If [x] denotes the greatest integer less than or equal to x, then the value of the integral ${\int }_0^2{\mathrm{x}}^2\left[\mathrm{x}\right]d\mathrm{x}$  equals

• $\frac{5}{3}$

• $\frac{7}{3}$

• $\frac{8}{3}$

• $\frac{4}{3}$

For non-zero vectors a and b, if $\left|\mathrm{a} + \mathrm{b}\right| < \left|\mathrm{a} - \mathrm{b}\right|$,  then a and b are

• collinear

• perpendicular to each other

• inclined at an acute angle

• inclined at an obtuse angle

General solution of $\mathrm{y}\frac{d\mathrm{y}}{d\mathrm{x}} + {\mathrm{by}}^2 = \mathrm{acos}\left(\mathrm{x}\right), 0 < \mathrm{x} < 1$ is

• ${\mathrm{y}}^2 = 2\mathrm{a}\left(2\mathrm{bsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right) + {\mathrm{ce}}^{- 2\mathrm{bx}}$

• $\left(4{\mathrm{b}}^2 + 1\right){\mathrm{y}}^2 = 2\mathrm{a}\left(\sin \left(\mathrm{x}\right) + 2\mathrm{bcos}\left(\mathrm{x}\right)\right) + {\mathrm{ce}}^{- 2\mathrm{bx}}$

• $\left(4{\mathrm{b}}^2 + 1\right){\mathrm{y}}^2 = 2\mathrm{a}\left(\sin \left(\mathrm{x}\right) + 2\mathrm{bcos}\left(\mathrm{x}\right)\right) + {\mathrm{ce}}^{2\mathrm{bx}}$

• ${\mathrm{y}}^2 = 2\mathrm{a}\left(2\mathrm{bsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right) + {\mathrm{ce}}^{- 2\mathrm{bx}}$

If $$\begin{gathered} \mathrm{\varphi }\left(\mathrm{t}\right) = \left\{1, \mathrm{for} 0 \le \mathrm{t} < 1 \\ 0, \mathrm{otherwise}\right\}, \mathrm{then} \end{gathered}$$

${\int }_{- 3000}^{3000}\left(\sum _{\mathrm{r}\text{'} = 2014}^{2016}\left(\mathrm{\varphi }\left(\mathrm{t} - \mathrm{r}\text{'}\right)\mathrm{\varphi }\left(\mathrm{t} - 2016\right)\right)\right)\mathrm{dt}$ is

• a real number

• 1

• 0

• does not exist

75.

If A, B are two events such that $\mathrm{P}(\mathrm{A} \cup \mathrm{B}) \ge \frac{3}{4} \mathrm{and} \frac{1}{8} \le \mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right) \le \frac{3}{8}$ then

• $\mathrm{P}\left(\mathrm{A}\right) + \mathrm{P}\left(\mathrm{B}\right) \le \frac{11}{8}$

• $\mathrm{P}\left(\mathrm{A}\right) . \mathrm{P}\left(\mathrm{B}\right) \le \frac{3}{8}$

• $\mathrm{P}\left(\mathrm{A}\right) + \mathrm{P}\left(\mathrm{B}\right) \ge \frac{7}{8}$

• None of the above