71.

$$\begin{gathered} \mathrm{A} \mathrm{and} \mathrm{B} \mathrm{are} \mathrm{events} \mathrm{of} \mathrm{a} \mathrm{random} \mathrm{experiment} \\ \mathrm{such} \mathrm{that} \mathrm{P}\left(\mathrm{A} \mathrm{U} \mathrm{B}\right) = \frac{4}{5}, \mathrm{P}\left(\overline{\mathrm{A}} \mathrm{U} \overline{\mathrm{B}}\right) = \frac{7}{10} \mathrm{and} \mathrm{P}\left(\mathrm{B}\right) = \frac{2}{5}, \mathrm{then} \mathrm{P}\left(\mathrm{A}\right) = ? \end{gathered}$$

• $\frac{9}{10}$

• $\frac{8}{10}$

• $\frac{7}{10}$

• $\frac{3}{5}$

The image of the point (3, 2, 1) in the plane 2x - y + 3z = 7 is

• (1, 2, 3)

• (2, 3, 1)

• (3, 2, 1)

• (2, 1, 3)

$\int \frac{\mathrm{dx}}{\left(\mathrm{x} + 1\right)\sqrt{4\mathrm{x} + 3}} = ?$

• ${\text{tan}}^{-1}\left(\sqrt{4x + 3}\right) + c$

• $3{\tan }^{-1}\left(\sqrt{4\mathrm{x} + 3}\right) +\hspace{0.17em}\mathrm{c}$

• $2{\tan }^{-1}\left(\sqrt{4\mathrm{x} + 3}\right) + \mathrm{c}$

• $4{\tan }^{-1}\left(\sqrt{4\mathrm{x} + 3}\right) + \mathrm{c}$

$\int \left(\frac{2 - \sin \left(2\mathrm{x}\right)}{1 - \cos \left(2\mathrm{x}\right)}\right){\mathrm{e}}^{\mathrm{x}}\mathrm{dx} = ?$

• $- \cot \left(\mathrm{x}\right) {\mathrm{e}}^{\mathrm{x}} +\hspace{0.17em}\mathrm{c}$

• $cot\left(x\right) e^x +\hspace{0.17em}c$

• $2\cot \left(\mathrm{x}\right) {\mathrm{e}}^{\mathrm{x}} + \mathrm{c}$

• $- 2\cot \left(\mathrm{x}\right) {\mathrm{e}}^{\mathrm{x}} + \mathrm{c}$

$\mathrm{If} {\mathrm{I}}_{\mathrm{n}} = \int {\sin }^{\mathrm{n}}\mathrm{xdx}, \mathrm{then} {\mathrm{nI}}_{\mathrm{n}} - \left(\mathrm{n} - 1\right){\mathrm{I}}_{\mathrm{n} - 2} = ?$

• ${\sin }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)$

• ${\cos }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)$

• $- {\sin }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)$

• $- {\cos }^{\mathrm{n} - 1}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right)$

${\int }_0^{\mathrm{\pi }}\frac{1}{1 + \sin \left(\mathrm{x}\right)}d\mathrm{x} =?$

• 1

• 2

• - 1

• - 2

$$\begin{gathered} \mathrm{The} \mathrm{line} \mathrm{x} = \frac{\mathrm{\pi }}{4} \mathrm{divides} \mathrm{the} \mathrm{area} \mathrm{of} \mathrm{the} \mathrm{region} \mathrm{bounded} \mathrm{by} \mathrm{y} = \sin \left(\mathrm{x}\right), \mathrm{y} = \cos \left(\mathrm{x}\right) \\ \mathrm{and} \mathrm{x}-\mathrm{axis} \left(0 \le \mathrm{x} \frac{\mathrm{\pi }}{2}\right) \mathrm{into} \mathrm{two} \mathrm{regions} \mathrm{of} \mathrm{areas} {\mathrm{A}}_1 \mathrm{and} {\mathrm{A}}_2. \mathrm{Then} {\mathrm{A}}_1, {\mathrm{A}}_2 \mathrm{equals} \end{gathered}$$

• 4 : 1

   

• 3 : 1

• 2 : 1

• 1 : 1

The velocity of a particle which starts from rest is given by the following table

The total distance travelled (in metre) by the particles in 10s, using trapezoidal rule is given by

• 113

• 226

• 143

• 246

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \sin \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right)\tan \left(\mathrm{x} + \mathrm{y}\right) - 1 \mathrm{is}$

• $\csc \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) + \tan \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) = \mathrm{x} + \mathrm{c}$

• $x +\hspace{0.17em}\csc \left(x + y\right) = c$

• $\mathrm{x} +\hspace{0.17em}\tan \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) = \mathrm{c}$

• $\mathrm{x} +\hspace{0.17em}\sec \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) = \mathrm{c}$

$$\begin{gathered} \mathrm{The} \mathrm{differential} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{family} \mathrm{y} = {\mathrm{ae}}^{\mathrm{x}} + {\mathrm{bxe}}^{\mathrm{x}} + {\mathrm{cx}}^2{\mathrm{e}}^{\mathrm{x}} \mathrm{of} \mathrm{curves}, \\ \mathrm{where} \mathrm{a}, \mathrm{b}, \mathrm{c} \mathrm{are} \mathrm{arbitrary} \mathrm{constants} , \mathrm{is} \end{gathered}$$

• $\mathrm{y}\text{'}\text{'}\text{'} + 3\mathrm{y}\text{'}\text{'} + 3\text{'} + \mathrm{y} = 0$

• $\mathrm{y}\text{'}\text{'}\text{'} + 3\mathrm{y}\text{'}\text{'} - 3\text{'} - \mathrm{y} = 0$

• $\mathrm{y}\text{'}\text{'}\text{'} - 3\mathrm{y}\text{'}\text{'} - 3\text{'} + \mathrm{y} = 0$

• $\mathrm{y}\text{'}\text{'}\text{'} - 3\mathrm{y}\text{'}\text{'} + 3\text{'} - \mathrm{y} = 0$