$\mathrm{The} \mathrm{equation} \left|1 - \mathrm{x}\right| + {\mathrm{x}}^2 = 5 \mathrm{has} :$

• a rational root and an irrational root

• two rational roots

• two irrational roots

• no real roots

$\mathrm{What} \mathrm{is} {\mathrm{i}}^{1000} + {\mathrm{i}}^{1001} + {\mathrm{i}}^{1002} + {\mathrm{i}}^{1003} \mathrm{equal} \mathrm{to} \left(\mathrm{where} \mathrm{i} = \sqrt{ - 1}\right) ?$

• 0

• i

•  - i

• 1

The modulus-amplitude form of $\sqrt{3} +\hspace{0.17em}\mathrm{i}, \mathrm{where} \mathrm{i} = \sqrt{ - 1} \mathrm{is} :$

• $2\left(\cos \left(\frac{\mathrm{\pi }}{3}\right) + \mathrm{isin}\left(\frac{\mathrm{\pi }}{3}\right)\right)$

• $2\left(\cos \left(\frac{\mathrm{\pi }}{6}\right) + \mathrm{isin}\left(\frac{\mathrm{\pi }}{6}\right)\right)$

• $4\left(\cos \left(\frac{\mathrm{\pi }}{3}\right) + \mathrm{isin}\left(\frac{\mathrm{\pi }}{3}\right)\right)$

• $4\left(\cos \left(\frac{\mathrm{\pi }}{6}\right) + \mathrm{isin}\left(\frac{\mathrm{\pi }}{6}\right)\right)$

Let [x] denote the greatest integer function. What is the number of solutions of the equation x² - 4x + [x] = 0 in the interval [0, 2] ?

• Zero (No solution)

• One

• Two

• Three

15.

$$\begin{gathered} \mathrm{If} \mathrm{\alpha } \mathrm{and} \mathrm{\beta }( \ne 0) \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{quadratic} \mathrm{equation} {\mathrm{x}}^2 + \mathrm{\alpha x} - \mathrm{B} = 0, \\ \mathrm{then} \mathrm{the} \mathrm{quadratic} \mathrm{expression} -{\mathrm{x}}^2 + \mathrm{\alpha x} + \mathrm{\beta } \mathrm{where} \mathrm{x} \in \mathrm{R} \mathrm{has} \end{gathered}$$

• $\mathrm{Least} \mathrm{value} - \frac{1}{4}$

• $\mathrm{Least} \mathrm{value} - \frac{9}{4}$

• $\mathrm{Greatest} \mathrm{value} \frac{1}{4}$

• $\mathrm{Greatest} \mathrm{value} \frac{9}{4}$

What is the value of ${\left(\frac{ - 1 + \mathrm{i}\sqrt{3}}{2}\right)}^{3\mathrm{n}} + {\left(\frac{ - 1 - \mathrm{i}\sqrt{3}}{2}\right)}^{3\mathrm{n}}$

$\mathrm{where} \mathrm{i} = \sqrt{ - 1} ?$

• 3

• 2

• 1

• 0

Suppose f(x) is such a quadratic expression that it is positive for all real x.

If g(x) = f(x) + f'(x) + f "(x), then for any real x

• $\mathrm{g}\left(\mathrm{x}\right) < 0$

• $\mathrm{g}\left(\mathrm{x}\right) > 0$

• g(x) = 0

• $\mathrm{g}\left(\mathrm{x}\right) \ge 0$

$$\begin{gathered} \mathrm{If} \cos \left(\mathrm{\alpha }\right) \mathrm{and} \cos \left(\mathrm{\beta }\right) (0 < \mathrm{\alpha } < \mathrm{\beta } < \mathrm{\pi }) \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{quadratic} \\ \mathrm{equation} 4{\mathrm{x}}^2 - 3 = 0, \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{the} \mathrm{value} \mathrm{of} \sec \left(\mathrm{\alpha }\right) \mathrm{x} \sec \left(\mathrm{\beta }\right) ? \end{gathered}$$

• $- \frac{4}{3}$

• $\frac{3}{4}$

• $\frac{3}{4}$

• $- \frac{3}{4}$

The ratio of roots of the equations ax² + bx + c = 0 and px² + qx + r = 0 are equal. If D₁ and D₂ are respective discriminants, then what is $\frac{{\mathrm{D}}_1}{{\mathrm{D}}_2}$ equal to ?

• $\frac{{\mathrm{a}}^2}{{\mathrm{p}}^2}$

• $\frac{{\mathrm{b}}^2}{{\mathrm{q}}^2}$

• $\frac{{\mathrm{c}}^2}{{\mathrm{r}}^2}$

• None of the above

The value of ${\mathrm{i}}^{2\mathrm{n}} + {\mathrm{i}}^{2\mathrm{n} + 1} + {\mathrm{i}}^{2\mathrm{n} + 2} + {\mathrm{i}}^{2\mathrm{n} + 3}$, where i = $\sqrt{ - 1}$ is

• 0

• 1

• i

• - i