41.

$\int \frac{\sin \left(\mathrm{x}\right)}{\sin \left(\mathrm{x} - \mathrm{a}\right)}\mathrm{dx}$ is equal to :

• $\mathrm{xcos}\left(\mathrm{a}\right) - \sin \left(\mathrm{a}\right)\log \left(\sin \left(\mathrm{x} - \mathrm{a}\right)\right) + \mathrm{c}$

• $\mathrm{xsin}\left(\mathrm{a}\right) + \mathrm{c}$

• $\mathrm{xsin}\left(\mathrm{a}\right) + \sin \left(\mathrm{a}\right)\log \left(\sin \left(\mathrm{x} - \mathrm{a}\right)\right) + \mathrm{c}$

• $\mathrm{xcos}\left(\mathrm{a}\right) + \sin \left(\mathrm{a}\right)\log \left(\sin \left(\mathrm{x} - \mathrm{a}\right)\right) + \mathrm{c}$

If $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{i}}\right)$ $+ \hat{\mathrm{j}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{j}}\right) + \hat{\mathrm{k}} \times \left(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{k}}\right)$ then length of $\overrightarrow{\mathrm{b}}$ is equal to :

• $\sqrt{12}$

• $2\sqrt{12}$

• $3\sqrt{14}$

• $2\sqrt{14}$

$\int \frac{\mathrm{f}\text{'}\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)\log \left[\mathrm{f}\left(\mathrm{x}\right)\right]}\mathrm{dx}$ is equal to :

• $\frac{\mathrm{f}\left(\mathrm{x}\right)}{\log \left(\mathrm{f}\left(\mathrm{x}\right)\right)}$

• f(x) . log(f(x)) + c

• $\log \left[\log \left(\mathrm{f}\left(\mathrm{x}\right)\right)\right] + \mathrm{c}$

• $\frac{1}{\log \left[\log \left(\mathrm{f}\left(\mathrm{x}\right)\right)\right]}$ + c

$\int \frac{{\mathrm{e}}^{\mathrm{x}}}{\left(2 + {\mathrm{e}}^{\mathrm{x}}\right)\left({\mathrm{e}}^{\mathrm{x}} + 1\right)}\mathrm{dx}$ is equal to :

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

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

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

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

If I_n = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\tan }^{\mathrm{n}}\left(\mathrm{\theta }\right)\mathrm{d\theta }$, then ${\mathrm{I}}_{\mathrm{\theta }} + {\mathrm{I}}_6$ is equal to :

• $\frac{1}{7}$

• $\frac{1}{4}$

• $\frac{1}{5}$

• $\frac{1}{6}$

The order of the differential equation

${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^3 = {\left(1 + \frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ is :

• 2

• 3

• $\frac{1}{2}$

• 4

The integrating factor of the differential equation $\cos \left(\mathrm{x}\right)\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + \mathrm{ysin}\left(\mathrm{x}\right)$ = 1 is :

• $\cos \left(\mathrm{x}\right)$

• $\tan \left(\mathrm{x}\right)$

• $\sin \left(\mathrm{x}\right)$

• $\sec \left(\mathrm{x}\right)$

Solution of the differential equation tan(y) . sec²(x)dx + tan(x) · sec²(y)dy = 0 is

• tan(x) + tan(y) = k

• tan(x) - tan(y) = k

• $\left(\raisebox{1ex}{$\tan \left(\mathrm{x}\right)$}\!\left/ \!\raisebox{-1ex}{$\tan \left(\mathrm{y}\right)$}\right.\right) = \mathrm{k}$

• $\tan \left(\mathrm{x}\right) . \tan \left(\mathrm{y}\right) = \mathrm{k}$

A class has 175 students. The following data shows the number of students opting one or more subjects. Mathematics 100; Physics 70; Chemistry 40; Mathematics and Physics 30; Mathematics and Chemistry 28; Physics and Chemistry 23; Mathematics, Physics and Chemistry 18. How many students have offered Mathematics alone ?

• 35

• 48

• 60

• 22