11.

The probability that a man will live 10 more years, is $\frac{1}{4}$ and the probability that his wife  will live 10 more years, is $\frac{1}{3}$. Then, what is the probability that neither will be alive in 10 yr ?

• $\frac{1}{2}$

• $\frac{3}{7}$

• $\frac{2}{3}$

• $\frac{1}{4}$

The probability that in a year of the 22nd century chosen at random, there will be 53 Sunday, is

• $\frac{3}{28}$

• $\frac{2}{28}$

• $\frac{7}{28}$

• $\frac{5}{28}$

$\int \frac{\mathrm{x} + 2}{\left({\mathrm{x}}^2 +\hspace{0.17em}3\mathrm{x} +\hspace{0.17em}3\right)\sqrt{\mathrm{x} + 1}}\mathrm{dx}$ is equa to

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

• $\frac{2}{\sqrt{3}}{\tan }^{-1}\left[\frac{\mathrm{x}}{\sqrt{3\left(\mathrm{x} + 1\right)}}\right] + \mathrm{C}$

• $\frac{2}{\sqrt{3}}{\tan }^{-1}\left[\frac{\mathrm{x}}{{\left(\mathrm{x} + 1\right)}^2}\right] + \mathrm{C}$

• None of these

$\int {\sin }^{-1}\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a} + \mathrm{x}}}\right)\mathrm{dx}$ is equal to

• $\left[{\tan }^{-1}\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}}} + \sqrt{\frac{\mathrm{x}}{\mathrm{a}}}\right)\right] + \mathrm{C}$

• $\mathrm{a}\left[{\tan }^{-1}\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}}} - \sqrt{\frac{\mathrm{x}}{\mathrm{a}}}\right)\right] + \mathrm{C}$

• $\mathrm{a}\left[{\tan }^{-1}\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}}} . \frac{\left(\mathrm{a} +\hspace{0.17em}\mathrm{x}\right)}{\mathrm{a}}\right)\right] + \mathrm{C}$

• $\mathrm{a}\left[{\tan }^{-1}\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}}} . \frac{\left(\mathrm{a} + \mathrm{x}\right)}{\mathrm{a}} - \sqrt{\frac{\mathrm{x}}{\mathrm{a}}}\right)\right] + \mathrm{C}$

A girl walks 4 km towards West, then she walks 3 km in a drection 30° East of North and stops. Then, the girl's displacement from herinitial point of departures is

• $- \frac{5}{2}\hat{\mathrm{i}} + \frac{3\sqrt{3}}{2}\hat{\mathrm{j}}$

• $\frac{1}{2}\hat{\mathrm{i}} + \frac{\sqrt{3}}{2}\hat{\mathrm{j}}$

• $- \frac{1}{2}\hat{\mathrm{i}} + \frac{3\sqrt{3}}{2}\hat{\mathrm{j}}$

• None of these

If a = $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$,b = $4\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ and c = $\hat{\mathrm{i}} + \mathrm{\alpha }\hat{\mathrm{j}} + \mathrm{\beta }\hat{\mathrm{k}}$ are linearly dependent vectors and $\left|\mathrm{c}\right| = \sqrt{3}$, then the value of $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are respectively

• $\pm 1, 1$

• $\pm 2, 1$

• $0, \pm 1$

• None of these

The projection of the vector a = $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ on  the vextor $\mathrm{b} = 4\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 7\hat{\mathrm{k}}$ is

• $\frac{9}{19}$

• $\frac{19}{9}$

• 9

• $\sqrt{19}$

Forces of magnitude 5 and 3 units acting in the directions $6\hat{\mathrm{i}} + 2\hat{\mathrm{j}} +\hspace{0.17em}3\hat{\mathrm{k}}$ and $3\hat{\mathrm{i}} - 2\hat{\mathrm{j}} +\hspace{0.17em}6\hat{\mathrm{k}}$ respectively act on a particle which is displaced from the point (2, 2, - 1) to (4, 3, 1). The work done by the forces is

• 148 units

• $\frac{148}{7} \mathrm{units}$

• $\frac{78}{7} \mathrm{units}$

• None of these

The solution of differential equation $\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right) - 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0$ is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = \mathrm{xC}$

• ${\mathrm{x}}^2 - {\mathrm{y}}^2 = \mathrm{xC}$

• x² + y² = C

• x² - y² = C

The solution of differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{x}\left(2\log \left(\mathrm{x}\right) + 1\right)}{\sin \left(\mathrm{y}\right) + \mathrm{ycos}\left(\mathrm{y}\right)}$ is

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

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

• $\mathrm{ysin}\left(\mathrm{y}\right) = {\mathrm{x}}^2 + \mathrm{C}$

• None of these