31.

If $\mathrm{xsin}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)\mathrm{dy} = \left[\mathrm{ysin}\left(\frac{\mathrm{y}}{\mathrm{x}}\right) - \mathrm{x}\right]\mathrm{dx}$ and $\mathrm{y}\left(1\right) = \frac{\mathrm{\pi }}{2}$, then the value of $\cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)$ is equal to :

• x

• $\frac{1}{\mathrm{x}}$

• $\log \left(\mathrm{x}\right)$

• e^x

The differential equation of the system of all circles of radius r in the xy plane is :

• ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^3\right]}^2 = {\mathrm{r}}^2{\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^2$

• ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^3\right]}^2 = {\mathrm{r}}^2{\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^3$

• ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^3 = {\mathrm{r}}^2{\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^2$

• ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^3 = {\mathrm{r}}^2{\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^3$

The general solution of the differential equation

$\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 2\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = 2{\mathrm{e}}^{3\mathrm{x}}$ is given by

• $\mathrm{y} = \left({\mathrm{c}}_1 + {\mathrm{c}}_2\mathrm{x}\right){\mathrm{e}}^{\mathrm{x}} + \frac{{\mathrm{e}}^{3\mathrm{x}}}{8}$

• $\mathrm{y} = \left({\mathrm{c}}_1 + {\mathrm{c}}_2\mathrm{x}\right){\mathrm{e}}^{- \mathrm{x}} + \frac{{\mathrm{e}}^{- 3\mathrm{x}}}{8}$

• $\mathrm{y} = \left({\mathrm{c}}_1 + {\mathrm{c}}_2\mathrm{x}\right){\mathrm{e}}^{- \mathrm{x}} + \frac{{\mathrm{e}}^{3\mathrm{x}}}{8}$

• $\mathrm{y} = \left({\mathrm{c}}_1 + {\mathrm{c}}_2\mathrm{x}\right){\mathrm{e}}^{\mathrm{x}} + \frac{{\mathrm{e}}^{- 3\mathrm{x}}}{8}$

The solution of the differential ydx + (x - y³)dy = 0 is:

• xy = $\frac{1}{3}{\mathrm{y}}^3 + \mathrm{c}$

• xy = y⁴ + c

• y⁴ = 4xy + c

• 4y = y³ + c

If the probability density function of a random variable X is f(x) = $\frac{\mathrm{x}}{2}$ in $0 \le \mathrm{x} \le 2$, then $\mathrm{P}\left(\mathrm{X} > 1.5 | \mathrm{x} > 1\right)$ is equal to :

• $\frac{7}{16}$

• $\frac{3}{4}$

• $\frac{7}{12}$

• $\frac{21}{64}$

If X follows a binomial distribution with parameters n = 100 and p = $\frac{1}{3}$ then P(X = r) 3 is maximum when r is equal to

• 16

• 32

• 33

• none of these

If $\overrightarrow{\mathrm{b}}$ is a unit vector, then $\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)\overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}}\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)$ is :

• ${\left|\overrightarrow{\mathrm{a}}\right|}^2 \overrightarrow{\mathrm{b}}$

• $\left|\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right|\overrightarrow{\mathrm{a}}$

• $\overrightarrow{\mathrm{a}}$

• $\overrightarrow{\mathrm{b}}$

If $\mathrm{\theta }$ is the angle between the lines AB and AC where A, B and C are the three points with coordinates (1, 2, - 1), (2, 0, 3), (3, - 1, 2) respectively, then $\sqrt{462} \cos \left(\mathrm{\theta }\right)$ is equal to

• 20

• 10

• 30

• 40

Let the pairs $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{d}}$ each determine a plane. Then the planes are parallel, if :

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

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

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

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

The area of a parallelogram with $3\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}} \mathrm{and} \hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ as diagonal is :

• $\sqrt{72}$

• $\sqrt{73}$

• $\sqrt{74}$

• $\sqrt{75}$