If a non-zero vector a is parallel to the line of intersection of the plane determined by the vectors $\hat{\mathrm{j}} - \hat{\mathrm{k}}, 3\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$ the plane determined by the vectors $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}, \widehat{\mathrm{i} }- 3\hat{\mathrm{j}}$ then the angle between the vectors a and $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$  is

• ${\sin }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

• ${\cos }^{-1}\left(\pm \frac{2}{\sqrt{3}}\right)$

• ${\tan }^{-1}\left(\sqrt{3}\right)$

• ${\cos }^{-1}\left( \pm \frac{1}{\sqrt{3}}\right)$

If three numbers are drawn at random successively without replacement from a set S = {1, 2, ... 10}, then the probability that the minimum of the chosen numbers is 3 or their maximum is 7

• $\frac{11}{40}$

• $\frac{5}{40}$

• $\frac{3}{40}$

• $\frac{1}{40}$

63.

$$\begin{gathered} \mathrm{For} {\mathrm{x}}^2 - 4 \ne 0, \mathrm{the} \mathrm{value} \mathrm{of} \\ \frac{\mathrm{d}}{\mathrm{dx}}\left[\log \left({\mathrm{e}}^{\mathrm{x}}{\left(\frac{\mathrm{x} - 2}{\mathrm{x} + 2}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\right)\right] \mathrm{at} \mathrm{x} = 3 \mathrm{is} \end{gathered}$$

• $\frac{8}{5}$

• 2

• 1

• $\frac{8{\mathrm{e}}^3}{5}$

$\mathrm{If} \mathrm{y} = \frac{{\sinh }^{-1}\left(\mathrm{x}\right)}{\sqrt{1 + {\mathrm{x}}^2}}, \mathrm{then} \left(1 + {\mathrm{x}}^2\right){\mathrm{y}}^2 + 3{\mathrm{xy}}_1 + \mathrm{y} = ?$

• 2

• 1

•  - 1 

• 0

$\int \frac{5\mathrm{x} + 3}{{\mathrm{x}}^2\left({\mathrm{x}}^2 - 2\right)}\mathrm{dx} = ?$

• $\frac{3}{2\mathrm{x}} + \frac{\sqrt{13}}{2\sqrt{2}}\log \left(\left|\frac{\sqrt{2} - \mathrm{x}}{\sqrt{2} + \mathrm{x}}\right|\right) + \mathrm{C}$

• $\frac{3}{2\mathrm{x}} + \frac{13}{4\sqrt{2}}\log \left(\left|\frac{x + \sqrt{2}}{x - \sqrt{2}}\right|\right) + \mathrm{C}$

• $\frac{3}{2\mathrm{x}} + \frac{13}{4\sqrt{2}}\log \left(\left|\frac{\mathrm{x} - \sqrt{2}}{\mathrm{x} + \sqrt{2}}\right|\right) + \mathrm{C}$

• $\frac{3}{5}x + \frac{5}{3\sqrt{2}}\log \left(\left|\frac{x + \sqrt{2}}{x - \sqrt{2}}\right|\right) + \mathrm{C}$

$\mathrm{If} \mathrm{y} = {\tan }^{-1}\left(\left\{\frac{\mathrm{x}}{1 + \sqrt{1 - {\mathrm{x}}^2}}\right\}\right) + \sin \left(2{\tan }^{-1}\left(\sqrt{\frac{1 - \mathrm{x}}{1 + \mathrm{x}}}\right)\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $\frac{1 - 2\mathrm{x}}{2\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{1 - 2\mathrm{x}}{\mathrm{x}\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{2\mathrm{x} + 1}{\mathrm{x}\sqrt{1 - \mathrm{x}}}$

• $\frac{2 - \mathrm{x}}{2\sqrt{1 - {\mathrm{x}}^2}}$

The equation of the plane through (4,4,0) and perpendicular to the planes 2x + y + 2z + 3 = 0 and 3x + 3y + 2z - 8 = 0

• 4x + 3y + 3z = 28

• 4x - 2y - 3z = 8

• 4x + 2y + 3z = 24

• 4x +2y - 3z = 24

The solution of the equation

$\left(\mathrm{x} - 4{\mathrm{y}}^3\right)\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y} = 0, \left(\mathrm{y} > 0\right) \mathrm{is}$

• x = y³ + cy

• x + 2y³ = cy

• y = x³ + cx

• y + 2x³ = cx

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{points} \mathrm{having} \mathrm{the} \mathrm{position} \mathrm{vectors} \\ 3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - \hat{\mathrm{k}}, 2\mathrm{i} + 3\hat{\mathrm{j}} - 4\hat{\mathrm{k}}, - \hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}} \mathrm{and} \\ 4\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + \mathrm{\lambda }\hat{\mathrm{k}} \mathrm{are} \mathrm{coplanar}, \mathrm{then} \mathrm{\lambda } = ? \end{gathered}$$

• $- \frac{146}{17}$

• 8

•  - 8

• $\frac{146}{17}$

If ${\int }_0^{10}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}= 5, \mathrm{then} \sum _{\mathrm{k} = 1}^{10}{\int }_0^1\mathrm{f}\left(\mathrm{k} - 1 + \mathrm{x}\right)\mathrm{dx}$ = ?

• 50

• 10

• 5

• 20