$$\begin{gathered} \mathrm{If} \mathrm{A} = \left\{\mathrm{x} \in \mathrm{Z} : {\mathrm{x}}^3 - 1 = 0\right\} \mathrm{and} \mathrm{B} = \left\{\mathrm{x} \in \mathrm{Z} :\hspace{0.17em}{\mathrm{x}}^2 + \mathrm{x} + 1 = 0\right\}, \\ \mathrm{where} \mathrm{Z} \mathrm{is} \mathrm{set} \mathrm{of} \mathrm{complex} \mathrm{numbers}, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{A} \cap \mathrm{B} = ? \end{gathered}$$

• Null set

• $\left\{\frac{ - 1 + \sqrt{3}\mathrm{i}}{4}, \frac{ - 1 - \sqrt{3}\mathrm{i}}{4}\right\}$

• $\left\{\frac{ - 1 + \sqrt{3}\mathrm{i}}{4}, \frac{ - 1 - \sqrt{3}\mathrm{i}}{4}\right\}$

• $\left\{\frac{1 + \sqrt{3}\mathrm{i}}{2}, \frac{1 - \sqrt{3}\mathrm{i}}{2}\right\}$

$$\begin{gathered} \mathrm{The} \mathrm{common} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equations} {\mathrm{z}}^3 + 2\mathrm{z} + 2\mathrm{z} + 1 = 0 \\ \mathrm{and} {\mathrm{z}}^{2017} + {\mathrm{z}}^{2018} + 1 = 0 \mathrm{are} : \end{gathered}$$

• $- 1, \mathrm{\omega }$

• $1, {\mathrm{\omega }}^2$

• $- 1, {\mathrm{\omega }}^2$

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

A complex number is given by $\mathrm{z} = \frac{1 + 2\mathrm{i}}{1 - {\left(1 - \mathrm{i}\right)}^2}.$ What is the modulus of z ?

• 4

• 2

• 1

• $\frac{1}{2}$

A complex number is given by z = $\frac{1 + 2\mathrm{i}}{1 - {\left(1 - \mathrm{i}\right)}^2}$. What is the principal argument of z ?

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

The equation px² + qx + r = 0 (where p, q, r, all are positive) has distinct real roots a and b. Which one of the following is correct ?

• $\mathrm{a} > 0, \mathrm{b} > 0$

• $\mathrm{a} < 0, \mathrm{b} < 0$

• $\mathrm{a} > 0, \mathrm{b} < 0$

• $\mathrm{a} < 0, \mathrm{b} > 0$

The number of real roots for the equation ${\mathrm{x}}^2 + 9\left|\mathrm{x}\right| + 20 = 0$ is : 

• Zero

• One

• Two

• Three

What is the principal argument of (- 1 - i), where i = $\sqrt{ - 1} ?$

• $\frac{\mathrm{\pi }}{4}$

• $- \frac{\mathrm{\pi }}{4}$

• $- \frac{3\mathrm{\pi }}{4}$

• $\frac{3\mathrm{\pi }}{4}$

Let $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ be real numbers and z be a complex number. If ${\mathrm{z}}^2 + \mathrm{\alpha z} + \mathrm{\beta } = 0$ has two distinct non-real roots with Re(z) = 1, then it is necessary that :

• $\mathrm{\beta } \in \left( - 1, 0\right)$

• $\left|\mathrm{\beta }\right| = 1$

• $\mathrm{\beta } \in \left(1, \infty \right)$

• $\mathrm{\beta } \in \left(0, 1\right)$

9.

The number of non-zero integral solutions of the equation   ${\left|1 - 2\mathrm{i}\right|}^{\mathrm{x}} = 5^{\mathrm{x}}$ is :

• Zero (No solution)

• One

• Two

• Three

$$\begin{gathered} \mathrm{If} \mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{are} \mathrm{different} \mathrm{complex} \mathrm{numbers} \mathrm{with} \left|\mathrm{\alpha }\right| = 1, \\ \mathrm{Then} \mathrm{what} \mathrm{is} \left|\frac{\mathrm{\alpha } - \mathrm{\beta }}{1 - \mathrm{\alpha \beta }}\right| = ? \end{gathered}$$

• $\left|\mathrm{\beta }\right|$

• 2

• 1

• 0