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

• - 1

• 1

• 2

• - 2

$\mathrm{In} ∆\mathrm{PQR}, \angle \mathrm{R} = \frac{\mathrm{\pi }}{4}, \tan \left(\frac{\mathrm{P}}{3}\right), \tan \left(\frac{\mathrm{Q}}{3}\right) \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{ax}}^2 + \mathrm{bx} + \mathrm{c} = 0, \mathrm{then}$

• a +b = c

• b + c = 0

• a + c = 0

• b = c

$\mathrm{The} \mathrm{product} \mathrm{of} \mathrm{real} \mathrm{of} \mathrm{the} \mathrm{equation} {\left|\mathrm{x}\right|}^{\raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} - 26{\left|\mathrm{x}\right|}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} - 27 = 0$

• - 3¹⁰

• $- 3^{12}$

• - 3^12/5

• - 3^12/5

If $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are the roots of the equation x³ + px² + qx + r = 0, then the coefficient of x in the cubic equation whose roots are $\mathrm{\alpha }\left(\mathrm{\beta } + \mathrm{\gamma }\right), \mathrm{\beta }\left(\mathrm{\gamma } + \mathrm{\alpha }\right) \mathrm{and} \mathrm{\gamma }\left(\mathrm{\alpha } + \mathrm{\beta }\right) \mathrm{is}$

• 2q

• q² + pr

• p² - qr

• r(pq - r)

If z is complex number such that $\left|\mathrm{z} - \frac{4}{\mathrm{z}}\right| = 2$, then the greatest value of $\left|\mathrm{z}\right|$ is

• $1 + \sqrt{2}$

• $\sqrt{2}$

• $\sqrt{3} + 1$

• $1 + \sqrt{5}$

$$\begin{gathered} \mathrm{If} \mathrm{\alpha } \mathrm{is} \mathrm{a} \mathrm{non}-\mathrm{real} \mathrm{root} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{x}}^6 - 1 = 0, \\ \mathrm{then} \frac{{\mathrm{\alpha }}^2 + {\mathrm{\alpha }}^3 + {\mathrm{\alpha }}^4 + {\mathrm{\alpha }}^5}{\mathrm{\alpha } + 1} = ? \end{gathered}$$

• $\mathrm{\alpha }$

• 1

• 0

• - 1

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of the equation ${\mathrm{x}}^2 - 2\mathrm{x} + 4 = 0$, then ${\mathrm{\alpha }}^9 + {\mathrm{\beta }}^9$ is equal to

• - 2⁸

• 2⁹

• - 2¹⁰

• 2¹⁰

258.

If a complex number z satisfied $\left|{\mathrm{z}}^2 - 1\right| = {\left|\mathrm{z}\right|}^2 + 1$, then z lies on

• the real axis

• the imaginary axis

• y = x

• a circle

The number of solutions for ${\mathrm{z}}^3 + \overline{\mathrm{z}} = 0$ , is

• 5

• 1

• 3

• 2

If x = p + q, y = $\mathrm{p\omega } + {\mathrm{q\omega }}^2 \mathrm{and} \mathrm{z} = {\mathrm{p\omega }}^2 + \mathrm{q\omega }$, where is a complex cube root of unity, then xyz equals to

• p³ + q³

• p³ - pq + q³

• 1 + p³ + q³

• p³ - q³