41.

Solve $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\tan \left(\mathrm{y}\right) = \sin \left(\mathrm{x} + \mathrm{y}\right) + \sin \left(\mathrm{x} - \mathrm{y}\right)$

• $\sec \left(\mathrm{x}\right) - \frac{1}{2}\tan \left(\mathrm{y}\right) = \mathrm{c}$

• $\log \left(\sin \left(\mathrm{x} + \mathrm{y}\right)\right) = \mathrm{c}$

• $\sec \left(\mathrm{x}\right) + \tan \left(\mathrm{y}\right) = \mathrm{c}$

• $\sec \left(\mathrm{y}\right) + 2\cos \left(\mathrm{x}\right) = \mathrm{c}$

The ratio rn which the XY- plane meets the line joining the points (- 3, 4, - 8)and (5, - 6, 4 ) is

• 2 : 3

• 2 : 1

• 4 : 5

• None of these

The unit vector prependicular to each of the vectors $3\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ and $2\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ is

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{3}}$

• None of these

If a, b c are coplanar vectors, then which of the following is not correct ?

• $\mathrm{a} . \left(\mathrm{b} \times \mathrm{c}\right) = 0$

• $\mathrm{a} \times \left(\mathrm{b} \times \mathrm{c}\right) = 0$

• [a + b, b + c, c + a] = 0

• a = pb + qc

$\int \frac{\mathrm{x} + \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\mathrm{xlog}\left(1 + \cos \left(\mathrm{x}\right)\right) + \mathrm{c}$

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

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

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

Find the differential equation of curves y = Ae^x + Be^-x fordifferent values of A and B

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 2\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} = \mathrm{y}$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} = 4\mathrm{y} + 3$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{y} = 0$

Solve $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\mathrm{y}}^2}{\mathrm{xy} - {\mathrm{x}}^2}$

• y = ce^x/y

• y = ce^-y/x + x

• y = ce^y/x

• xy = ce^y/x

Solve $\mathrm{xcos}\left(\mathrm{x}\right)\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + \mathrm{y}\left(\mathrm{xsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right) = 1$

• $\mathrm{y} = \mathrm{xtan}\left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}$

• $\mathrm{x} = \mathrm{ytan}\left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{yxsec}\left(\mathrm{x}\right) = \tan \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{xycos}\left(\mathrm{x}\right) = \mathrm{x} +\hspace{0.17em}\mathrm{c}$

Find the equation of plane through the line $\frac{\mathrm{x} - 2}{2} = \frac{\mathrm{y} - 3}{3} = \frac{\mathrm{z} - 4}{5}$ and parallel to X-axis.

• 2x + 3y + 5z = 1

• 2x - 3z - 3 = 0

• 5y - 3z - 3 = 0

• 3y + 4z = 0

Find the moment of the force $5\hat{\mathrm{i}} + 10\hat{\mathrm{j}} + 16\hat{\mathrm{k}}$ acting at the point $2\hat{\mathrm{i}} - 7\hat{\mathrm{j}} + 10\hat{\mathrm{k}}$, about the point $- 5\hat{\mathrm{i}} + 6\hat{\mathrm{j}} - 10\hat{\mathrm{k}}$

• $41\hat{\mathrm{i}} - 8\hat{\mathrm{j}} + 55\hat{\mathrm{k}}$

• $- 408\hat{\mathrm{i}} - 12\hat{\mathrm{j}} + 135\hat{\mathrm{k}}$

• $- 36\hat{\mathrm{i}} + 14\mathrm{j} - 35\hat{\mathrm{k}}$

• None of the above