1) Based on the information obtainable from the graph in Figure 2, show, with appropriate reasoning, that the function:
$$f ( x ) = \sqrt { 2 } - \frac { e ^ { x } + e ^ { - x } } { 2 } \quad x \in \mathbb { R }$$
adequately represents the profile of the platform for $x \in [ - a ; a ]$; also determine the value of the endpoints $a$ and $-a$ of the interval.

Ministry of Education, University and Research

To visualize the complete profile of the platform on which the bicycle will be able to move, several copies of the graph of the function $f(x)$ relating to the interval $[-a; a]$ are placed side by side, as shown in Figure 3.
[Figure]
Figure 3

2) For the bicycle to proceed smoothly on the platform it is necessary that: – to the left and right of the points of non-differentiability the sections of the graph are orthogonal; – the length of the side of the square wheel equals the length of a "bump", that is, the arc of the curve with equation $y = f(x)$ for $x \in [-a; a]$.
Establish whether these conditions are satisfied.${ } ^ { 1 }$

3) Considering the similarity of the right triangles ACL and ALM in Figure 4, and recalling the geometric meaning of the derivative, verify that the value of the ordinate $d$ of the centre of the wheel remains constant during motion. Therefore, it seems to the cyclist that they are moving on a flat surface.
[Figure]
Figure 4
\footnotetext{${ } ^ { 1 }$ In general, the length of the arc of a curve with equation $y = \varphi ( x )$ between the abscissae $x _ { 1 }$ and $x _ { 2 }$ is given by $\int _ { x _ { 1 } } ^ { x _ { 2 } } \sqrt { 1 + \left( \varphi ^ { \prime } ( x ) \right) ^ { 2 } } d x$. }

Ministry of Education, University and Research

The graph of the function:
$$f ( x ) = \frac { 2 } { \sqrt { 3 } } - \frac { e ^ { x } + e ^ { - x } } { 2 } , \quad \text { for } x \in \left[ - \frac { \ln ( 3 ) } { 2 } ; \frac { \ln ( 3 ) } { 2 } \right]$$
if replicated several times, can also represent the profile of a platform suitable for being traversed by a bicycle with very particular wheels, having the shape of a regular polygon.

5. Given the points $A(-2,3,1), B(3,0,-1), C(2,2,-3)$, determine the equation of the line $r$ passing through $A$ and $B$ and the equation of the plane $\pi$ perpendicular to $r$ and passing through $C$.

6. Determine the real number $a$ so that the value of
$$\lim _ { x \rightarrow 0 } \frac { \sin ( x ) - x } { x ^ { a } }$$
is a non-zero real number.

7. Determine the coordinates of the centres of the spheres with radius $\sqrt{6}$ tangent to the plane $\pi$ with equation:
$$x + 2 y - z + 1 = 0$$
at its point $P$ with coordinates $(1,0,2)$.

8. A die has the shape of a regular dodecahedron with faces numbered from 1 to 12. The die is loaded so that the face marked with the number 3 appears with a probability $p$ double that of each other face. Determine the value of $p$ as a percentage and calculate the probability that in 5 rolls of the die the face number 3 comes up at least 2 times.

9. Prove that the equation:
$$\arctan ( x ) + x ^ { 3 } + e ^ { x } = 0$$
has one and only one real solution.

10. Given the function:
$$f ( x ) = \left| 4 - x ^ { 2 } \right|$$
verify that it does not satisfy all the hypotheses of Rolle's theorem in the interval $[-3; 3]$ and that nevertheless there exists at least one point in the interval $[-3; 3]$ where the first derivative of $f(x)$ vanishes. Does this example contradict Rolle's theorem? Justify your answer thoroughly.
\footnotetext{Maximum duration of the examination: 6 hours. The use of scientific and/or graphing calculators is permitted provided they are not equipped with symbolic computation capability (O.M. no. 257 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. }