51.

The equation of the plane in which the lines $\frac{\mathrm{x} - 5}{4} = \frac{\mathrm{y} - 7}{4} = \frac{\mathrm{z} + 3}{- 5} \mathrm{and} \frac{\mathrm{x} - 8}{7} = \frac{\mathrm{y} - 4}{1} = \frac{\mathrm{z} - 5}{3}$ lie, is

• 17x - 47y - 24z + 172 = 0

• 17x + 47y - 24z + 172 = 0

• 17x + 47y + 24z + 172 = 0

• 17x - 47y + 24z + 172 = 0

If the vectors $\mathrm{a}\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and $3\hat{\mathrm{i}} + 6\hat{\mathrm{j}} - 5\hat{\mathrm{k}}$ are perpendicularto each other, then a is given by

• 9

• 16

• 25

• 36

If position vectors of points A,B,C are $\hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}}$ respectively and AB = CX, then position vector of point X is

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

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

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

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

$\int {\mathrm{e}}^{\mathrm{x}}\sin \left({\mathrm{e}}^{\mathrm{x}}\right)\mathrm{dx}$ is equal to

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

• $\cos \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{c}$

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

• None of these

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

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

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

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

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

The differential equation of all straight lines passing through the point (1, - 1), is

• $\mathrm{y} = \left(\mathrm{x} + 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 1$

• $\mathrm{y} = \left(\mathrm{x} + 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} - 1$

• $\mathrm{y} = \left(\mathrm{x} - 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 1$

• $\mathrm{y} = \left(\mathrm{x} - 1\right)\frac{\mathrm{dy}}{\mathrm{dx}} - 1$

Integrating factor of differential equation $\cos \left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{ysin}\left(\mathrm{x}\right) = 1$ is

• $\cos \left(\mathrm{x}\right)$

• $\tan \left(\mathrm{x}\right)$

• $\sec \left(\mathrm{x}\right)$

• $\sin \left(\mathrm{x}\right)$

The order and degree of the differential equation $\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right) + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} + {\mathrm{x}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.} = 0$ are respectively

• 2, 3

• 3, 3

• 2, 6

• 2, 4