The domain of the function f(x) = $\sqrt{{\log }_{0.5}\left(\mathrm{x}!\right)} \mathrm{is}$

• $\left\{0, 1, 2, 3, ...\right\}$

• $\left\{0, 1, 2, 3, ...\right\}$

• $\left(0, \infty \right)$

• $\left\{0, 1\right\}$

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 1\right| + \left|\mathrm{x} - 2\right| + \left|\mathrm{x} - 3\right|, 2 < \mathrm{x} < 3, \mathrm{then} \mathrm{f} \mathrm{is}$

• an onto function but not one-one

• one-one function but not onto

• a bijection

• neither one-one nor onto

The greatest positive integer which divides (n + 16)(n + 17)(n + 18)(n +19), for all positive integers n, is

• 6

• 24

• 28

• 20

4.

If x_1, x_2, x₃ as well as y_1, y_2, y₃ are in geometric progression with the same common ratio,then the points, $\left({\mathrm{x}}_1, {\mathrm{y}}_1\right), \left({\mathrm{x}}_2, {\mathrm{y}}_2\right), \left({\mathrm{x}}_3, {\mathrm{y}}_3\right)$ are

• vertices of an equilateral triangle

• vertices of a right angled triangle

• vertices of a right angled isosceles triangle

• collinear

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

 $$\begin{gathered} \mathrm{If} 1, {\mathrm{z}}_1, {\mathrm{z}}_2, ..., {\mathrm{z}}_{\mathrm{n} - 1} \mathrm{are} \mathrm{the} \mathrm{nth} \mathrm{roots} \mathrm{of} \mathrm{unity}, \\ \mathrm{then} \left(1 - {\mathrm{z}}_1\right)\left(1 - {\mathrm{z}}_2\right) ... \left(1 - {\mathrm{z}}_{\mathrm{n} - 1}\right) = ? \end{gathered}$$

• 0

• n - 1

• n

• 1

$\mathrm{If} {12}^{4 + 2{\mathrm{x}}^2} = {\left(24\sqrt{3}\right)}^{3{\mathrm{x}}^2 - 2}, \mathrm{then} \mathrm{x} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\pm \sqrt{\frac{13}{12}}$

• $\pm \sqrt{\frac{14}{5}}$

• $\pm \sqrt{\frac{12}{13}}$

• $\pm \sqrt{\frac{5}{14}}$

The product and sum of the roots of the equation $\left|{\mathrm{x}}^2\right| - 5\left|\mathrm{x}\right| - 24 = 0$ are respectively

• - 64, 0

• - 24, 5

• 5, - 24

• 0, 72

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