Academic Year 2022 Actual

Mathematics entry test – Calculus, geometry, and algebra.

1. Let $ω$ the differential form: $ω = (x^2 + 3y)dx −y^3dy$. Check if $ω$ is an exact differential form or not on $\mathbb{N^2}$.

2. Calculate the integral of $ω = y^2dx + x^2dy$ on the line segment delimited by the points A in $(1, 0)$ and B in $(0, 1)$.

3. Let $i$ the imaginary number such that $i^2 = −1.$ Calculate $i \sqrt i$.

4. Let $u_n$, such that $u_{n+1}\,=\,-\frac{3}{2}u_{n}^{2}+\frac{5}{2}u_{n}+1$ with $u_0 = 1$. Check if the sequence $u_n$ is periodic, and if so give the period.

5. Calculate the integral $\int_0^{\infty}e^{-\alpha x^2}\mathrm{d}x$ with $\alpha>0$ being a real number.

6. Consider a cube $(A,B,C,D,E,F,G,H)$ of side length $\ell$, and such that the surface $(ABCD)$ is parallel to the surface $(EFGH)$. Let M a point of the segment $[AH]$ such that $\overrightarrow{A M}=\,a\overrightarrow{A H},$ with $0\le a\le1,$ a real number. Calculate the angle $\widehat{BME}$.

7. Consider a cube $(A,B,C,D,E,F,G,H)$ of side length $\ell$, and such that the surface $(ABCD)$ is parallel to the surface $(EFGH)$. Let M a point of the segment $[AH]$ such that $\overrightarrow{A M}=\,a\overrightarrow{A H},$ with $0\le a\le1,$ a real number. Calculate the scalar product $\overrightarrow{MB}\cdot{\overline{}\overrightarrow{ME}}$

8. Let x a discrete random variable that takes the following values 0, 3, 6, 9 with probabilities $\frac{1}{6}, \frac{1}{3}, \frac{1}{3},$ and $\frac{1}{6},$ respectively. Calculate the standard deviation.

9. Let X the discrete random variable that follows the probability mass function:

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

where $n\in\mathbb{N},k=0,1,2,..,n$ is an integer, and $p\in [0, 1]$ a real parameter. Calculate the expectation value.

10. Calculate the integral $\int_{t}^{\infty}{\frac{1}{x^{\alpha}}}\operatorname{d}x,$ with $t > 1$ and $\alpha > 1$ being real numbers.

11. Calculate the sum $\sum_{n=1}^{\infty}{\frac{(-1)^{n-1}}{n}}.$

12. Consider the circles $C_1$ of radius 8 cm and $C_2$ of radius 3 cm. The distance between the centers of the two circles is 13 cm. The circles $C_1$ and $C_2$ have a common tangent. Calculate the distance between the two points of the tangent that are common with $C_1$ and $C_2$ respectively.

13. Find the eigenvalues of the matrix $\Alpha$:

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

14. Find the eigenvectors of the 4 ×4 matrix $\Alpha$:

$$A={\left(\begin{array}{l l l l}{-2}&{0}&{0}&{0}\\ {0}&{-2}&{0}&{0}\\ {24}&{-12}&{2}&{0}\\ {0}&{0}&{0}&{2}\end{array}\right)}$$

15. Let the vector field $\mathcal{F}=y^{2}z\vec{i}+y^{3}\vec{j}+x z\vec{k}$ expressed in the canonical orthonormal basis of $\mathbb{R}^{3} : ({\vec{i}}\!:{\vec{j}}\!:{\vec{k}})$. Let $S$ be the surface of a cube such that $-1\le x\le1，-1\le y\le1，$ and $0\le z\le2.$ Calculate the flow of the vector field $\mathcal{F}$ through the closed surface $S$:

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

where $\vec{n}$ is the unit vector normal to the surface element $dS\,$.