41.

The angle between the planes 3x-  4y + 5z = 0 and 2x - y - 2z = 5 is

• $\frac{\mathrm{\pi }}{2}$

• None of these

The distance between the lines $\frac{\mathrm{x} - 1}{3} = \frac{\mathrm{y} + 2}{- 2} = \frac{\mathrm{z} - 1}{2}$ and the plane 2x + 2y - z = 6 is

• 9

• 3

• 2

• 1

The cosine of the angle between any two diagonals of a cube is

• $\frac{1}{3}$

• $\frac{2}{3}$

• $- \frac{2}{3}$

• $\frac{1}{2}$

${\left(\mathrm{r} . \hat{\mathrm{i}}\right)}^2 + {\left(\mathrm{r} . \hat{\mathrm{j}}\right)}^2 + {\left(\mathrm{r} . \hat{\mathrm{k}}\right)}^2$ is equal to

• 0

• 1

• r²

• 3r²

The component of $\hat{\mathrm{i}} + \hat{\mathrm{j}}$ along $\hat{\mathrm{j}} \mathrm{and} \hat{\mathrm{k}}$ will be

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{2}$

• $\frac{\hat{\mathrm{j}} + \hat{\mathrm{k}}}{2}$

• $\frac{\hat{\mathrm{k}} + \hat{\mathrm{i}}}{2}$

• None of these

If $\mathrm{a} = 2\hat{\mathrm{i}} + 5\hat{\mathrm{j}}$ and $\mathrm{b} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}}$, then the unit vector along a + b will be

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}}}{\sqrt{2}}$

• $\hat{\mathrm{i}} + \hat{\mathrm{j}}$

• $\sqrt{2}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{\sqrt{2}}$

For any vectors, a, b, c $\mathrm{a} \times \left(\mathrm{b} + \mathrm{c}\right) + \mathrm{b} \times \left(\mathrm{c} + \mathrm{a}\right) + \mathrm{c} \times \left(\mathrm{a} +\hspace{0.17em}\mathrm{b}\right)$ = ?

• 0

• a + b + c

• [a, b, c]

• $\mathrm{a} \times \mathrm{b} \times \mathrm{c}$

$\int \left(\frac{2018{\mathrm{x}}^{2017} + {2018}^{\mathrm{x}} {\log }_{\mathrm{e}}\left(2018\right)}{{\mathrm{x}}^{2018} + {2018}^{\mathrm{x}}}\right)\mathrm{dx}$ =

• $\log \left|{2018}^{\mathrm{x}} + {\mathrm{x}}^{2018}\right| + \mathrm{c}$

• ${\left({2018}^{\mathrm{x}} + {\mathrm{x}}^{2018}\right)}^{-1} + \mathrm{c}$

• $\left|{2018}^{\mathrm{x}} + {\mathrm{x}}^{2018}\right| + \mathrm{c}$

• ${2018}^{\mathrm{x}} + {\mathrm{x}}^{2018} + \mathrm{c}$

$\int \frac{\mathrm{dx}}{{\mathrm{x}}^2 + 4\mathrm{x} +\hspace{0.17em}5}$ =

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

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

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

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

${\int }_0^{\frac{\mathrm{\pi }}{2}}\left({\cos }^2\left(\frac{\mathrm{x}}{2}\right) - {\sin }^2\left(\frac{\mathrm{x}}{2}\right)\right)$ =

• 0

• 1

• - 1

• None of the above