Complex Numbers and Quadratic Equations

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

• - 2i

• 2i

• 2

•  - 2

The number of real roots of the equation ${\mathrm{x}}^5 +\hspace{0.17em}3{\mathrm{x}}^3 + 4\mathrm{x} + 30 = 0$ is

• 1

• 2

• 3

• 5

If the coefficients of the equation whose roots are k times the roots of the equation ${\mathrm{x}}^3 + \frac{1}{4}{\mathrm{x}}^2 - \frac{1}{16}\mathrm{x} + \frac{1}{144} = 0$, are integers, then a possible value of k is

• 3

• 12

• 9

• 4

The harmonic mean of two numbers is $- \frac{8}{5}$ and their geometric mean is 2. The quadratic equation whose roots are twice those numbers is

• ${\mathrm{x}}^2 + 5\mathrm{x} + 4 = 0$

• ${\mathrm{x}}^2 + 10\mathrm{x} + 16 = 0$

• ${\mathrm{x}}^2 - 10\mathrm{x} + 16 = 0$

• ${\mathrm{x}}^2 - 5\mathrm{x} + 4 = 0$

$\mathrm{If} \mathrm{z} \mathrm{is} \mathrm{a} \mathrm{complex} \mathrm{number} \mathrm{with} \left|\mathrm{z}\right| \ge 5. \mathrm{Then} \mathrm{the} \mathrm{least} \mathrm{value} \mathrm{of} \left|\mathrm{z} + \frac{2}{\mathrm{z}}\right| \mathrm{is}$

• $\frac{24}{5}$

• $\frac{26}{5}$

• $\frac{23}{5}$

• $\frac{29}{5}$

$\mathrm{If} \mathrm{\alpha } \mathrm{is} \mathrm{a} \mathrm{non}-\mathrm{real} \mathrm{root} \mathrm{of} {\mathrm{x}}^7 = 1, \mathrm{then} \mathrm{\alpha }(1 + \mathrm{\alpha }) (1 + {\mathrm{\alpha }}^2 + {\mathrm{\alpha }}^4) =$

• 1

• 2

• - 1

• - 2

$$\begin{gathered} \mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} {\mathrm{x}}^3 + {\mathrm{px}}^2 + \mathrm{qx} +\hspace{0.17em}\mathrm{r} = 0 \\ \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \left(1 + {\mathrm{\alpha }}^2\right)\left(1 + {\mathrm{\beta }}^2\right)\left(1 + {\mathrm{\gamma }}^2\right) \mathrm{is} \end{gathered}$$

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

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

• ${\left(\mathrm{r}\hspace{0.17em}+ \mathrm{p}\right)}^2 + {\left(\mathrm{q} + 1\right)}^2$

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

The imaginary part of

${\left(3 +\hspace{0.17em}2\sqrt{ - 54}\right)}^{\frac{1}{2}} - {\left(3 - 2\sqrt{ - 54}\right)}^{\frac{1}{2}} \mathrm{can} \mathrm{be}$

• $\sqrt{ - 6}$

• $\sqrt{6}$

• $- 2\sqrt{6}$

• 6

$$\begin{gathered} \mathrm{Let} \mathrm{\alpha }, \mathrm{\beta } \mathrm{are} \mathrm{roots} \mathrm{of} {\mathrm{x}}^2 + \mathrm{px} + 2 = 0 \mathrm{and} \frac{1}{\mathrm{\alpha }}, \frac{1}{\mathrm{\beta }} \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} 2{\mathrm{x}}^2 - 2\mathrm{qx} + 1 = 0. \\ \mathrm{Then} \mathrm{find} \mathrm{the} \mathrm{value} \mathrm{of} \left(\mathrm{\alpha } + \frac{1}{\mathrm{\beta }}\right)\left(\mathrm{\beta } + \frac{1}{\mathrm{\alpha }}\right)\left(\mathrm{\alpha } - \frac{1}{\mathrm{\alpha }}\right)\left(\mathrm{\beta } - \frac{1}{\mathrm{\beta }}\right) \end{gathered}$$

• $\frac{9}{4}\left(9 - {\mathrm{p}}^2\right)$

• $\frac{9}{4}\left(9 + {\mathrm{p}}^2\right)$

• $\frac{4}{9}\left(9 - {\mathrm{q}}^2\right)$

• $\frac{9}{4}\left(9 - {\mathrm{q}}^2\right)$

If z₁, z₂ are complex numbers such that Re(z₁) = |z₁ = 1| and Re(z₂) = |z₂ = 1| and arg (z₁ – z₂) = $\frac{\mathrm{\pi }}{6}$, then Im (z₁ + z₂) is equal to :

• $2\sqrt{3}$

• $\frac{\sqrt{3}}{2}$

• $\frac{1}{\sqrt{3}}$

• $\frac{2}{\sqrt{3}}$