71.

An objective type test paper has 5 questions. Out of these 5 questions, 3 questions have four options each (a, b, c, d) with one option being the correct answer. The other 2 questions have two options each, namely true and false. A candidate randomly ticks the options. Then, the probability that he/she will tick the correct option in atleast four questions, is

• $\frac{5}{32}$

• $\frac{3}{128}$

• $\frac{3}{256}$

• $\frac{3}{64}$

The solution of the differential equation $\left({\mathrm{y}}^2 + 2\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}} = \mathrm{y}$ satisfies x = 1, y = 1. Then, the solution is

• $\mathrm{x} = {\mathrm{y}}^2\left(1 + {\log }_{\mathrm{e}}\left(\mathrm{y}\right)\right)$

• $\mathrm{y} = {\mathrm{x}}^2\left(1 + {\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$

• $\mathrm{x} = {\mathrm{y}}^2\left(1 - {\log }_{\mathrm{e}}\left(\mathrm{y}\right)\right)$

• $\mathrm{y} = {\mathrm{x}}^2\left(1 + {\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$

A family of curves is such that the length intercepted on the y-axis between the origin and the tangent at a point is three times the ordinate of the point of contact. The family of curves is

• xy = C, C is a constant

• xy² = C, C is a constant

• x²y = C, C is a constant

• x²y² = C, C is a constant

The solution of the differential equation $\mathrm{ysin}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)\mathrm{dx} = \left\{\mathrm{xsin}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) - \mathrm{y}\right\}\mathrm{dy}$ satisfying $\mathrm{y}\left(\frac{\mathrm{\pi }}{4}\right)$ = 1 is

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

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

• $\sin \left(\frac{\mathrm{x}}{\mathrm{y}}\right) = {\log }_{\mathrm{e}}\left(\mathrm{x}\right) - \frac{1}{\sqrt{2}}$

• $\cos \left(\frac{\mathrm{x}}{\mathrm{y}}\right) = - {\log }_{\mathrm{e}}\left(\mathrm{x}\right) - \frac{1}{\sqrt{2}}$

If [a] denote the greatest integer which is less than or equal to a. Then, the value of the integral ${\int }_{- \frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\left[\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)\right]d\mathrm{x}$ is

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• $- \mathrm{\pi }$

• $- \frac{\mathrm{\pi }}{2}$

The value of integral ${\int }_{\frac{\mathrm{\pi }}{6}}^{\frac{\mathrm{\pi }}{3}}\frac{\left(\sin \left(\mathrm{x}\right) - \mathrm{xcos}\left(\mathrm{x}\right)\right)}{\mathrm{x}\left(\mathrm{x} + \sin \left(\mathrm{x}\right)\right)}d\mathrm{x}$

• ${\log }_{\mathrm{e}}\left(\frac{2\left(\mathrm{\pi } + 3\right)}{\left(2\mathrm{\pi } + 3\sqrt{3}\right)}\right)$

• ${\log }_e\left(\frac{\mathrm{\pi } + 3}{2\left(2\mathrm{\pi } + 3\sqrt{3}\right)}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{2\mathrm{\pi } + 3\sqrt{3}}{2\left(\mathrm{\pi } + 3\right)}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{2\left(2\mathrm{\pi } + 3\sqrt{3}\right)}{\mathrm{\pi } +\hspace{0.17em}3}\right)$

If $\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}, \mathrm{x} \ge 0$. Then, the area of the region enclosed by the curve y = f (x) and the three lines y = x, x = 1and x = 8 is

• $\frac{63}{2}$

• $\frac{93}{5}$

• $\frac{105}{7}$

• $\frac{129}{10}$

If f(x) = $\mathrm{x}\left(\frac{1}{\mathrm{x} - 1} + \frac{1}{\mathrm{x}} + \frac{1}{\mathrm{x} + 1}\right), \mathrm{x}> 1$. Then,

• $\mathrm{f}\left(\mathrm{x}\right) \le 1$

• $1 < \mathrm{f}\left(\mathrm{x}\right) \le 2$

• $2 < \mathrm{f}\left(\mathrm{x}\right) \le 3$

• $\mathrm{f}\left(\mathrm{x}\right) > 3$

If F(x) = ${\int }_0^{\mathrm{x}}\frac{\cos \left(\mathrm{t}\right)}{\left(1 +\hspace{0.17em}{\mathrm{t}}^2\right)}d\mathrm{t}, 0 \le \mathrm{x} \le 2\mathrm{\pi }$. Then,

• F is decreasing in $\left(\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2}\right)$ and decreasing in $\left(0, \frac{\mathrm{\pi }}{2}\right) \mathrm{and} \left(\frac{3\mathrm{\pi }}{2}, 2\mathrm{\pi }\right)$

• F is increasing in (0, $\mathrm{\pi }$) and decreasing in ( $\mathrm{\pi }, 2\mathrm{\pi }$).

• Fis increasing in ($\mathrm{\pi }, 2\mathrm{\pi }$) and decreasing in ( $0, \mathrm{\pi }$).

• Fis increasing in (0, $\frac{\mathrm{\pi }}{2}$) and $\left(\frac{3\mathrm{\pi }}{2}, 2\mathrm{\pi }\right)$ and decreasing in ( $\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2}$).

The area of the region enclosed between parabola y² = x and the line y = mx is $\frac{1}{48}$. Then, the value of m is

• - 2

• - 1

• 1

• 2