathrm{in} \mathrm{the} \mathrm{set} \\ \mathrm{S} \mathrm{is} \mathrm{equal} \mathrm{to}. \end{gathered}$$

$$\begin{gathered} \mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 2\right| \mathrm{and} \mathrm{g}\left(\mathrm{x}\right) = \mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right), \mathrm{x} \in \left[0, 4\right]. \mathrm{Then} \\ {\int }_0^3\left(\mathrm{g}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{x}\right)\right)d\mathrm{x} = ? \end{gathered}$$

• 0

• $\frac{1}{2}$

• $\frac{3}{2}$

• 1

$\mathrm{The} \mathrm{integral} \int {\left(\frac{\mathrm{x}}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}\right)}^2\mathrm{dx} = ? \left(\mathrm{Where} \mathrm{Cis} \mathrm{a} \mathrm{constant} \mathrm{of} \mathrm{integration}\right) :$

• $\tan \left(\mathrm{x}\right) + \frac{\mathrm{xsec}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$

• $\tan \left(\mathrm{x}\right) - \frac{\mathrm{xsec}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$

• $\sec \left(\mathrm{x}\right) + \frac{\mathrm{xtan}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$

• $\sec \left(\mathrm{x}\right) - \frac{\mathrm{xsec}\left(\mathrm{x}\right)}{\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)} + \mathrm{C}$

Let [t] denote the greatest integer $\le$ t. Then the equation in x, [x]² + 2[x + 2] – 7 = 0 has :

• no integral solution

• infinitely many solutions

• exactly two solutions

• exactly four integral solutions

$\mathrm{Let} \mathrm{f}\left(\mathrm{x}\right) = \int \frac{\sqrt{\mathrm{x}}}{{\left(1 + \mathrm{x}\right)}^2}\mathrm{dx} \left(\mathrm{x} \ge 0\right). \mathrm{Then} \mathrm{f}\left(3\right) - \mathrm{f}\left(1\right) = ?$

• $- \frac{\mathrm{\pi }}{6} + \frac{1}{2} + \frac{\sqrt{3}}{4}$

• $- \frac{\mathrm{\pi }}{12} + \frac{1}{2} + \frac{\sqrt{3}}{4}$

• $\frac{\mathrm{\pi }}{12} + \frac{1}{2} - \frac{\sqrt{3}}{4}$

• $\frac{\mathrm{\pi }}{6} + \frac{1}{2} - \frac{\sqrt{3}}{4}$

$$\begin{gathered} \mathrm{Let} \mathrm{f} \mathrm{be} \mathrm{the} \mathrm{twice} \mathrm{differential} \mathrm{function} \mathrm{on} \left(1, 6\right). \mathrm{If} \mathrm{f}\left(2\right) = 8, \mathrm{f}\text{'}\left(2\right) = 5, \\ \mathrm{f}\text{'}\left(\mathrm{x}\right) \ge 1 \mathrm{and} \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right) \ge 4, \mathrm{for} \mathrm{all} \mathrm{x} \in \left(1, 6\right), \mathrm{then} : \end{gathered}$$

• $\mathrm{f}\left(5\right) \hspace{0.17em}+ \mathrm{f}\text{'}\left(5\right) \ge 28$

• $\mathrm{f}\text{'}\left(5\right) + \mathrm{f}\text{'}\text{'}\left(5\right) \le 20$

• $\mathrm{f}\left(5\right) \le 10$

• $\mathrm{f}\left(5\right) + \mathrm{f}\text{'}\left(5\right) \le 26$

If the system of equations

x – 2y + 3z = 9

2x + y + z = b

x – 7y + az = 24,

has infinitely many solutions, then a – b is equal to :

$\mathrm{The} \mathrm{integral} {\int }_{\frac{\mathrm{\pi }}{6}}^{\frac{\mathrm{\pi }}{3}}{\tan }^3\left(\mathrm{x}\right) . {\sin }^2\left(3\mathrm{x}\right)\left(2{\sec }^2\left(\mathrm{x}\right){\sin }^2\left(3\mathrm{x}\right) + 3\tan \left(\mathrm{x}\right)\sin \left(6\mathrm{x}\right)\right)\mathrm{dx} = ?$

• $- \frac{1}{9}$

• $\frac{9}{2}$

• $- \frac{1}{18}$

• $\frac{7}{18}$

$$\begin{gathered} \mathrm{Let} \left\{\mathrm{x}\right\} \mathrm{and} \left[\mathrm{x}\right] \mathrm{denote} \mathrm{the} \mathrm{fractional} \mathrm{part} \mathrm{of} \mathrm{x} \mathrm{and} \mathrm{the} \mathrm{greatest} \mathrm{integer} \mathrm{r} \le \mathrm{x} \mathrm{respectively} \mathrm{of} \mathrm{a} \mathrm{real} \mathrm{number} \mathrm{x}. \\ \mathrm{if} {\int }_0^{\mathrm{n}}\left\{\mathrm{x}\right\}d\mathrm{x} \mathrm{and} 10({\mathrm{n}}^2 – \mathrm{n}), (\mathrm{n} \in \mathrm{N}, \mathrm{n} > 1) \mathrm{are} \mathrm{three} \mathrm{consecutive} \mathrm{terms} \mathrm{of} \mathrm{a} \mathrm{G}.\mathrm{P}. \mathrm{then} \mathrm{n} \mathrm{is} \mathrm{equal} \mathrm{to} ...... \end{gathered}$$

$\mathrm{If} \overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} {\left|\hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right)\right|}^2 + {\left|\hat{\mathrm{j}}\left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}\right)\right|}^2 + {\left|\hat{\mathrm{k}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right)\right|}^2 = ?$