The value of $\underset{\mathrm{x}\to 0}{\lim }\frac{{\int }_0^{{\mathrm{x}}^2}\cos \left({\mathrm{t}}^2\right)d\mathrm{x}}{\mathrm{xsin}\left(\mathrm{x}\right)}$

• 1

• - 1

• 2

• ${\log }_{\mathrm{e}}\left(2\right)$

The curve $\mathrm{y} = {\left(\cos \left(\mathrm{x}\right) \hspace{0.17em}+ \mathrm{y}\right)}^{1/2}$ satisfies the differential equation

• $\left(2\mathrm{y} - 1\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + 2{\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2 +\hspace{0.17em} \cos \left(\mathrm{x}\right) = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + 2{\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2 + \hspace{0.17em}\cos \left(\mathrm{x}\right) = 0$

• $\left(2\mathrm{y} - 1\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - 2{\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2 +\hspace{0.17em} \cos \left(\mathrm{x}\right) = 0$

• $\left(2\mathrm{y} - 1\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - {\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2 + \hspace{0.17em}\cos \left(\mathrm{x}\right) = 0$

The solution of the differential equation

$\frac{d\mathrm{y}}{d\mathrm{x}} + \frac{\mathrm{y}}{{\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{x}\right)} = \frac{1}{\mathrm{x}}$

under the condition y = 1 when x = e is

• $2\mathrm{y} = {\log }_{\mathrm{e}}\left(\mathrm{x}\right) +\hspace{0.17em}\frac{1}{{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}$

• $\mathrm{y} = {\log }_{\mathrm{e}}\left(\mathrm{x}\right) +\hspace{0.17em}\frac{2}{{\log }_{\mathrm{e}}\left(\mathrm{x}\right)}$

• ${\mathrm{ylog}}_{\mathrm{e}}\left(\mathrm{x}\right) = {\log }_{\mathrm{e}}\left(\mathrm{x}\right) +\hspace{0.17em}1$

• $\mathrm{y} = {\log }_{\mathrm{e}}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{e}$

Let f(x) = $\max \left\{\mathrm{x} +\hspace{0.17em}\left|\mathrm{x}\right|, \mathrm{x} - \left[\mathrm{x}\right]\right\}$, where [x] denotes the greatest integer $\le \mathrm{x}$. Then, the values of ${\int }_{- 3}^3\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is

• 0

• 51/2

• 21/2

• 1

Suppose $\mathrm{M} = {\int }_0^{\mathrm{\pi }/2}\frac{\cos \left(\mathrm{x}\right)}{\mathrm{x} + 2}d\mathrm{x}, \mathrm{N} = {\int }_0^{\mathrm{\pi }/4}\frac{\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)}{{\left(\mathrm{x} + 1\right)}^2}d\mathrm{x}$. Then, the values of (M - N) equals

• $\frac{3}{\mathrm{\pi } + 2}$

• $\frac{2}{\mathrm{\pi } - 4}$

• $\frac{4}{\mathrm{\pi } - 2}$

• $\frac{2}{\mathrm{\pi } + 4}$

If u(x) and u(x) are two independent solutions of the differential equation

$\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + \mathrm{b} \frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{cy} = 0,$

then additional solution(s) of the given differential equation is(are)

• y = 5u(x) + 8v(x)

• y = c₁{u(x) - v(x)} + c₂v(x), c₁ and c₂ are arbitrary constants

• y = c₁u(x)v(x) + c₂u(x)v(x), c₁ and c₂ are arbitrary constant

• y = u(x)v(x)

77.

For two events A and B, let P(A) = 0.7 and P(B) = 0.6. The necessarily false statement(s) is/are

• $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right) = 0.35$

• $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right) = 0.45$

• $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right) = 0.65$

• $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right) = 0.28$