11.

The negation of the Boolean expression $\mathrm{x} \leftrightarrow ~ \mathrm{y}$ is equivalent to : 

• $\left(~ \mathrm{x} \wedge \mathrm{y}\right) \vee \left(~ \mathrm{x} \wedge ~ \mathrm{y}\right)$

• $\left(\mathrm{x} \wedge \mathrm{y}\right) \vee \left(~ \mathrm{x} \wedge ~ \mathrm{y}\right)$

• $\left(\mathrm{x} \wedge ~ \mathrm{y}\right) \vee \left(~ \mathrm{x} \wedge \mathrm{y}\right)$

• $\left(\mathrm{x} \wedge \mathrm{y}\right) \wedge \left(~ \mathrm{x} \vee ~ \mathrm{y}\right)$

The mean and variance of 7 observations are 8 and 16, respectively. If five observation are 2, 4, 10, 12, 14 then the absolute difference of the remaining two observations is :

• 1

• 2

• 3

• 4

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{k}}_1{\left(\mathrm{x} - \mathrm{\pi }\right)}^2, \mathrm{x} \le \mathrm{\pi } \\ {\mathrm{k}}_2\cos \left(\mathrm{x}\right), \mathrm{x} > \mathrm{\pi }\right\} \mathrm{is} \mathrm{twice} \mathrm{differentiable}, \\ \mathrm{then} \mathrm{the} \mathrm{ordered} \mathrm{pair} ({\mathrm{k}}_1, {\mathrm{k}}_2) \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\left(\frac{1}{2}, - 1\right)$

• $\left(\frac{1}{2}, 1\right)$

• (1, 0)

• (1, 1)

$$\begin{gathered} \mathrm{If} 3^{2\sin \left(2\mathrm{\alpha }\right) - 1}, 14 \mathrm{and} 3^{4 - 2\sin \left(2\mathrm{\alpha }\right)} \mathrm{are} \mathrm{the} \mathrm{first} \mathrm{three} \mathrm{terms} \mathrm{of} \mathrm{an} \mathrm{A}.\mathrm{P}. \\ \mathrm{for} \mathrm{some} \mathrm{\alpha }, \mathrm{then} \mathrm{the} \mathrm{sixth} \mathrm{term} \mathrm{of} \mathrm{this} \mathrm{A}.\mathrm{P}. \mathrm{is} \end{gathered}$$

• 81

• 65

• 66

• 78

If the minimum and the maximum values of the function

$\mathrm{f} :\hspace{0.17em}\left[\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{2}\right] \to \mathrm{R}, \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{\theta }\right) = \left|\begin{array}{ccc} - {\sin }^2\left(\mathrm{\theta }\right)& - 1 - {\sin }^2\left(\mathrm{\theta }\right)& 1\\ {\cos }^2\left(\mathrm{\theta }\right)& - 1 - {\cos }^2\left(\mathrm{\theta }\right)& 1\\ 12& 10& - 2\end{array}\right|$

are m and M respectively, then the ordered pair (m, M) = : 

• ( - 4, 4)

• $\left(0, 2\sqrt{2}\right)$

• ( - 4, 0)

• (0, 4)

$$\begin{gathered} \mathrm{Let} \mathrm{\lambda } \in \mathrm{R}. \mathrm{The} \mathrm{system} \mathrm{of} \mathrm{linear} \mathrm{equations} \\ 2{\mathrm{x}}_1 - 4{\mathrm{x}}_2 + {\mathrm{\lambda x}}_3 = 1 \\ {\mathrm{x}}_1 - 6{\mathrm{x}}_2 + {\mathrm{x}}_3 = 2 \\ {\mathrm{\lambda x}}_1 - 10{\mathrm{x}}_2 + 4{\mathrm{x}}_3 = 3 \\ \mathrm{is} \mathrm{inconsistant} \mathrm{for} : \end{gathered}$$

• every value of $\mathrm{\lambda }$

• exactly two values of $\mathrm{\lambda }$

• $\mathrm{exactly} \mathrm{one} \mathrm{positive} \mathrm{value} \mathrm{of \lambda }$

• exactly one negative value of $\mathrm{\lambda }$

$$\begin{gathered} \mathrm{If} \mathrm{y} = \mathrm{y}\left(\mathrm{x}\right) \mathrm{is} \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \frac{5 + {\mathrm{e}}^{\mathrm{x}}}{2 + \mathrm{y}}\frac{\mathrm{dy}}{\mathrm{dx}} {\mathrm{e}}^{\mathrm{x}} = 0 \mathrm{satisfying} \\ \mathrm{y}\left(0\right) = 1, \mathrm{then} \mathrm{a} \mathrm{value} \mathrm{of} {\mathrm{ylog}}_{\mathrm{e}}\left(13\right) \mathrm{is} :\hspace{0.17em} \end{gathered}$$

•  - 1

• 0

• 2

• 1

$$\begin{gathered} \mathrm{If} \int \left({\mathrm{e}}^{2\mathrm{x}} + 2{\mathrm{e}}^{\mathrm{x}} - {\mathrm{e}}^{ - \mathrm{x}} - 1\right){\mathrm{e}}^{\left({\mathrm{e}}^{\mathrm{x}} +{\mathrm{e}}^{ - \mathrm{x}}\right)}\mathrm{dx} = \mathrm{g}\left(\mathrm{x}\right){\mathrm{e}}^{\left({\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{ - \mathrm{x}}\right)} +\hspace{0.17em}\mathrm{C}, \\ \mathrm{where} \mathrm{c} \mathrm{is} \mathrm{a} \mathrm{constant} \mathrm{of} \mathrm{integration}, \mathrm{then} \mathrm{g}\left(0\right) \mathrm{is} \end{gathered}$$

• 1

• e

• e²

• 2

If the volume of a parallelopiped, whose conterminous edges are given by the vectors $$\begin{gathered} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \mathrm{n}\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} +\hspace{0.17em}4\hat{\mathrm{j}} - \mathrm{n}\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \mathrm{n}\hat{\mathrm{j}} + 3\hat{\mathrm{k}} \left(\mathrm{n} \ge 0\right), \\ \mathrm{is} 158 \mathrm{cu}-\mathrm{units}, \mathrm{then} :\hspace{0.17em} \end{gathered}$$

• n = 7

• n = 9

• $\overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} = 10$

• $\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{c}} = 17$

The value of ${\int }_{\raisebox{1ex}{$ - \mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{1}{1 +{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}d\mathrm{x} \mathrm{is} :$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{2}$

• $\mathrm{\pi }$