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

• - 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

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

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

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

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

If p and q are distinct prime numbers and if the equation x² - px + q = 0 has positive integers as its roots, then the roots of the equation are

• 1, - 1

• 2, 3

• 1, 2

• 3, 1

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

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

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

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

$$\begin{gathered} \mathrm{If} \mathrm{\omega } \mathrm{is} \mathrm{a} \mathrm{complex} \mathrm{cube} \mathrm{root} \mathrm{of} \mathrm{unity}, \mathrm{then} \\ {\mathrm{\omega }}^{\left(\frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \infty \right)} + \mathrm{\omega }\left(\frac{1}{2} + \frac{3}{8} + \frac{9}{32} + \infty \right) = ? \end{gathered}$$

• 1

• - 1

• $\mathrm{\omega }$

• i

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

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

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

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

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

If the harmonic mean of the roots of $\sqrt{2}{\mathrm{x}}^2 - \mathrm{bx} + \left(8 - 2\sqrt{\mathrm{d}}\right) = 0$ is 4, then the value of b is

• 2

• 3

• $4 - \sqrt{5}$

• $4 + \sqrt{5}$

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

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

• $\left[\frac{1}{2}, 2\right]$

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

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

269.

The locus of the point representing the complex number z for which ${\left|\mathrm{z} + 3\right|}^2 - {\left|\mathrm{z} - 3\right|}^2 = 15 \mathrm{is}$

• a circle

• a parabola

• a straight line

• an ellipse

$\frac{{\left(1 + \mathrm{i}\right)}^{2016}}{{\left(1 - \mathrm{i}\right)}^{2014}} = ?$

• - 2i

• 2i

• 2

•  - 2