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Concours Commun Mines-Ponts - independent subject practice study materials
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Concours Commun Mines-Ponts - independent subject practice practice questions

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Question 1Advanced analysis, convergence and multivariable reasoning

Consider the smooth function $f: \mathbb{R}^2 \to \mathbb{R}$ defined by $f(x, y) = x^3 + y^3 - 3xy$. What is the nature of the critical point located at $(1, 1)$?

  1. A strict local minimum with critical value $f(1, 1) = -1$ Correct answer.
  2. A saddle point (point col) because the Hessian has mixed signs
  3. A strict local maximum with critical value $f(1, 1) = -1$
  4. A degenerate non-isolated critical point with zero Hessian determinant

Explanation

The gradient is $\nabla f(x, y) = (3x^2 - 3y, 3y^2 - 3x)$, which vanishes at $(0, 0)$ and $(1, 1)$. The Hessian matrix of second partial derivatives is $H_f(x, y) = \begin{pmatrix} 6x & -3 \\ -3 & 6y \end{pmatrix}$. At $(1, 1)$, $H_f(1, 1) = \begin{pmatrix} 6 & -3 \\ -3 & 6 \end{pmatrix}$. Its determinant is $\det(H_f) = 36 - 9 = 27 > 0$, and the trace is $\text{Tr}(H_f) = 12 > 0$ with $f_{xx} = 6 > 0$. Thus, $H_f(1, 1)$ is symmetric positive-definite, proving $(1, 1)$ is a strict local minimum with $f(1, 1) = 1 + 1 - 3 = -1$.

Question 2Advanced analysis, convergence and multivariable reasoning

Using the method of Lagrange multipliers, what is the minimum value of $f(x, y, z) = x^2 + y^2 + z^2$ subject to the affine constraint $x + 2y + 3z = 14$ on $\mathbb{R}^3$?

  1. 7
  2. 14 Correct answer.
  3. 28
  4. 196

Explanation

Let $g(x, y, z) = x + 2y + 3z - 14 = 0$. The Lagrange multiplier condition $\nabla f = \lambda \nabla g$ gives $(2x, 2y, 2z) = \lambda(1, 2, 3)$, so $x = \lambda/2$, $y = \lambda$, $z = 3\lambda/2$. Substituting into the constraint gives $\frac{\lambda}{2} + 2\lambda + \frac{9\lambda}{2} = 7\lambda = 14$, so $\lambda = 2$. This yields the minimizer $(x, y, z) = (1, 2, 3)$. The minimum value is $f(1, 2, 3) = 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14$. Geometrically, this is the square of the distance from the origin to the plane: $\frac{|-14|^2}{1^2 + 2^2 + 3^2} = \frac{196}{14} = 14$.

Question 3Advanced analysis, convergence and multivariable reasoning

Let $S$ be the two-dimensional submanifold of $\mathbb{R}^3$ defined by the level set $S = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 - z^2 = 1\}$. What is the Cartesian equation of the affine tangent plane to $S$ at the point $M_0 = (1, 1, 1)$?

  1. $x + y + z = 3$
  2. $x + y - z = 0$
  3. $x + y - z = 1$ Correct answer.
  4. $2x + 2y - z = 3$

Explanation

The surface is defined as $g(x, y, z) = 1$ with $g(x, y, z) = x^2 + y^2 - z^2$. The gradient vector is $\nabla g(x, y, z) = (2x, 2y, -2z)$. At $M_0 = (1, 1, 1)$, $\nabla g(1, 1, 1) = (2, 2, -2)$, which is non-zero, confirming $(1, 1, 1)$ is a regular point. The affine tangent plane equation is $\nabla g(1, 1, 1) \cdot ((x, y, z) - (1, 1, 1)) = 0 \iff 2(x - 1) + 2(y - 1) - 2(z - 1) = 0 \iff 2x + 2y - 2z = 2 \iff x + y - z = 1$.

