41.

The vector equation of the straight line  is

• $\overrightarrow{\mathrm{r}} = \left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} = \left(3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - \hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} = \left(3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(\hat{\mathrm{i}} - \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$

The probability distribution of a random variable X is given as

x	- 5	- 4	- 3	- 2	- 1	0	1	2	3	4	5
P(X = x)	p	2p	3p	4p	5p	7p	8p	9p	10p	11p	12p

Then, the value of p is

• $\frac{1}{72}$

• $\frac{3}{73}$

• $\frac{5}{72}$

• $\frac{1}{74}$

The angle between the curves, y = x and y² - x = 0 at the point (1, 1) is

• $\frac{\mathrm{\pi }}{2}$

• ${\tan }^{-1}\left(\frac{4}{3}\right)$

• $\frac{\mathrm{\pi }}{3}$

• ${\tan }^{-1}\left(\frac{3}{4}\right)$

If $\int \frac{\mathrm{x} + 2}{2{\mathrm{x}}^2 + 6\mathrm{x} + 5}\mathrm{dx}$ = P $\int \frac{4\mathrm{x} + 6}{2{\mathrm{x}}^2 + 6\mathrm{x} + 5}\mathrm{dx}$ + $\frac{1}{2}\int \frac{{\displaystyle \mathrm{dx}}}{{\displaystyle 2{\mathrm{x}}^2 + 6\mathrm{x} + 5}}$, then the values of P is

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{1}{4}$

• 2

$\int \left(\mathrm{x} + 1\right){\left(\mathrm{x} + 2\right)}^7\left(\mathrm{x} + 3\right)\mathrm{dx}$ is equal to

• $\frac{{\left(\mathrm{x} + 2\right)}^{10}}{10} - \frac{{\displaystyle {\left(\mathrm{x} + 2\right)}^8}}{8} + \mathrm{c}$

• $\frac{{\left(\mathrm{x} + 1\right)}^{10}}{2} - \frac{{\displaystyle {\left(\mathrm{x} + 2\right)}^8}}{8} - \frac{{\left(\mathrm{x} + 3\right)}^2}{2}+ \mathrm{c}$

• $\frac{{\left(\mathrm{x} + 2\right)}^{10}}{10} + \mathrm{c}$

• $\frac{{\left(\mathrm{x} + 1\right)}^2}{2} + \frac{{\left(\mathrm{x} + 2\right)}^8}{8} + \frac{{\displaystyle {\left(\mathrm{x} + 3\right)}^2}}{2} + \mathrm{c}$

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

• $\frac{{\left(\mathrm{x} + 1\right)}^{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{7} - 2\frac{{\displaystyle {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{5} + 2\frac{{\displaystyle {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{3} + \mathrm{c}$

• $2\left[\frac{{\left(\mathrm{x} + 1\right)}^{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{7} - 2\frac{{\displaystyle {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{5} + 2\frac{{\displaystyle {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{3}\right] + \mathrm{c}$

• $\frac{{\left(\mathrm{x} + 1\right)}^{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{7} - 2\frac{{\displaystyle {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{5} + \mathrm{c}$

• ${\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + {\left(\mathrm{x} + 1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + \mathrm{c}$

$\int \frac{1 + \mathrm{x}}{\mathrm{x} + {\mathrm{e}}^{- \mathrm{x}}}\mathrm{dx}$ is equal to

• $\log \left|\left(\mathrm{x} - {\mathrm{e}}^{- \mathrm{x}}\right)\right|$

• $\log \left|\left(\mathrm{x} + {\mathrm{e}}^{- \mathrm{x}}\right)\right|$

• $\log \left|\left(1 + {\mathrm{xe}}^{\mathrm{x}}\right)\right| + \mathrm{c}$

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

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

• ${\left[\log \left(\mathrm{x} + \sqrt{1 + {\mathrm{x}}^2}\right)\right]}^2 + \mathrm{c}$

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

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

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

$\int \frac{\mathrm{dx}}{\sqrt{1 - {\mathrm{e}}^{2\mathrm{x}}}}$ is equal to

• $\log \left|{\mathrm{e}}^{- \mathrm{x}} + \sqrt{{\mathrm{e}}^{- 2\mathrm{x}} - 1}\right| + \mathrm{c}$

• $\log \left|{\mathrm{e}}^{\mathrm{x}} + \sqrt{{\mathrm{e}}^{2\mathrm{x}} - 1}\right| + \mathrm{c}$

• $- \log \left|{\mathrm{e}}^{- \mathrm{x}} + \sqrt{{\mathrm{e}}^{- 2\mathrm{x}} - 1}\right| + \mathrm{c}$

• $- \log \left|{\mathrm{e}}^{- 2\mathrm{x}} + \sqrt{{\mathrm{e}}^{- 2\mathrm{x}} - 1}\right| + \mathrm{c}$

$\int \frac{\cos \left(\mathrm{x}\right) + \mathrm{xsin}\left(\mathrm{x}\right)}{{\mathrm{x}}^2 \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\log \left|\frac{\sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\right| + \mathrm{c}$

• $\log \left|\frac{\sin \left(\mathrm{x}\right)}{\mathrm{x} + \cos \left(\mathrm{x}\right)}\right| + \mathrm{c}$

• $\log \left|\frac{2\sin \left(\mathrm{x}\right)}{\mathrm{x} + \cos \left(\mathrm{x}\right)}\right| + \mathrm{c}$

• $\log \left|\frac{\mathrm{x}}{\mathrm{x} + \cos \left(\mathrm{x}\right)}\right| + \mathrm{c}$