C

9000072907

Level: 
C
A homogeneous cube with the side \(10\, \mathrm{cm}\) is in the water. The bottom side is parallel to the water surface \(10\, \mathrm{cm}\) below the surface. Find the work required to move the cube to the position when the bottom side just touches the surface of the water. The mass density of the cube is \(2\: 000\, \mathrm{kg\, m}^{-3}\), the mass density of the water is \(1\: 000\, \mathrm{kg\, m}^{-3}\) and the standard acceleration of gravity is \(g = 10\, \mathrm{m\, s}^{-2}\).
\(1.5\, \mathrm{J}\)
\(2\, \mathrm{J}\)
\(1\, \mathrm{J}\)

9000070505

Level: 
C
The dimensions of the box form a geometric sequence. The volume of the box is \(27\, \mathrm{cm}^{3}\) and the length of the shortest side is \(2\, \mathrm{cm}\). Find the surface area of the box.
\(57\, \mathrm{cm}^{2}\)
\(28.5\, \mathrm{cm}^{2}\)
\(27\, \mathrm{cm}^{2}\)
\(35\, \mathrm{cm}^{2}\)
\(45\, \mathrm{cm}^{2}\)

9000071208

Level: 
C
Evaluate the following integral on the interval \((0,+\infty)\). \[ \int x^{2}\ln x\, \mathrm{d}x \]
\(\frac{x^{3}} {3} \left (\ln x -\frac{1} {3}\right ) + c,\ c\in \mathbb{R}\)
\(\frac{x^{2}} {3} + c,\ c\in \mathbb{R}\)
\(x^{2}\left (\frac{x\ln x} {3} -\frac{1} {2}\right ) + c,\ c\in \mathbb{R}\)

9000066006

Level: 
C
Evaluate the following integral on the interval \((0,+\infty)\). \[ \int x\ln x\, \mathrm{d}x \]
\(\frac{1} {2}x^{2}\ln x -\frac{1} {4}x^{2} + c,\ c\in \mathbb{R}\)
\(x\ln x -\frac{1} {2}x^{2} + c,\ c\in \mathbb{R}\)
\(x\ln x - x + c,\ c\in \mathbb{R}\)
\(\frac{1} {2}x^{2} + \frac{1} {|x|} + c,\ c\in \mathbb{R}\)