$\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^7 + 1\right)}$ is equal to

• $\log \left(\frac{{\mathrm{x}}^7}{{\mathrm{x}}^7 + 1}\right)$

• $\frac{1}{7}\log \left(\frac{{\mathrm{x}}^7}{{\mathrm{x}}^7 + 1}\right) + \mathrm{c}$

• $\log \left(\frac{{\mathrm{x}}^{7 }+ 1}{{\mathrm{x}}^7}\right) + \mathrm{c}$

• $\frac{1}{7}\log \left(\frac{{\mathrm{x}}^7 +\hspace{0.17em}1}{{\mathrm{x}}^7}\right)$

The solution of the differential equation $\frac{\mathrm{dx}}{\mathrm{x}} + \frac{\mathrm{dy}}{\mathrm{y}} = 0$ is

• xy = c

• x + y = c

• log(x)log(y) = c

• x² + y² = c

23.

The differential equation obtained by eliminating arbitrary constants from y = a . e^bx, is

• $\mathrm{y}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• $\mathrm{y}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

• $\mathrm{y}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = 0$

• $\mathrm{y}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2 = 0$

f(x) is a polynomial of degree 2, f(0) = 4, f'(0) = 3 and f''(0) = 4, then f(- 1) is equal to

• 3

• - 2

• 2

• - 3

Solution of differential equation sec(x)dy - cosec(y)dx = 0 is

• cos(x) + sin(y) = c

• sin(x) + cos(y) = 0

• sin(y) - cos(x) = c

• cos(y) - sin(x) = c

The point P(9/2, 6) lies on the parabola y² = 4ax, then parameter of the point P is

• $\frac{3\mathrm{a}}{2}$

• $\frac{2}{3\mathrm{a}}$

• $\frac{2}{3}$

• $\frac{3}{2}$

Using Trapezoidal rule and following table ${\int }_0^8\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

x	0	0	4	6	8
f(x)	2	5	10	17	26

• 184

• 92

• 46

• - 36

Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ be vectors with magnitudes 3, 4 and 5 respectively and $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$, then $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}}$ is

• 47

• 25

• 50

• - 25

Unbiased die is thrown, probability that outcome is greater than 4, is

• 3/4

• 4/5

• 1/3

• 5/6