11.

$$\begin{gathered} \frac{{\mathrm{x}}^2 + \mathrm{x} + 1}{\left(\mathrm{x} - 1\right)\left(\mathrm{x} - 2\right)\left(\mathrm{x} - 3\right)} = \frac{\mathrm{A}}{\mathrm{x} - 1} + \frac{\mathrm{B}}{\mathrm{x} - 2} + \frac{\mathrm{C}}{\mathrm{x} - 3} \\ \Rightarrow \mathrm{A} + \mathrm{C} = \end{gathered}$$

• 4

• 5

• 6

• 8

$\sum _{\mathrm{n} = 1}^{\infty } \frac{2\mathrm{n}}{\left(2\mathrm{n} + 1\right){\displaystyle !}} = ?$

• $\frac{1}{\mathrm{e}}$

• $\frac{\mathrm{e}}{2}$

• e

• 2e

If a > 0 and b² - 4ac = 0, then the curve y = ax² + bx + c

• cuts the x-axis

• touches the x-axis and lies below it

• lies entirely above the x-axis

• touches the x-axis and lies above it

If tan(A) and tan(B) are the roots of the quadratic equation x² - px + q = 0, then sin²(A + B) is equal to

• $\frac{{\mathrm{p}}^2}{{\mathrm{p}}^2 + {\mathrm{q}}^2}$

• $\frac{{\mathrm{p}}^2}{{\mathrm{p}}^2 +\hspace{0.17em}{\mathrm{q}}^2}$

• $1 - \frac{ \mathrm{p}}{{\left(1 - \mathrm{q}\right)}^2}$

• $\frac{{\mathrm{p}}^2}{{\mathrm{p}}^2 + {\left(1 - \mathrm{q}\right)}^2}$

The value of a for which the equations x³ + ax + 1 = 0 and x⁴ + ax² + 1 = 0 have acommon root is

• - 2

• - 1

• 1

• 2

$\mathrm{Let} \mathrm{z} = \mathrm{a} - \frac{\mathrm{i}}{2}; \mathrm{a} \in \mathrm{R}. \mathrm{Then} {\left|\mathrm{i} + \mathrm{z}\right|}^2 - {\left|\mathrm{i} - \mathrm{z}\right|}^2 = ?$

• 2

• - 2

• 4

• - 4

$\mathrm{The} \mathrm{locus} \mathrm{of} \mathrm{the} \mathrm{complex} \mathrm{number} \mathrm{z} \mathrm{such} \mathrm{that} \arg \left(\frac{\mathrm{z} - 2}{\mathrm{z} +\hspace{0.17em}2}\right) = \frac{\mathrm{\pi }}{3} \mathrm{is}$

• a circle 

• a straight line 

• a parabola 

• an ellipse

$\frac{{\left(1 +\hspace{0.17em}\mathrm{i}\right)}^{2011}}{{\left(1 - \mathrm{i}\right)}^{2009}} = ?$

• - 1

• 1

• 2

• - 2

$\mathrm{If} \mathrm{R} \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = 7 + \cos \left(5\mathrm{x} +\hspace{0.17em}3\right) \mathrm{for} \mathrm{x} \in \mathrm{R}, \mathrm{then} \mathrm{the} \mathrm{period} \mathrm{of} \mathrm{f} \mathrm{i}$

• $2\mathrm{\pi }$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{5}$

• $\frac{2\mathrm{\pi }}{5}$

$\cos \left(\mathrm{A}\right) = \frac{3}{4} \Rightarrow 32\sin \left(\frac{\mathrm{A}}{2}\right)\sin \left(\frac{5\mathrm{A}}{2}\right) = ?$

• 7

• 8

• 13

• 11