1. With reference to the example, determine the expression of the function $y = f(x)$ and the equation of the curve $\Lambda$, so as to be able to perform a test and verify the operation of the machine. You are asked to construct a tile with a more elaborate design that, in addition to respecting conditions a), b) and c) described above, has $f'(0) = 0$ and the area of the coloured part equal to 55\% of the area of the entire tile. For this purpose, consider polynomial functions of second and third degree.
2. After verifying that it is not possible to achieve what is required using a second-degree polynomial function, determine the coefficients $a, b, c, d \in \mathbb{R}$ of the function $f(x)$ polynomial of third degree that satisfies the stated conditions. Finally, represent the resulting tile in a Cartesian plane. Two different types of design are proposed to a customer, derived respectively from the functions $a_n(x) = 1 - x^n$ and $b_n(x) = (1-x)^n$, considered for $x \in [0,1]$, with $n$ a positive integer.
3. Verify that as $n$ varies, all these functions respect conditions a), b) and c). Let $A(n)$ and $B(n)$ be the areas of the coloured parts of the tiles obtained from such functions $a_n$ and $b_n$, calculate $\lim_{n \rightarrow +\infty} A(n)$ and $\lim_{n \rightarrow +\infty} B(n)$ and interpret the results in geometric terms. The customer decides to order 5,000 tiles with the design derived from $a_2(x)$ and 5,000 with the one derived from $b_2(x)$. The painting is carried out by a mechanical arm that, after depositing the colour, returns to the initial position by flying over the tile along the diagonal. Due to a malfunction, during the production of the 10,000 tiles there is a 20\% probability that the mechanical arm drops a drop of colour at a random point along the diagonal, thus staining the newly produced tile.
5. With a fence 2 metres long, one wants to enclose a surface having the shape of a rectangle topped by a semicircle, as in the figure: [Figure] Determine the dimensions of the sides of the rectangle that allow enclosing the surface of maximum area.
6. Determine the equation of the spherical surface $S$, with centre on the line $r: \left\{ \begin{array}{l} x = t \\ y = t \\ z = t \end{array} \right. t \in \mathbb{R}$ tangent to the plane $\pi: 3x - y - 2z + 14 = 0$ at the point $T(-4, 0, 1)$.
8. In a two-player game, each game won earns 1 point and the winner is the first to reach 10 points. Two players who in each game have the same probability of winning challenge each other. What is the probability that one of the two players wins in a number of games less than or equal to 12?
9. Given in three-dimensional space the points $A(3, 1, 0)$, $B(3, -1, 2)$, $C(1, 1, 2)$. After verifying that $ABC$ is an equilateral triangle and that it is contained in the plane $\alpha$ with equation $x + y + z - 4 = 0$, establish which are the points $P$ such that $ABCP$ is a regular tetrahedron.
10. Determine which are the values of the parameter $k \in \mathbb{R}$ for which the function $y(x) = 2e^{kx+2}$ is a solution of the differential equation $y'' - 2y' - 3y = 0$. \footnotetext{Maximum duration of the examination: 6 hours. The use of scientific and/or graphical calculators is permitted provided they are not equipped with symbolic calculation capacity (O.M. no. 350 Art. 18 paragraph 8). The use of a bilingual dictionary (Italian–language of the country of origin) is permitted for candidates whose native language is not Italian. It is not permitted to leave the Institute before 3 hours have elapsed from the dictation of the theme.}