41.

The distance of the point of intersection of the line  and the plane x - y + z = 5 from the point (- 1, - 5, - 10)is

• 13

• 12

• 11

• 8

If the direction cosines of a line are $\left(\frac{1}{\mathrm{c}}, \frac{1}{\mathrm{c}}, \frac{1}{\mathrm{c}}\right)$, then

• 0 < c < 1

• c > 2

• $\mathrm{c} = \pm \sqrt{2}$

• $\mathrm{c} = \pm \sqrt{3}$

The vector form of the sphere 2(x² + y² + z²) - 4x + 6y + 8z - 5 = 0 is

• $\overrightarrow{\mathrm{r}} . \left[\overrightarrow{\mathrm{r}} - \left(2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)\right] = \frac{2}{5}$

• $\overrightarrow{\mathrm{r}} . \left[\overrightarrow{\mathrm{r}} - \left(2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)\right] = \frac{1}{2}$

• $\overrightarrow{\mathrm{r}} . \left[\overrightarrow{\mathrm{r}} - \left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)\right] = \frac{5}{2}$

• $\overrightarrow{\mathrm{r}} . \left[\overrightarrow{\mathrm{r}} - \left(2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)\right] = \frac{5}{2}$

If the lines $\frac{1 - \mathrm{x}}{3} = \frac{\mathrm{y} - 2}{2\mathrm{\alpha }} = \frac{\mathrm{z} - 3}{2}$$\frac{\mathrm{x} - 1}{3\mathrm{\alpha }} = \mathrm{y} - 1 = \frac{6 - \mathrm{z}}{5}$ are perpendicular, then the value of $\mathrm{\alpha }$ is

• $\frac{- 10}{7}$

• $\frac{10}{7}$

• $\frac{- 10}{11}$

• $\frac{10}{11}$

The distance between the lines $\overrightarrow{\mathrm{r}} = \left(4\hat{\mathrm{i}} - 7\hat{\mathrm{j}} - 9\hat{\mathrm{k}}\right) + \mathrm{t}\left(3\hat{\mathrm{i}} - 7\hat{\mathrm{j}} + 4\hat{\mathrm{k}}\right)$ and $\overrightarrow{\mathrm{r}} = \left(7\hat{\mathrm{i}} - 14\hat{\mathrm{j}} - 5\hat{\mathrm{k}}\right) + \mathrm{s}\left(- 3\hat{\mathrm{i}} + 7\hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)$ is equal to

• 1

• $\frac{1}{2}$

• $\frac{3}{4}$

• 0

$\int \left(\sqrt[3]{\mathrm{x}}\right)\left(\sqrt[5]{1 + \sqrt[3]{{\mathrm{x}}^4}}\right)\mathrm{dx}$ is equal to

• ${\left(1 + {\mathrm{x}}^{\frac{3}{4}}\right)}^{\frac{6}{5}} + \mathrm{c}$

• ${\left(1 + {\mathrm{x}}^{\frac{4}{3}}\right)}^{\frac{6}{5}} + \mathrm{c}$

• $\frac{5}{8}{\left(1 + {\mathrm{x}}^{\frac{4}{3}}\right)}^{\frac{6}{5}} + \mathrm{c}$

• $\frac{1}{6}{\left(1 + {\mathrm{x}}^{\frac{4}{3}}\right)}^6 + \mathrm{c}$

If u = - $\mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right)\sin \left(\mathrm{\theta }\right) + \mathrm{f}\text{'}\left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right)$ and v = $\mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right)\cos \left(\mathrm{\theta }\right) + \mathrm{f}\text{'}\left(\mathrm{\theta }\right)\sin \left(\mathrm{\theta }\right)$, then $\int {\left[{\left(\frac{\mathrm{du}}{\mathrm{d\theta }}\right)}^2 + {\left(\frac{\mathrm{dv}}{\mathrm{d\theta }}\right)}^2\right]}^{\frac{1}{2}}\mathrm{d\theta }$ is equal to

• $\mathrm{f}\left(\mathrm{\theta }\right) - \mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right) + \mathrm{c}$

• $\mathrm{f}\left(\mathrm{\theta }\right) + \mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right) + \mathrm{c}$

• $\mathrm{f}\text{'}\left(\mathrm{\theta }\right) + \mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right) + \mathrm{c}$

• $\mathrm{f}\text{'}\left(\mathrm{\theta }\right) - \mathrm{f}\text{'}\text{'}\left(\mathrm{\theta }\right) + \mathrm{c}$

$\int \frac{{\mathrm{e}}^{6{\log }_{\mathrm{e}}\left(\mathrm{x}\right)} - {\mathrm{e}}^{5{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}}{{\mathrm{e}}^{4{\log }_{\mathrm{e}}\left(\mathrm{x}\right)} - {\mathrm{e}}^{3{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}}\mathrm{dx}$ is equal to

• $\frac{{\mathrm{x}}^3}{3} + \mathrm{c}$

• $\frac{{\mathrm{x}}^2}{2} + \mathrm{c}$

• $\frac{{\mathrm{x}}^2}{3} + \mathrm{c}$

• $\frac{- {\mathrm{x}}^3}{3} + \mathrm{c}$

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

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

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

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

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

$\int \frac{{\mathrm{x}}^4 - 1}{{\mathrm{x}}^2{\left({\mathrm{x}}^4 + {\mathrm{x}}^2 + 1\right)}^{{\displaystyle \frac{1}{2}}}}\mathrm{dx}$ is equal to

• $\sqrt{\frac{{\mathrm{x}}^4 + {\mathrm{x}}^2 + 1}{\mathrm{x}}} + \mathrm{c}$

• $\frac{{\mathrm{x}}^2}{\sqrt{{\mathrm{x}}^4 + {\mathrm{x}}^2 + 1}} + \mathrm{c}$

• $\mathrm{x}{\left({\mathrm{x}}^4 + {\mathrm{x}}^2 + 1\right)}^{\frac{3}{2}} + \mathrm{c}$

• $\frac{\sqrt{{\mathrm{x}}^4 + {\mathrm{x}}^2 + 1}}{\mathrm{x}} + \mathrm{c}$