61.

Select the correct statement from (a), (b), (c), (d). The function f(x) = xe^{1 - x}

• strictly increases in the interval $\left(\frac{1}{2}, 2\right)$

• increases in the interval (0, $\infty$)

• decreases in the interval (0, 2)

• strictly decreases in the interval (1, $\infty$)

The function f(x) = ${\mathrm{e}}^{\mathrm{ax}} + {\mathrm{e}}^{- \mathrm{ax}}$, a > 0 is monotonically increasing for

• - 1 < x < 1

• x < - 1

• x > - 1

• x > 0

If the matrix $\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$ is commutative with the matrix  $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$, then

• a = 0, b = c

• b = 0, c = d

• c = 0, d = a

• d = 0, a = b

If 1, w, w² are the cube roots of unity, then

$\left|\begin{array}{ccc}1& {\mathrm{w}}^{\mathrm{n}}& {\mathrm{w}}^{2\mathrm{n}}\\ {\mathrm{w}}^{2\mathrm{n}}& 1& {\mathrm{w}}^{\mathrm{n}}\\ {\mathrm{w}}^{\mathrm{n}}& {\mathrm{w}}^{2\mathrm{n}}& 1\end{array}\right|$ has value

• 0

• w

• w²

• w + w²

If I = ${\int }_{- \mathrm{\pi }}^{\mathrm{\pi }}\frac{{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}{{\mathrm{e}}^{\sin \left(\mathrm{x}\right)} +\hspace{0.17em}{\mathrm{e}}^{- \sin \left(\mathrm{x}\right)}}d\mathrm{x}$, then I equals

• $\frac{\mathrm{\pi }}{2}$

• $2\mathrm{\pi }$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

If h(x) = ${\int }_0^{\mathrm{x}}{\sin }^4\left(\mathrm{t}\right)d\mathrm{t}, \mathrm{then} \mathrm{h}\left(\mathrm{x} + \mathrm{\pi }\right)$ equals

• $\frac{\mathrm{h}\left(\mathrm{x}\right)}{\mathrm{h}\left(\mathrm{\pi }\right)}$

• $\mathrm{h}\left(\mathrm{x}\right)\mathrm{h}\left(\mathrm{\pi }\right)$

• $\mathrm{h}\left(\mathrm{x}\right) - \mathrm{h}\left(\mathrm{\pi }\right)$

• $\mathrm{h}\left(\mathrm{x}\right) + \mathrm{h}\left(\mathrm{\pi }\right)$

The degree of the differential equation

${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = \frac{d^2\mathrm{y}}{d{\mathrm{x}}^2}$ is

• 1

• 5

• $\frac{10}{3}$

• 3

The differential equation of all parabolas whose axes are parallel to y-axis, is

• $\frac{d^3\mathrm{y}}{d{\mathrm{x}}^3} = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + \frac{d\mathrm{y}}{d\mathrm{x}} = 0$

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} + \frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = 0$

The solution of the differential equation $\frac{d\mathrm{y}}{d\mathrm{x}} = {\mathrm{e}}^{\mathrm{y} + \mathrm{x}} + {\mathrm{e}}^{\mathrm{y} - \mathrm{x}}$ is

• e^{- y} = e^x - e^{- x} + c, c integrating constant

• e^{- y} = e^- x - e^x + c, c integrating constant

• e^{- y} = e^x + e^{- x} + c, c integrating constant

• e^{- y} + e^x - e^{- x} = c, c integrating constant

The value of the integral ${\int }_0^2\left|{\mathrm{x}}^2 - 1\right|d\mathrm{x}$ is

• 0

• 2

• - $\frac{1}{3}$

• - 2