11.

$\mathrm{Area} \mathrm{enclosed} \mathrm{by} 0 \le \mathrm{y} \le {\mathrm{x}}^2 + 1, 0 \le \mathrm{y} \le \mathrm{x} + 1, \frac{1}{2} \le \mathrm{x} \le 2 \mathrm{is} :$

• $\frac{1}{12}$

• $\frac{1}{6}$

• 1

• $\frac{1}{3}$

A pair of dice is thrown and sum of dice come up multiple of 4 then find probability that at least one dice shows 4

• $\frac{2}{7}$

• $\frac{4}{9}$

• $\frac{1}{9}$

• $\frac{5}{8}$

The equation of curve satisfying differential equation $\left(1 + {\mathrm{y}}^2\right)\left({\mathrm{e}}^{\mathrm{x}} + 1\right)\mathrm{dy} = {\mathrm{e}}^{\mathrm{x}}{\mathrm{y}}^2\mathrm{dx} \mathrm{and} \mathrm{also} \mathrm{passes} \mathrm{through} \mathrm{the} \mathrm{point} \left(0, 1\right) \mathrm{is}$

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

• ${\mathrm{y}}^2 - 1 = \mathrm{ylog}\left(\frac{1 + {\mathrm{e}}^{\mathrm{x}}}{2}\right)$

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

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

$\mathrm{If} {\mathrm{y}}^2 + \log \left({\cos }^2\left(\mathrm{x}\right)\right) = \mathrm{y} \mathrm{then}$

• $\left|\mathrm{y}\text{'}\text{'}\left(0\right)\right| = 2$

• $\left|\mathrm{y}\text{'}\left(0\right)\right| +\left| \mathrm{y}\text{'}\text{'}\left(0\right)\right| = 1$

• $\left|\mathrm{y}\text{'}\left(0\right) \right| + \left|\mathrm{y}\text{'}\text{'}\left(0\right)\right| = 3$

• None of these

$\mathrm{Evaluate} {\int }_{ - \mathrm{\pi }}^{\mathrm{\pi }}\left|\mathrm{\pi } - \left|\mathrm{x}\right|\right|d\mathrm{x}$

• ${\mathrm{\pi }}^2$

• $\frac{{\mathrm{\pi }}^2}{2}$

• $\frac{{\mathrm{\pi }}^2}{3}$

• $\frac{{\mathrm{\pi }}^2}{4}$

$\left(2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} + 5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}\right) \mathrm{has} \mathrm{integral} \mathrm{terms} \mathrm{find} \mathrm{least} \mathrm{value} \mathrm{of} \mathrm{n} \mathrm{is}$

• 256

• 257

• 258

• 259

$\mathrm{The} \mathrm{area} \left(\mathrm{in} \mathrm{sq} \mathrm{unit}\right) \mathrm{of} \mathrm{the} \mathrm{region} \left\{\left(\mathrm{x}, \mathrm{y}\right); \frac{1}{2} \le \mathrm{y} \le \sin \left(\mathrm{x}\right), 0 \le \mathrm{x} \le \mathrm{\pi }\right\}\mathrm{is} \mathrm{equal} \mathrm{to}$

• $3 - 2\mathrm{\pi }$

• $\sqrt{3} - \frac{\mathrm{\pi }}{6}$

• 3 - $\frac{\mathrm{\pi }}{3}$

• $\sqrt{3} - \frac{\mathrm{\pi }}{3}$

A bag contains 6 red and 10 green balls, 3 balls are drawn from it one by one without replacement. If the third ball drawn is red, then the probability, that first two balls are green is

• $\frac{3}{7}$

• $\frac{9}{149}$

• $\frac{9}{56}$

• $\frac{3}{8}$

$\underset{\mathrm{x}\to 0}{\lim }\frac{\left(1 - \cos \left({\displaystyle \frac{{\mathrm{x}}^2}{2}}\right)\right)\left(1 - \cos \left({\displaystyle \frac{{\mathrm{x}}^2}{4}}\right)\right)}{{\mathrm{x}}^8} = 2^{ - \mathrm{k}}, \mathrm{find} \mathrm{k}$

$\mathrm{The} \mathrm{value} \mathrm{of} 0 . {16}^{{\log }_{2 . 5}\left(\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + ... \infty \right)} \mathrm{is}$