31.

The ${\overrightarrow{\mathrm{r}}}^2 - \overrightarrow{\mathrm{r}} . \overrightarrow{\mathrm{c}} + \mathrm{h} = 0$, $\left|\overrightarrow{\mathrm{c}}\right| > \sqrt{\mathrm{h}}$, represents

• circle

• ellipse

• cone

• sphere

If I = $\int \frac{{\mathrm{x}}^5}{\sqrt{1 + {\mathrm{x}}^3}}\mathrm{dx}$, then I is equal to

• $\frac{2}{9}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{5}{2}} + \frac{2}{3}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{3}{2}} + \hspace{0.17em}\mathrm{c}$

• $\log \left|\sqrt{\mathrm{x}} + \sqrt{1 + {\mathrm{x}}^3}\right| + \mathrm{c}$

• $\log \left|\sqrt{\mathrm{x}} - \sqrt{1 + {\mathrm{x}}^3}\right| + \mathrm{c}$

• $\frac{2}{9}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{3}{2}} - \frac{2}{3}{\left(1 + {\mathrm{x}}^3\right)}^{\frac{1}{2}} + \hspace{0.17em}\mathrm{c}$

The value of ${\int }_0^{\mathrm{a}}\sqrt{\frac{\mathrm{a} - \mathrm{x}}{\mathrm{x}}}d\mathrm{x}$ is

• $\frac{\mathrm{a}}{2}$

• $\frac{\mathrm{a}}{4}$

• $\frac{\mathrm{\pi a}}{2}$

• $\frac{\mathrm{\pi a}}{4}$

The differential equation of all straight lines touching the circle x² + y² = a² is

• ${\left(\mathrm{y} - \frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = {\mathrm{a}}^2\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]$

• ${\left(\mathrm{y} - \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = {\mathrm{a}}^2\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]$

• $\left(\mathrm{y} - \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}\right) = {\mathrm{a}}^2\left[1 + \frac{\mathrm{dy}}{\mathrm{dx}}\right]$

• $\left(\mathrm{y} - \frac{\mathrm{dy}}{\mathrm{dx}}\right) = {\mathrm{a}}^2\left[1 - \frac{\mathrm{dy}}{\mathrm{dx}}\right]$

The differential equation $\left|\frac{\mathrm{dy}}{\mathrm{dx}}\right| + \left|\mathrm{y}\right| + 3 = 0$ admits

• infinite number of solutions

• no solutions

• a unique solution

• many solutions

Solution of the differential equation $\mathrm{xdy} - \mathrm{ydx} - \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2}\mathrm{dx} = 0$

• $\mathrm{y} - \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{cx}}^2$

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

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

• $\mathrm{x} - \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{cy}}^2$

The probability that a certain kind of component will survive a given shock test is $\frac{3}{4}$· The probability that exactly 2 of the next 4 components tested survive is

• $\frac{9}{41}$

• $\frac{25}{128}$

• $\frac{1}{5}$

• $\frac{27}{128}$

A random variable X follows binomial distribution with mean $\mathrm{\alpha }$ and variance $\mathrm{\beta }$ Then

• $0 < \mathrm{\alpha } < \mathrm{\beta }$

• $0 < \mathrm{\beta } < \mathrm{\alpha }$

• $\mathrm{\alpha } < 0 < \mathrm{\beta }$

• $\mathrm{\beta } < 0 < \mathrm{\alpha }$

The plane through the point (- 1, - 1, - 1) and containing the line of intersection of the planes $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)$ = 0 and $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$ = 0 is

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 3\hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + 5\hat{\mathrm{j}} - 5\hat{\mathrm{k}}\right)$

• $\overrightarrow{\mathrm{r}} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$

$\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ are one of the sides and medians respectively, of a triangle through the same vertex, then area of the triangle is

• $\frac{1}{2}\sqrt{83}$

• $\sqrt{83}$

• $\frac{1}{2}\sqrt{85}$

• $\sqrt{86}$