$\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)

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

• $\mathrm{\alpha }$

• 1

• 0

• - 1

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

$$\begin{gathered} \mathrm{If} {\mathrm{Z}}_{\mathrm{R}} = \cos \left(\frac{\mathrm{\pi }}{2^{\mathrm{r}}}\right) + \mathrm{isin}\left(\frac{\mathrm{\pi }}{2^{\mathrm{r}}}\right) \mathrm{for} \mathrm{r} = 1, 2, 3, ..., \\ \mathrm{then} {\mathrm{Z}}_1{\mathrm{Z}}_2{\mathrm{Z}}_3 ... \infty = ? \end{gathered}$$

• - 2

• 1

• 2

• - 1

If x₁ and x₂ are the real roots of the equation x² - kx + c = 0, then the distance between the points A(x₁, 0) and B(x₂, 0) is

• $\sqrt{{\mathrm{k}}^2 + 4\mathrm{c}}$

• $\sqrt{{\mathrm{k}}^2 - \mathrm{c}}$

• $\sqrt{\mathrm{c} - {\mathrm{k}}^2}$

• $\sqrt{{\mathrm{k}}^2 - 4\mathrm{c}}$

The cubic equation whose roots are the squares of the roots of x³ - 2x² + 10x - 8 = 0, is

• ${\mathrm{x}}^3 + 16{\mathrm{x}}^2 + 68\mathrm{x} - 64 = 0$

• ${\mathrm{x}}^3 + 8{\mathrm{x}}^2 + 68\mathrm{x} - 64 = 0$

• ${\mathrm{x}}^3 + 16{\mathrm{x}}^2 - 68\mathrm{x} - 64 = 0$

229.

The common roots of the equations ${\mathrm{z}}^3 +\hspace{0.17em}2{\mathrm{z}}^2 + 2\mathrm{z} + 1 = 0, {\mathrm{z}}^{2014} + {\mathrm{z}}^{2015} + 1 = 0 \mathrm{are}$

• $\mathrm{\omega }, {\mathrm{\omega }}^2$

• $1, \mathrm{\omega }, {\mathrm{\omega }}^2$

• $- 1, \mathrm{\omega }, {\mathrm{\omega }}^2$

• $- \mathrm{\omega }, - {\mathrm{\omega }}^2$

$\mathrm{For} \mathrm{real} \mathrm{value} \mathrm{of} \mathrm{x}, \mathrm{the} \mathrm{rane} \mathrm{of} \frac{{\mathrm{x}}^2 + 2\mathrm{x} +\hspace{0.17em}1}{{\mathrm{x}}^2 + 2\mathrm{x} - 1} \mathrm{is}$

• $\left(- \infty , 0\right) \cup \left(1, \infty \right)$

• $\left[\frac{1}{2}, 2\right]$

• $\left(- \infty , \frac{- 2}{9}\right) \cup \left(1, \infty \right)$

• $\left(- \infty , - 6\right) \cup \left(- 2, \infty \right)$