31.

The chances of defective screws in three boxes A, B, C are $\frac{1}{5}, \frac{1}{6}, \frac{1}{7}$ respectively. A box 5 6 7 is selected at random and a screw drawn from it at random is found to be defective. Then, the probability that it came from box A, is

• $\frac{16}{29}$

• $\frac{1}{15}$

• $\frac{27}{59}$

• $\frac{42}{107}$

The value of $\frac{\cos \left(\mathrm{\theta }\right)}{1 + \sin \left(\mathrm{\theta }\right)}$ is equal to

• $\tan \left(\frac{\mathrm{\theta }}{2} - \frac{\mathrm{\pi }}{4}\right)$

• $\tan \left(- \frac{\mathrm{\pi }}{4} - \frac{\mathrm{\theta }}{2}\right)$

• $\tan \left(\frac{\mathrm{\pi }}{4} - \frac{\mathrm{\theta }}{2}\right)$

• $\tan \left(\frac{\mathrm{\pi }}{4} + \frac{\mathrm{\theta }}{2}\right)$

If $\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right), \mathrm{then} \int {\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^2\mathrm{dx}$ is equal to

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

• ${\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^3$

• $\frac{{\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^3}{3}$

• ${\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}}^2$

The area bounded by the curve y = | sin(x) |, x-axis and the lines | x | = π, is

• 2 sq unit

• 1 sq unit

• 4 sq unit

• None of these

$\int {\sin }^{-1}\left\{\frac{\left(2\mathrm{x} +\hspace{0.17em}2\right)}{\sqrt{4{\mathrm{x}}^2 + 8\mathrm{x} + 13}}\right\}\mathrm{dx}$ is equal to

• $\left(\mathrm{x} + 1\right){\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{4}\log \left(\frac{4{\mathrm{x}}^2 + 8\mathrm{x} + 13}{9}\right) + \mathrm{c}$

• $\frac{3}{2}{\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{4}\mathrm{og}\left(\frac{{\displaystyle 4{\mathrm{x}}^2 + 8\mathrm{x} + 13}}{{\displaystyle 9}}\right) + \mathrm{c}$

• $\left(\mathrm{x} + 1\right){\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{2}\mathrm{l}\mathrm{og}\left(4{\mathrm{x}}^2 + 8\mathrm{x} + 13\right) + \mathrm{c}$

• $\frac{3}{2}\left(\mathrm{x} + 1\right){\tan }^{-1}\left(\frac{2\mathrm{x} + 2}{3}\right) - \frac{3}{4}\log \left(4{\mathrm{x}}^2 + 8\mathrm{x} + 13\right) + \mathrm{c}$

If ${\int }_1^{\mathrm{x}}\frac{d\mathrm{t}}{\left|\mathrm{t}\right|\sqrt{{\mathrm{t}}^2 - 1}} = \frac{\mathrm{\pi }}{6}$, then x can be equal to

• $\frac{2}{\sqrt{3}}$

• $\sqrt{3}$

• 2

• None of these

The degree of the differential equation of all curves having normal of constant length c is

• 1

• 3

• 4

• None of these

If $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}} = - \hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = 3\hat{\mathrm{i}} + \hat{\mathrm{j}}$ then $\overrightarrow{\mathrm{a}} + \mathrm{t}\overrightarrow{\mathrm{b}}$ is perpendicular to c, if t is equal to

• 2

• 4

• 6

• 8

The distance between the line $\overrightarrow{\mathrm{r}} = 2\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}} + \mathrm{\lambda }\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$ and the plane $\overrightarrow{\mathrm{r}} . \left( \hat{\mathrm{i}} + 5\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = 5$, is

• $\frac{10}{3}$

• $\frac{10}{\sqrt{3}}$

• $\frac{10}{3\sqrt{3}}$

• $\frac{10}{9}$

If A(- 2, 1), B(2, 3) and C( - 2, - 4) are three points. Then the angle between BA and BC is

• ${\tan }^{-1}\left(\frac{2}{3}\right)$

• ${\tan }^{-1}\left(\frac{3}{2}\right)$

• ${\tan }^{-1}\left(\frac{1}{3}\right)$

• ${\tan }^{-1}\left(\frac{1}{2}\right)$