Academic Year 2023

Mathematics entry test – session 3

1. Let the discrete random variable $\mathrm{X}$ have the probability mass function

$$p(X=k)= \frac {n!}{k!(n-k)!} p^{k} (1-p)(n-k)$$

where $k \in \mathbb{N}$. The moment generating function $M(t)$ of this mass function is:

(a) $M(t)=k;$ (b) $M(t) = npk + e^{tk};$ (c) $M(t)= (1-p+pe^ {tk})^ {n} ;$ (d) $M(t)= ne^ {2t} / (2k).$

2. Let $u_{n}$, with $n \in \mathbb{N}$, such that $u_0 = a$, with $a$ being a real positive number, and $u_n + 1 = f(u_n)$with $f (x) = (x + 16)/(x + 7)$. Let $v_n = (u_n −2)/(u_n + 8)$. The relationship between $v_n$ and $v_{n + 1}$ reads:

(a) $v_{n+1} = − v_n /9;$ (b) $v_{n+1} = f (v_n );$ (c) $v_{n+1} = av_n;$ (d) $v_{n+1} = constant.$

3. Let I defined as:

$$I = \int_{0}^{10} \frac {dx}{x-2}$$

Which of the following statement is true:

(a) I cannot be calculated; (b) $ln 4;$ (c) $I \rightarrow + \infty;$ (d) $ln2.$

4. Let the vector field $\mathcal{F}=2x\vec{i}+y^{2}\vec{j}+z^{2}\vec{k}$ expressed in the canonical orthonormal basis of $\mathbb{R}^{3} : ({\vec{i}}\!:{\vec{j}}\!:{\vec{k}})$. Let $S\,=\,\{(x,y,z)\,\in\,\mathbb{R}^{\mathbf{{3}}}\,:x^{2}+y^{2}+z^{2}\,=\,1\}$ the unit sphere. The flow of the vector field $\mathcal{F}$ through the closed surface $S\,$of the sphere $S\,$ is given by:

$${\oiint_{S}}\mathcal{F}\cdot\vec{n}\mathrm{d}S,$$

where $\vec{n}$ is the unit vector normal to the surface element $dS\,$. The result of the calculation is:

(a) $→∞;$

(b) $4π/3;$

(c) $8π/3;$

(d) $1.$

5. Let A and B two anticommuting matrices such that $A^{2}=B^{2}=\textsf{\textbf{1}}$, with $\textsf{\textbf{1}}$ being the identity matrix, and $[A,B]=2i C$ with $i^2 = \mathbb{-1}$. The commutator $[A,B]$ is equal to:

(a) $0$;

(b) $2iA$;

(c) $-A$;

(d) $i\textsf{\textbf{1}}\times B.$

6. Consider two circles $C1$ and $C2$ both have the same radius: $10 cm$. The circles $C1$ and $C2$ overlap in such a fashion that their centers are $10 cm$ apart. The calculation of the surface of the overlapping area is approximately, up to one decimal:

(a) $314.2 \, cm^2;$ (b) $122.8 \, cm^2;$ (c) $100.0 \, cm^2;$ (d) $\mathbb{0}$ since the distance between the centers is equal to the radius of each circle.

7. Let the matrix $\Alpha$:

$$A={\left(\begin{array}{l l l}{\sqrt{2}\cos x}&{i\sin x}&{0}\\ {i\sin x}&{0}&{-i\sin x}\\ {0}&{-i\sin x}&{-{\sqrt{2}}\cos x}\end{array}\right)}$$

where $x$ is a real number in the interval $[0;2\pi],$ and $i$ is the complex number such that $i^2 = −1$. The matrix $\Alpha$ is diagonalizable on the following condition:

(a) for all $x\in[0;\pi/4[\cup]\pi/4;3\pi/4[\cup]3\pi/4;2\pi]$;

(b) for all $x\in[0;2\pi];$

(c) only for $x = \pi/4$ and $x = 3\pi/4;$

(d) for $x\neq\pi/2$ and $x\neq3\pi/2.$

8. Let $x$ a discrete random variable that takes the following values: 0, 1, and 2, with probabilities $\frac{1}{2}, \frac{1}{4},$ and $\frac{1}{4}$ respectively. The standard deviation is equal to:

(a) $1;$

(b) $1/4;$

(c) ${\sqrt{11}}/2;$

(d) $0.$

9. Let the 3 ×3 matrix $\Alpha$

$$A=\left( \begin{array}{c c c} {{0}} & {{1}} & {{0}} \\ {{-1}} & {{0}} & {{0}} \\ {{0}} & {{0}} & {{2}} \\ \end{array}\right)$$

The matrix $\Alpha$ is diagonalizable. Which of the following is the set of eigenvectors?

(a) $\left\{\; \left[\begin{array}{} {{1}} \\ {{0}} \\ {{-i}} \end{array}\right];\;\; \left[\begin{array}{} {{-1}} \\ {{0}} \\ {{i}} \end{array}\right];\;\; \left[\begin{array}{} {{1}} \\ {{1}} \\ {{0}} \end{array}\right]\; \right\};$

(b) $\left\{\; \left[\begin{array}{} {{0}} \\ {{0}} \\ {{1}} \end{array}\right];\;\; \left[\begin{array}{} {{1}} \\ {{i}} \\ {{0}} \end{array}\right];\;\; \left[\begin{array}{} {{1}} \\ {{-i}} \\ {{0}} \end{array}\right]\; \right\};$

(c) $\left\{\; \left[\begin{array}{} {{-i}} \\ {{1}} \\ {{0}} \end{array}\right];\;\; \left[\begin{array}{} {{-1}} \\ {{1}} \\ {{0}} \end{array}\right];\;\; \left[\begin{array}{} {{1}} \\ {{i}} \\ {{1}} \end{array}\right]\; \right\};$

(d) $\left\{\; \left[\begin{array}{} {{-1}} \\ {{1}} \\ {{i}} \end{array}\right];\;\; \left[\begin{array}{} {{-1}} \\ {{0}} \\ {{1}} \end{array}\right];\;\; \left[\begin{array}{} {{-i}} \\ {{0}} \\ {{1}} \end{array}\right]\; \right\}.$

10. Consider the harmonic series $S_1$ and the alternating series $S_2$:

$$S_{1}=\sum_{n=1}^{\infty}{\frac{1}{n}}\quad{\mathrm{and}}\quad S_{2}=\sum_{n=1}^{\infty}{\frac{(-1)^{n-1}}{n}}$$

Which of the following is true?

(a) $S_1$ is divergent and $S_2 = 0$;

(b) $S_1$ is divergent and $S_2 = ln2$;

(c) $S_1 = 1 −S_2$;

(d) $S_1 = \pi ^ 2 /6$ and $S_2 = \pi/4$