1.

The system $\left(\begin{array}{ccc}1& - 1& 2\\ 3& 5& - 3\\ 2& 6& \mathrm{a}\end{array}\right)\left(\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right) = \left(\begin{array}{c}3\\ \mathrm{b}\\ 2\end{array}\right)$ has no solutions, if

• a = - 5, $\mathrm{b} \ne 5$

• a = - 5, b = 5

• $\mathrm{a} \ne - 5, \mathrm{b} = 5$

• $\mathrm{a} \ne - 5, \mathrm{b} \ne 5$

The equation of displacement of a particle is x(t) = 5t² - 7t + 3. The acceleration at the moment when its velocity becomes 5 m/sec is

• 3 m/sec²

• 7 m/sec²

• 10 m/sec²

• 8 m/sec²

The range of x for which the formula $3{\sin }^{-1}\left(\mathrm{x}\right) = {\sin }^{-1}\left[\mathrm{x}\left(3 - 4{\mathrm{x}}^2\right)\right]$ hold is

• $- \frac{1}{2} \le \mathrm{x} \le \frac{1}{2}$

• $- \frac{1}{4} \le \mathrm{x} \le \frac{2}{3}$

• $- \frac{1}{3} \le \mathrm{x} \le 1$

• $- \frac{2}{3} \le \mathrm{x} \le \frac{2}{3}$

The mean value of the function $\mathrm{f}\left(\mathrm{x}\right) = \frac{2}{{\mathrm{e}}^{\mathrm{x}} + 1}$ on the interval [0, 2] is

• $2 - {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 + 1}\right)$

• $2 + {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 + 1}\right)$

• $2 + {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 - 1}\right)$

• $- 2 + {\log }_{\mathrm{e}}\left(\frac{2}{{\mathrm{e}}^2 - 1}\right)$

If $\frac{7}{2} \mathrm{and} 1$ are the roots of the equation $\left|\begin{array}{ccc}2\mathrm{x}& 3& 7\\ 2& 2\mathrm{x}& 2\\ 7& 6& 2\mathrm{x}\end{array}\right|$ = 0, then third root is

• $- \frac{7}{2}$

• $- \frac{9}{2}$

• $- \frac{3}{2}$

• $- \frac{5}{2}$

The function $\mathrm{y} = \sqrt{2\mathrm{x} - {\mathrm{x}}^2}$

• increases in (0, 1) but decreases in (1, 2)

• decreases in (0, 2)

• increases m (1, 2) but decreases in (0, 1)

• increases in (0, 2)

The interval in which the function $\mathrm{y} = \mathrm{x} - 2\sin \left(\mathrm{x}\right)$; $0 \le \mathrm{x} \le 2\mathrm{\pi }$ increases throughout is

• $\left(\frac{5\mathrm{\pi }}{3}, 2\mathrm{\pi }\right)$

• $\left(0, \frac{\mathrm{\pi }}{3}\right)$

• $\left(\frac{\mathrm{\pi }}{3}, \frac{5\mathrm{\pi }}{3}\right)$

• $\left(0, \frac{\mathrm{\pi }}{4}\right)$

The inverse ofthe function y = $\frac{2^{\mathrm{x}}}{1 + 2^{\mathrm{x}}}$ is

• $\mathrm{x} = {\log }_2\left(\frac{1}{1 - 2^{\mathrm{y}}}\right)$

• $\mathrm{x} = {\log }_2\left(1 - \frac{1}{\mathrm{y}}\right)$

• $\mathrm{x} = {\log }_2\left(\frac{1}{1 - \mathrm{y}}\right)$

• $\mathrm{x} = {\log }_2\left(\frac{\mathrm{y}}{1 - \mathrm{y}}\right)$

The domain of the definition of the function

$\mathrm{y} = \frac{1}{{\log }_{10}\left(1 - \mathrm{x}\right)} + \sqrt{\left(\mathrm{x} +\hspace{0.17em}2\right)}$ is

• $\mathrm{x} \ge - 2$

• $- 3 < \mathrm{x} \le - 2$

• $- 2 \le \mathrm{x} < 0$

• $- 2 \le \mathrm{x} < 1$

Let f(x) = $$\begin{gathered} \left\{- 2\sin \left(\mathrm{x}\right) \mathrm{if} \mathrm{x} \le - \frac{\mathrm{\pi }}{2} \\ \mathrm{Asin}\left(\mathrm{x}\right) + \mathrm{B} \mathrm{if} - \frac{\mathrm{\pi }}{2} < \mathrm{x} < \frac{\mathrm{\pi }}{2} \\ \cos \left(\mathrm{x}\right) \mathrm{if} \mathrm{x} \ge \frac{\mathrm{\pi }}{2}\right\} \end{gathered}$$

For what values of A and B, the function f(x) is continuous throughout the real line ?

• A = - 1, B = 1

• A = - 1, B = - 1

• A = 1, B = - 1

• A = 1, B = 1