11.

If A and B are independent events such that P(A) > 0 and P(B) > 0, then

• $\mathrm{P}\left(\mathrm{A} \cup \mathrm{B}\right) = \mathrm{P}\left(\mathrm{A}\right) . \mathrm{P}\left(\mathrm{B}\right)$

• $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right) = \mathrm{P}\left(\mathrm{A}\right) . \mathrm{P}\left(\mathrm{B}\right)$

• $\mathrm{P}\left(\mathrm{A}|\mathrm{B}\right) = \mathrm{P}\left(\mathrm{A}\right)$

• $\mathrm{P}\left(\mathrm{B}|\mathrm{A}\right) = \mathrm{P}\left(\mathrm{B}\right)$

The value of interal I = $\int \frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\sqrt{1 + \sin \left(2\mathrm{x}\right)}}\mathrm{dx}$ is

• $\sqrt{1 + \cos \left(2\mathrm{x}\right)}$

• $\sqrt{\mathrm{x}}$

• x

• $\sqrt{1 + 2\mathrm{x}}$

The value of interal ${\int }_{- \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sin }^3\left(\mathrm{x}\right){\cos }^3\left(\mathrm{x}\right)\mathrm{dx}$ is

• 0

• 1/2

• 1

• None of these

The solution of differential equation

$4\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3{\left(1 +\hspace{0.17em}\mathrm{x}\right)}^2\left(1 + {\mathrm{y}}^2\right)}{\left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)}$ is

• $\log \left(1 + \mathrm{y}\right) = \log \left(\mathrm{x}\right) + 2\tan \left(\mathrm{x}\right) + \mathrm{C}$

• $\log \left(1 + {\mathrm{y}}^2\right) = 3\log \left(\frac{1}{x}\right) + 6{\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

• $\log \left(1 + {\mathrm{y}}^2\right) = 3\log \left(\mathrm{x}\right) + 6{\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{C}$

• None of the above

The area of a parallelogram whose adjacent sides are represented by the vectors $\mathrm{a} = - \hat{\mathrm{i}} - 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ and $\mathrm{b} = - \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$ is

• $\sqrt{14}$

• $\sqrt{6}$

• $\frac{49}{36}$

• $4\sqrt{10}$

The resultant of two forces of 3 kg-wt and 4 kg-wt which are inclined at 30° angles to each other is

• $\sqrt{25 + 6\sqrt{3}}$

• $\sqrt{25 + 12\sqrt{3}}$

• $\sqrt{37}$

• $\sqrt{13}$

Let two like parallel forces are acting, at points A and B such that the force at A is double than the force at B. If C is the point of action ofthe resultant, then AC/BC is

• 2

• 1/2

• 1/3

• 1/4

Let a, b and c represent vector quantities. Which of the following points are collinear?

• a - 2b + 3c, 2a + 3b - 4c, - 7b + 10c

• - 2a + 3b + 5c, a + 2b + 3c, 7a - c

• a, b, 3a - 2b

• a + b - c, 3a - 4c, 2a + b + 3c

The values of $\mathrm{\lambda } \mathrm{and} \mathrm{\mu }$ for which the vectors a = $2\hat{\mathrm{i}} + \mathrm{\lambda }\hat{\mathrm{j}} - \hat{\mathrm{k}}$ is perpendicular to the vector b = $3\hat{\mathrm{i}} + \hat{\mathrm{j}} + \mathrm{\mu }\hat{\mathrm{k}}$ with $\left|\mathrm{a}\right| = \left|\mathrm{b}\right|$ are

• $\mathrm{\lambda } = \frac{41}{12}, \mathrm{\mu } = \frac{31}{12}$

• $\mathrm{\lambda } = \frac{41}{12}, \mathrm{\mu } = - \frac{31}{12}$

• $\mathrm{\lambda } = - \frac{41}{12}, \mathrm{\mu } = \frac{31}{12}$

• None of these

Two parallel unlike forces of magnitudes 15N and 10N are acting at points A and B respectively. If C is the point of action of resultant, then AB/BC is

• 2/1

• 1/2

• 2/3

• 3/2