11.

A ray of light coming from the point (2, 2$\sqrt{3}$) is incident at an angle 30° on the line x = 1 at the point A. The ray gets reflected on the line x = 1 and meets x-axis at the point B. Then, the line AB passes through the point ;

• $\left(4, - \frac{\sqrt{3}}{2}\right)$

• $\left(3, - \sqrt{3}\right)$

• $(4, - \sqrt{3})$

• $\left(3, - \frac{1}{\sqrt{3}}\right)$

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be two roots of the equation x² – 64x + 256 = 0. Then the value of ${\left(\frac{{\mathrm{\alpha }}^3}{{\mathrm{\beta }}^5}\right)}^{\frac{1}{8}} + {\left(\frac{{\mathrm{\beta }}^3}{{\mathrm{\alpha }}^5}\right)}^{\frac{1}{8}} = ?$

• 2

• 4

• 4

• 3

Let m and M be respectively the minimum and maximum value values of $\left|\begin{array}{ccc}{\cos }^2\left(\mathrm{x}\right)& 1 + {\sin }^2\left(\mathrm{x}\right)& \sin \left(2\mathrm{x}\right)\\ 1 + {\cos }^2\left(\mathrm{x}\right)& {\sin }^2\left(\mathrm{x}\right)& \sin \left(2\mathrm{x}\right)\\ {\cos }^2\left(\mathrm{x}\right)& {\sin }^2\left(\mathrm{x}\right)& 1 + \sin \left(2\mathrm{x}\right)\end{array}\right|$

Then the ordered pair (m, M) = ?

• ( - 3, 3)

• (1, 3)

• ( - 3, - 1)

• ( - 4, - 1)

The values of $\mathrm{\lambda } \mathrm{and} \mathrm{\mu }$ for which the system of linear equations 

x + y + z = 2

x + 2y + 3z = 5

x + 3y + $\mathrm{\lambda z} = \mathrm{\mu }$

has infinitely many solutions are respectively :

• 5 and 8

• 4 and 9

• 6 and 8

• 5 and 7

The position of moving car at time tis given by f(t) = at² + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average speed of the car over the time interval [t₁, t₂] is attained at the point

• $\frac{{\mathrm{t}}_1 + {\mathrm{t}}_2}{2}$

• $\frac{{\mathrm{t}}_2 - {\mathrm{t}}_1}{2}$

• $2\mathrm{a}\left({\mathrm{t}}_1 +\hspace{0.17em} {\mathrm{t}}_2\right) \hspace{0.17em}+ \mathrm{b}$

• $\mathrm{a}\left({\mathrm{t}}_2 - {\mathrm{t}}_1\right) + \mathrm{b}$

$\mathrm{The} \mathrm{area} (\mathrm{in} \mathrm{sq}. \mathrm{units}) \mathrm{of} \mathrm{the} \mathrm{region} \mathrm{A} = \left\{\left(\mathrm{x}, \mathrm{y}\right) :\hspace{0.17em}\left|\mathrm{x}\right| + \left|\mathrm{y}\right| \le 1, 2{\mathrm{y}}^2 \ge \left|\mathrm{x}\right|\right\} :\hspace{0.17em}$

• $\frac{5}{6}$

• $\frac{7}{6}$

• $\frac{1}{3}$

• $\frac{1}{6}$

The general solution of the differential equation

$\sqrt{1 + {\mathrm{x}}^2 + {\mathrm{y}}^2 +{\mathrm{x}}^2{\mathrm{y}}^2} + \mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}} = 0 \left(\mathrm{where} \mathrm{C} \mathrm{is} \mathrm{constant} \mathrm{of} \mathrm{integration} \right)$

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

• $\sqrt{1 + {\mathrm{y}}^2} + \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 \hspace{0.17em}+ {\mathrm{x}}^2} +\hspace{0.17em}1}{\sqrt{1 +{\mathrm{x}}^2} - 1}\right) +\hspace{0.17em}\mathrm{C}$

• $\sqrt{1 + {\mathrm{y}}^2} - \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 \hspace{0.17em}+ {\mathrm{x}}^2} -\hspace{0.17em}1}{\sqrt{1 +{\mathrm{x}}^2 + 1} }\right) +\hspace{0.17em}\mathrm{C}$

• $\sqrt{1 + {\mathrm{y}}^2} - \sqrt{1 + {\mathrm{x}}^2} = \frac{1}{2}{\log }_{\mathrm{e}}\left(\frac{\sqrt{1 \hspace{0.17em}+ {\mathrm{x}}^2} +\hspace{0.17em}1}{\sqrt{1 +{\mathrm{x}}^2 - 1} }\right) +\hspace{0.17em}\mathrm{C}$

The shortest distance between the lines $\frac{\mathrm{x} - 1}{0} = \frac{\mathrm{y} + 1}{ - 1} = \frac{\mathrm{z}}{1} \mathrm{and} \mathrm{x} + \mathrm{y} + \mathrm{z} + 1 = 0, 2\mathrm{x} - \mathrm{y} + \mathrm{z} + 3 = 0 \mathrm{is}$

• $\frac{1}{2}$

• 1

• $\frac{1}{\sqrt{3}}$

• $\frac{1}{\sqrt{2}}$

$\mathrm{If} {\mathrm{I}}_1 = {\int }_0^1{\left(1 - {\mathrm{x}}^{50}\right)}^{100}d\mathrm{x} \mathrm{and} {\mathrm{I}}_2 = {\int }_0^1{\left(1 - {\mathrm{x}}^{50}\right)}^{101}d\mathrm{x} {\mathrm{I}}_2 = {\mathrm{\alpha I}}_1 \mathrm{Then} \mathrm{\alpha } = ?$

• $\frac{5049}{5050}$

• $\frac{5051}{5050}$

• $\frac{5050}{5051}$

• $\frac{5050}{5049}$

$\mathrm{If} \left(\mathrm{x} + \mathrm{y}\right) = \mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right) \mathrm{and} \sum _{\mathrm{x} = 1}^{\infty } \mathrm{f}\left(\mathrm{x}\right) = 2, \mathrm{x}, \mathrm{y} \in \mathrm{N} \mathrm{is} \mathrm{the} \mathrm{set} \mathrm{of} \mathrm{all} \mathrm{natural} \mathrm{numbers}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \frac{\mathrm{f}\left(4\right)}{\mathrm{f}\left(2\right)} \mathrm{is} :$

• $\frac{2}{3}$

• $\frac{1}{3}$

• $\frac{1}{9}$

• $\frac{4}{9}$