The minimum value of 2^sin(x)+ 2^cos(x) is :

• $2^{ - 1 + \frac{1}{\sqrt{2}}}$

• $2^{ - 1 + \sqrt{2}}$

• $2^{1 - \sqrt{2}}$

• $2^{1 - \frac{1}{\sqrt{2}}}$

If a and b are real numbers suchthat (2 + $\mathrm{\alpha }$)⁴ = a + b$\mathrm{\alpha }$, where $\mathrm{\alpha } = \frac{ - 1 + \mathrm{i}\sqrt{3}}{2}$ then a + b is equal to :

• 33

• 24

• 9

• 57

$$\begin{gathered} \mathrm{Let} \mathrm{\lambda } \ne 0 \mathrm{be} \mathrm{in} \mathrm{R}. \mathrm{If} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation}, {\mathrm{x}}^2 – \mathrm{x} + 2\mathrm{\lambda } = 0 \mathrm{and} \\ \mathrm{\alpha } \mathrm{and} \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation}, 3{\mathrm{x}}^2 – 10\mathrm{x} + 27\mathrm{\lambda } = 0, \mathrm{then} \frac{\mathrm{\beta \gamma }}{\mathrm{\lambda }} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• 27

• 36

• 9

• 18

$$\begin{gathered} \mathrm{The} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{\mathrm{\pi }}{4} + {\tan }^{-1}\left(\mathrm{x}\right), \left|\mathrm{x}\right| \le 1 \\ \frac{1}{2}\left(\left|\mathrm{x}\right| - 1\right), \left|\mathrm{x}\right| > 1 \right\} \mathrm{is} : \end{gathered}$$

• $\mathrm{continuous} \mathrm{on} \mathrm{R} – \{ – 1\} \mathrm{and} \mathrm{differentiable} \mathrm{on} \mathrm{R} – \left\{ – 1, 1\right\}$

• $\mathrm{both} \mathrm{continuous} \mathrm{and} \mathrm{differentiable} \mathrm{on} \mathrm{R} – \left\{– 1\right\}$

• $\mathrm{both} \mathrm{continuous} \mathrm{and} \mathrm{differentiable} \mathrm{on} \mathrm{R} – \left\{1\right\}$

• $\mathrm{continuous} \mathrm{on} \mathrm{R} – 1 \mathrm{and} \mathrm{differentiable} \mathrm{on} \mathrm{R} – \left\{– 1, 1\right\}$

If the system of equations

$$\begin{gathered} \mathrm{x} + \mathrm{y} + \mathrm{z} = 2 \\ 2\mathrm{x} + 4\mathrm{y} – \mathrm{z} = 6 \\ 3\mathrm{x} + 2\mathrm{y} + \mathrm{\lambda z} = \mathrm{\mu } \end{gathered}$$

• $\mathrm{\lambda } + 2\mathrm{\mu } = 14$

• $2\mathrm{\lambda } - \mathrm{\mu } = 5$

• $2\mathrm{\lambda } + \mathrm{\mu } = 14$

• $\mathrm{\lambda } - 2\mathrm{\mu } = - 5$

$$\begin{gathered} \mathrm{The} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \frac{\mathrm{dy}}{\mathrm{dx}} - \frac{\mathrm{y} + 3\mathrm{x}}{{\log }_{\mathrm{e}}\left(\mathrm{y} + 3\mathrm{x}\right)} + 3 = 0 \mathrm{is} \\ (\mathrm{where} \mathrm{C} \mathrm{is} \mathrm{a} \mathrm{constant} \mathrm{of} \mathrm{integration}) \end{gathered}$$

• $\mathrm{x} - 2{\log }_{\mathrm{e}}\left(\mathrm{y} + 3\mathrm{x}\right) = \mathrm{C}$

• $\mathrm{x} - {\log }_{\mathrm{e}}\left(\mathrm{y} = 3\mathrm{x}\right) = \mathrm{C}$

• $\mathrm{y} + 3\mathrm{x} - \frac{1}{2}{\left({\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)}^2 = \mathrm{C}$

• $\mathrm{y} - \frac{1}{2}{\left({\log }_{\mathrm{e}}\left(\mathrm{y} + 3\mathrm{x}\right)\right)}^2 = \mathrm{C}$

In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of scores on the two dice, in each throw is noted. A wins the game if he throws a total of 6 before B throws a total of 7 and B wins the game if he throws a total of 7 before A throws a total of six. The game stops as soon as either of the players wins. The probability of A winning the game is :

• $\frac{30}{61}$

• $\frac{5}{6}$

• $\frac{5}{31}$

• $\frac{31}{61}$

18.

Suppose the vectors x₁, x₂ and x₃ are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b₁, b₂ and b₃ respectively. If

${\mathrm{x}}_1 = \left[\begin{array}{c}1\\ 1\\ 1\end{array}\right], {\mathrm{x}}_2 = \left[\begin{array}{c}0\\ 2\\ 1\end{array}\right], {\mathrm{x}}_3 = \left[\begin{array}{c}0\\ 0\\ 1\end{array}\right], {\mathrm{b}}_1 = \left[\begin{array}{c}1\\ 0\\ 0\end{array}\right], {\mathrm{b}}_2 = \left[\begin{array}{c}0\\ 2\\ 0\end{array}\right]$,

then the determinant of A is equal of A is

• $\frac{3}{2}$

• 4

• $\frac{1}{2}$

• 2

$\mathrm{The} \mathrm{integral} {\int }_{\frac{\mathrm{\pi }}{6}}^{\frac{\mathrm{\pi }}{3}}{\tan }^3\left(\mathrm{x}\right) . {\sin }^2\left(3\mathrm{x}\right)\left(2{\sec }^2\left(\mathrm{x}\right){\sin }^2\left(3\mathrm{x}\right) + 3\tan \left(\mathrm{x}\right)\sin \left(6\mathrm{x}\right)\right)\mathrm{dx} = ?$

• $- \frac{1}{9}$

• $\frac{9}{2}$

• $- \frac{1}{18}$

• $\frac{7}{18}$

The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x² – 1 below the x-axis, is

• $\frac{1}{3\sqrt{3}}$

• $\frac{4}{3}$

• $\frac{4}{3\sqrt{3}}$

• $\frac{2}{3\sqrt{3}}$