Academic Year 2022 Sample

Mathematics entry test sample

1. Let $ω_1$ the differential form: $\omega_{1}\,=\,(y^{3}-6x y^{2})\mathrm{d}x+(3x y^{2}-6x^{2}y)\mathrm{d}y$. and $ω_2$ the differential form: $\omega_{2}=(3x y^{2}-6x^{2}y)\mathrm{d}x+(y^{3}-6x y^{2})\mathrm{d}y$. Which of these two forms $ω_1$ or $ω_2$ is an exact differential form on $\R^2$?

(a) $ω_1$ ;

(b) $ω_2$ ;

(c) both $ω_1$ and $ω_2$;

(d) neither of these two is a differential form.

2. The integral of ω1 on the arc circle delimited by the points $\Alpha$ in $(1, 2)$ and $\Beta$ in $(3, 4)$ is equal to:

(a) 0;

(b) -236;

(c) tends to $∞$;

(d) $π/2$.

3. Let $i$ the imaginary number such that $i^2 = −1.$ The result of the calculation of $i^i$ is:

(a) $−1$;

(b) $\cos i$;

(c) nothing as the calculation is impossible;

(d) $\exp(−\pi/2)$.

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

(a) $v_{n+1} = −v_n / 9$;

(b) $v_{n+1} = \,f\left({v}_{n}\right)$;

(c) $v_{n+1} = av_n$ ;

(d) $v_{n+1} = constant$.

5. The integral $\int_{0}^{\infty}x^{\alpha-1}e^{-x}\mathrm{d}x$ with $α = −3/2$ is equal to:

(a) $\sqrt[3]{\pi}/{4}$;

(b) −1/2;

(c) $\sqrt[4]{\pi}/{3}$;

(d) 6.

6. Consider a pyramid with the following characteristics: it has a square base $(ABCD)$ of side length 1, and its summit $S$ is such that the segment $[SA]$ is perpendicular to the plane $(ABC)$ and $SA = 1$. Let M a point of the segment $[SC]$ such that $\overrightarrow{S M}=\,a\overrightarrow{S C},$ with $0\le a\le1,$ a real number. See a representation in Fig. 6. Consider the angle $\widehat{BMD}$; which of the following statement is correct:

(a) $\widehat{BMD}=a\frac{\pi}{2}$

(b) $\cos(\widehat{\mathrm{BMD}})=1-\frac{1}{3a^{2}-4a+2}$

(c) $\sin(\widehat{\mathrm{BMD}})=1+\frac{1}{3a^{2}-4a+2}$

(d) None of the above.

7. Consider the same geometrical figure as above, with the same characteristics, and representated in Fig. 6. The scalar product $\overrightarrow{MB}\cdot{\overline{}\overrightarrow{MD}}$ is equal to:

(a) $a/2$;

(b) $-3a^{2}+4a+2;$

(c) $0$;

(d) $3a^2 − 4a + 2$.

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 X the discrete random variable have the probability mass function:

$$p(X=K)=e^{-\lambda}\lambda^{k}/k!$$

where $k\in\mathbb{N}$, and $\lambda \in \R ^ {+*}\,, E(X)$ and $V(X)$denote the expectation value and the variance respectively. Which of the following statement is true?

(a) ${E(X)\to\infty\ \mathrm \, \text{and} \, \ V(X)=\lambda;}$

(b) ${E(X)=\lambda\ \mathrm \, \text{and}\ V(X)\to\infty;}$

(c) ${E(X)=\lambda\ \mathrm \, \text{and}\ V(X)\ = \lambda ^{2}}$

(d) $E(X)=V(X)=\lambda$

10. The integral $\int_{0}^{\infty}{\frac{sinx}{x}}\operatorname{dx},$ is equal to:

(a) $0$;

(b) $\to \infin$;

(c) $1$;

(d) $\pi / 2$

11. The sum $\sum_{n=1}^{\infty}{\frac{1}{2^n}}$is equal to:

(a) $\to \infin;$

(b) $1$

(c) $0$

(d) $\pi ^ 2 / 6$

12. Consider two circles $C_1$ and $C_2$; both have the same radius 10 cm. The circles $C_1$ and $C_2$ 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) $100.0 \, cm ^ 2$

(c) $122.8 \, cm ^ 2$

(d) $0$ since the distance between the centers is equal to the radius of each circle.

13. Let the matrix $\Alpha$:

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

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) only for $x = \pi/4 \, \text{and} \, x = 3\pi/4$;

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

(c) for all $x \in [0 \text{;} \, \pi/4 \, [\,\cup\,] \, \pi / 4 \, ;\, 3\pi /4 \, [\,\cup\,] \, 3\pi / 4 \,;\, 2\pi$

(d) for $x\neq\pi/2 \, \text{and} \, x\neq3\pi/2$

14. Let 3 × 3 matrix $\Alpha$:

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

The matrix $\Alpha$ is diagonalizable. One eigenvalue has multiplicity 1 and the other has multiplicity 2. Which of the following is the set of eigenvectors?

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

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

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

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

15. 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) $\to \infin$

(b) $4\pi / 3$

(c) $8 \pi / 3$

(d) $1$