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

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

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

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

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

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

35.

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

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

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

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

Out of thirty points in a plane, eight of them are collinear. The number of straight lines that can be formed by joining these points, is

• 296

• 540

• 408

• 348

If n is an integer with 0 $\le \mathrm{n} \le$ 11, then the minimum value of n!(11 - 1)! is attained when a value of n equals to

• 11

• 5

• 7

• 9

$\mathrm{If} {\left(\mathrm{a} + \mathrm{bx}\right)}^{ - 3} = \frac{1}{27} + \frac{1}{3}\mathrm{x} + ... , \mathrm{then} \mathrm{the} \mathrm{ordered} \mathrm{pair} \left(\mathrm{a}, \mathrm{b}\right) = ?$

• (3, - 27)

• $\left(1, \frac{1}{3}\right)$

• (3, 9)

• (3, - 9)

The term independent of x in the expansions of${\left(\sqrt{\mathrm{x}} - \frac{2}{\sqrt{\mathrm{x}}}\right)}^{18} \mathrm{is}$

• $- {}^{18}\mathrm{C}_9{2}^9$

• ${}^{18}\mathrm{C}_9{2}^{12}$

• ${}^{18}\mathrm{C}_9{2}^6$

• ${}^{18}\mathrm{C}_6{2}^8$

$\mathrm{If} \frac{2{\mathrm{x}}^3 + {\mathrm{x}}^2 - 5}{{\mathrm{x}}^4 - 25} = \frac{\mathrm{Ax} + \mathrm{B}}{{\mathrm{x}}^2 - 5} + \frac{\mathrm{Cx} \hspace{0.17em}+ 1}{{\mathrm{x}}^2 +\hspace{0.17em}5}, \mathrm{then} \left(\mathrm{A}, \mathrm{B}, \mathrm{C}\right) = ?$

• (1, 1, 1)

• (1, 1, 0)

• (1, 0, 1)

• (1, 2, 1)