1. A triangle $A B C$ is given with sides $A B = a$ and $B C = \sqrt { 3 } a$. Which of the following statements is correct? – If $A \hat { C } B = \frac { \pi } { 6 }$, then the triangle is right-angled; – If the triangle is right-angled, then $A \hat { C } B = \frac { \pi } { 6 }$. Justify your answers.

2. In a piggy bank there are 15 coins, of which 9 are 1 euro coins and the other 6 are 2 euro coins. 6 coins are drawn simultaneously. – What is the probability that the total value of the coins drawn is exactly 10 euros? – What is the probability that the total value of the coins drawn is at most 10 euros?

3. Verify that the points $O ( 0,0,0 ) , A ( 1,4,8 ) , B ( - 6,0,12 )$ and $C ( - 7 , - 4,4 )$ are coplanar. Calculate the area and perimeter of the quadrilateral $O A B C$ and classify it.

4. Determine the domain of the function $f ( x ) = \ln \left( \frac { a x - 7 } { x ^ { 2 } } \right)$, with $a$ a positive real parameter. Subsequently, identify the value of $a$ for which the hypotheses of Rolle's theorem are satisfied on the interval [1; 7] and the coordinates of the point that verifies the conclusion.

5. Determine the values of the real parameters $a$ and $b$ of the function $f ( x ) = \frac { a x ^ { 2 } + b x + 3 } { 2 x ^ { 2 } + 5 x - 1 }$ so that it has the line $y = 2$ as a horizontal asymptote and a stationary point at $x = 1$. For the values found, determine whether $f ( x )$ has further asymptotes.

6. In a Cartesian coordinate system $O x y$, consider the equilateral hyperbola with equation $x y = k$, with $k$ a non-zero real parameter. Let $t$ be the tangent line to the hyperbola at a point $P$ of it. Let $A$ and $B$ be the points where $t$ intersects the axes of the reference frame. Prove that the triangles $A P O$ and $B P O$ are equivalent and that their area does not depend on the choice of $P$.

7. A resistor with resistance $R$ is traversed by a current varying in time with intensity $I ( t ) = I _ { 0 } \frac { a } { t }$, with $t > 0$ and the positive constants $I _ { 0 }$ and $a$ expressed, respectively, in amperes and in seconds. Knowing that the power dissipated in the resistor due to the Joule effect is $P ( t ) = R I ^ { 2 } ( t )$, determine its average value on the interval $[ 2 a ; 3 a ]$.

8. Leonardo Sinisgalli writes, in a passage from Furor Mathematicus: ``I had in mind a chapter on the laws of chance: I wanted to find the relationships between Tartaglia's triangle, relating to the coefficients of the polynomial $( a + b ) ^ { n }$ and Pascal's arithmetic triangle, which gives us the probability of getting $m$ tails in $n$ games played at heads and tails''. Describe the relationship existing between binomial coefficients and the calculation of probabilities.
\footnotetext{Maximum duration of the exam: 6 hours. The use of scientific or graphical calculators is allowed provided they are not equipped with symbolic algebraic processing capability and do not have Internet connectivity. The use of a bilingual dictionary (Italian–language of the country of origin) is allowed for candidates whose native language is not Italian. It is not allowed to leave the Institute before 3 hours have elapsed from the delivery of the exam text. }