If w is a cube root of unity, then $\left|\begin{array}{ccc}1& \mathrm{w}& {\mathrm{w}}^2\\ \mathrm{w}& {\mathrm{w}}^2& 1\\ {\mathrm{w}}^2& 1& \mathrm{w}\end{array}\right|$ is equal to

• 1

• w

• w²

• 0

The domain of definition of function $\mathrm{f}\left(\mathrm{x}\right) = \frac{1 +\hspace{0.17em}2{\left(\mathrm{x} +\hspace{0.17em}4\right)}^{- 0.5}}{2 - {\left(\mathrm{x} +\hspace{0.17em}4\right)}^{- 0.5}} + {\left(\mathrm{x} + 4\right)}^{0.5} + 4{\left(\mathrm{x} + 4\right)}^{0.5}$ is

• R

• (- 4, 4)

• R⁺

• $\left(- 4, 0\right) \cup \left(0, \infty \right)$

13.

The set of points where the functton f(x) = $\mathrm{x}\left|\mathrm{x}\right|$ is differentiable is

• $\left(- \infty , \infty \right)$

• $\left(- \infty , 0\right) \cup \left(0, \infty \right)$

• $\left(0, \infty \right)$

• $\left[0, \infty \right]$

If $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{1 - \sin \left(\mathrm{x}\right)}{{\left(\mathrm{\pi } - 2\mathrm{x}\right)}^2} . \frac{\log \left(\sin \left(\mathrm{x}\right)\right)}{\log \left(1 + {\mathrm{\pi }}^2 - 4\mathrm{\pi x} + 4{\mathrm{x}}^2\right)}, \mathrm{x} \ne \frac{\mathrm{\pi }}{2} \\ \mathrm{k} , \mathrm{x} = \frac{\mathrm{\pi }}{2}\right\} \end{gathered}$$ is continuous at x = $\frac{\mathrm{\pi }}{2}$, then k is equal to

• $- \frac{1}{16}$

• $- \frac{1}{32}$

• $- \frac{1}{64}$

• $- \frac{1}{28}$

If y = 4x - 5 is a tangent to the curve y² = px³ + q at (2, 3), then

• p = 2, q = - 7

• p = - 2, q = 7

• p = - 2, q = - 7

• p = 2, q = 7

On which of the following intervals is the function f(x) = 2x² - $\log \left|\mathrm{x}\right|, \mathrm{x} \ne 0$ increasing 2.

• $\left(\frac{1}{2}, \infty \right)$

• $\left(- \infty , - \frac{1}{2}\right) \cup \left(\frac{1}{2}, \infty \right)$

• $\left(- \infty , - \frac{1}{2}\right) \cup \left(0, \frac{1}{2}\right)$

• $\left(- \frac{1}{2}, 0\right) \cup \left(\frac{1}{2}, \infty \right)$

The greatest value of f(x) = (x + 1)^1/3 - (x - 1)^1/3 on [0, 1] is

• 1

• 2

• 3

• $\frac{1}{3}$

$\int \frac{{\mathrm{x}}^{\mathrm{e} - 1} + {\mathrm{e}}^{\mathrm{x} - 1}}{{\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}}\mathrm{dx}$ is equal to

• $\frac{1}{\mathrm{e}}\log \left({\mathrm{x}}^{\mathrm{e}} - {\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{c}$

• $\frac{1}{\mathrm{e}}\log \left({\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{c}$

• $\frac{1}{\mathrm{e}}\log \left({\mathrm{e}}^{\mathrm{x}} - {\mathrm{x}}^{\mathrm{e}}\right) + \mathrm{c}$

• None of the above

${\int }_0^1\frac{\mathrm{dx}}{\sqrt{1 + \mathrm{x}} + \sqrt{\mathrm{x}}}$ is equal to

• $\frac{4}{3}\left(\sqrt{2} - 1\right)$

• $\frac{3}{4}\left(\sqrt{2} - 1\right)$

• $\frac{4}{3}\left(1 - \sqrt{2}\right)$

• $\frac{3}{4}\left(1 - \sqrt{2}\right)$

${\int }_0^1\left|5\mathrm{x} - 3\right|\mathrm{dx}$ is equal to

• $\frac{10}{13}$

• $\frac{31}{10}$

• $\frac{13}{10}$

• None of these