Mathematics : Integrals

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

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

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

• $\mathrm{f}\left(5\right) \le 10$

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

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 :

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

• $- \frac{1}{9}$

• $\frac{9}{2}$

• $- \frac{1}{18}$

• $\frac{7}{18}$

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

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

$$\begin{gathered} \mathrm{If} \int \left({\mathrm{e}}^{2\mathrm{x}} + 2{\mathrm{e}}^{\mathrm{x}} - {\mathrm{e}}^{ - \mathrm{x}} - 1\right){\mathrm{e}}^{\left({\mathrm{e}}^{\mathrm{x}} +{\mathrm{e}}^{ - \mathrm{x}}\right)}\mathrm{dx} = \mathrm{g}\left(\mathrm{x}\right){\mathrm{e}}^{\left({\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{ - \mathrm{x}}\right)} +\hspace{0.17em}\mathrm{C}, \\ \mathrm{where} \mathrm{c} \mathrm{is} \mathrm{a} \mathrm{constant} \mathrm{of} \mathrm{integration}, \mathrm{then} \mathrm{g}\left(0\right) \mathrm{is} \end{gathered}$$

• 1

• e

• e²

• 2

The value of ${\int }_{\raisebox{1ex}{$ - \mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{1}{1 +{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}d\mathrm{x} \mathrm{is} :$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

$\mathrm{If} \int \frac{\cos \left(\mathrm{\theta }\right)}{5 + 7\sin \left(\mathrm{\theta }\right) - 2{\cos }^2\left(\mathrm{\theta }\right)}\mathrm{d\theta } = {\mathrm{Alog}}_{\mathrm{e}}\left|\mathrm{B}\left(\mathrm{\theta }\right)\right| + \mathrm{C}, \mathrm{where} \mathrm{C} \mathrm{is} \mathrm{a} \mathrm{constant} \mathrm{of} \mathrm{integration}, \mathrm{then} \frac{\mathrm{B}\left(\mathrm{\theta }\right)}{\mathrm{A}}\mathrm{can} \mathrm{be}:$

• $\frac{2\sin \left(\mathrm{\theta }\right) + 1}{5\left(\sin \left(\mathrm{\theta }\right) + 3\right)}$

• $\frac{5\left(2\sin \left(\mathrm{\theta }\right) + 1\right)}{\sin \left(\mathrm{\theta }\right) + 3}$

• $\frac{2\sin \left(\mathrm{\theta }\right) + 1}{\sin \left(\mathrm{\theta }\right) + 3}$

• $\frac{5\left(\sin \left(\mathrm{\theta }\right) + 3\right)}{2\sin \left(\mathrm{\theta }\right) + 1}$

The general solution of the differential equation

$\sqrt{1 + {\mathrm{x}}^2 + {\mathrm{y}}^2 +{\mathrm{x}}^2{\mathrm{y}}^2} + \mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0 \left(\mathrm{where} \mathrm{C} \mathrm{is} \mathrm{constant} \mathrm{of} \mathrm{integration} \right)$

• $\sqrt{1 + {\mathrm{y}}^2} + \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 + {\mathrm{x}}^2} - 1}{\sqrt{1 + {\mathrm{x}}^2 +1}}\right) +\mathrm{C}$

• $\sqrt{1 + {\mathrm{y}}^2} + \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 \hspace{0.17em}+ {\mathrm{x}}^2} +\hspace{0.17em}1}{\sqrt{1 +{\mathrm{x}}^2} - 1}\right) +\hspace{0.17em}\mathrm{C}$

• $\sqrt{1 + {\mathrm{y}}^2} - \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 \hspace{0.17em}+ {\mathrm{x}}^2} -\hspace{0.17em}1}{\sqrt{1 +{\mathrm{x}}^2 + 1} }\right) +\hspace{0.17em}\mathrm{C}$

• $\sqrt{1 + {\mathrm{y}}^2} - \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 \hspace{0.17em}+ {\mathrm{x}}^2} +\hspace{0.17em}1}{\sqrt{1 +{\mathrm{x}}^2 - 1} }\right) +\hspace{0.17em}\mathrm{C}$

$\mathrm{If} {\mathrm{I}}_1 = {\int }_0^1{\left(1 - {\mathrm{x}}^{50}\right)}^{100}d\mathrm{x} \mathrm{and} {\mathrm{I}}_2 = {\int }_0^1{\left(1 - {\mathrm{x}}^{50}\right)}^{101}d\mathrm{x} {\mathrm{I}}_2 = {\mathrm{\alpha I}}_1 \mathrm{Then} \mathrm{\alpha } = ?$

• $\frac{5049}{5050}$

• $\frac{5051}{5050}$

• $\frac{5050}{5051}$

• $\frac{5050}{5049}$