Question 4Advanced analysis, convergence and multivariable reasoning

Consider the sequence of functions $(f_n)_{n \ge 1}$ defined on $[0, 1]$ by $f_n(x) = \frac{nx}{1 + n^2 x^2}$. Which statement correctly describes the convergence of $(f_n)$ on $[0, 1]$ and on compact subintervals $[a, 1]$ with $a > 0$?

  1. $(f_n)$ converges uniformly to 0 on $[0, 1]$
  2. $(f_n)$ converges pointwise to 0 on $[0, 1]$, but does not converge uniformly on any interval $[a, 1]$ with $a > 0$
  3. $(f_n)$ converges uniformly on $[0, 1]$ to the discontinuous function $f(x) = 1/2$ for $x > 0$ and $f(0) = 0$
  4. $(f_n)$ converges pointwise to 0 on $[0, 1]$, does not converge uniformly on $[0, 1]$, but converges uniformly on $[a, 1]$ for every $a > 0$ Correct answer.

Explanation

For $x = 0$, $f_n(0) = 0$. For $x > 0$, $f_n(x) \sim \frac{nx}{n^2 x^2} = \frac{1}{nx} \to 0$ as $n \to \infty$. So $(f_n)$ converges pointwise to the zero function $f \equiv 0$ on $[0, 1]$. However, \|f_n\|_\infty = \sup_{x \in [0, 1]} |f_n(x)| = f_n(1/n) = \frac{1}{1 + 1} = \frac{1}{2} \not\to 0$, so convergence is not uniform on $[0, 1]$. For any $a > 0$, once $n > 1/a$, the function $f_n$ is strictly decreasing on $[a, 1]$, so \sup_{x \in [a, 1]} |f_n(x)| = f_n(a) = \frac{na}{1 + n^2 a^2} \to 0$, ensuring uniform convergence on $[a, 1]$.

Question 5Advanced analysis, convergence and multivariable reasoning

What is the radius of convergence $R$ of the complex power series $\sum_{n=1}^\infty \frac{(2n)!}{(n!)^2} z^n$?

  1. $R = \frac{1}{4}$ Correct answer.
  2. $R = \frac{1}{2}$
  3. $R = 1$
  4. $R = 4$

Explanation

Let $a_n = \frac{(2n)!}{(n!)^2} = \binom{2n}{n}$. We apply d'Alembert's ratio test: $\frac{a_{n+1}}{a_n} = \frac{(2n+2)!}{((n+1)!)^2} \frac{(n!)^2}{(2n)!} = \frac{(2n+2)(2n+1)}{(n+1)^2} = \frac{2(2n+1)}{n+1} \to 4$ as $n \to \infty$. Therefore, the radius of convergence is $R = \frac{1}{\lim_{n \to \infty} \frac{a_{n+1}}{a_n}} = \frac{1}{4}$.

Question 6Advanced analysis, convergence and multivariable reasoning

Consider the series of functions $\sum_{n=1}^\infty u_n(x)$ where $u_n(x) = \frac{\cos(nx)}{n^2 + x^2}$ defined on $\mathbb{R}$. Which of the following statements is strictly correct regarding its convergence and sum function $S(x)$?

  1. The series converges pointwise on $\mathbb{R}$, but fails to converge uniformly because $n^2 + x^2$ depends on $x$
  2. The series converges normally on $\mathbb{R}$, and its sum function $S$ is continuous and bounded on $\mathbb{R}$ Correct answer.
  3. The series converges normally on every compact interval, but not on $\mathbb{R}$
  4. The series does not converge absolutely for $x = 0$

Explanation

For all $x \in \mathbb{R}$ and all $n \ge 1$, we have $|u_n(x)| = \frac{|\cos(nx)|}{n^2 + x^2} \le \frac{1}{n^2 + x^2} \le \frac{1}{n^2}$. Thus, the uniform norm satisfies \|u_n\|_\infty = \sup_{x \in \mathbb{R}} |u_n(x)| \le \frac{1}{n^2}$. Because the numerical series \sum_{n=1}^\infty \frac{1}{n^2} converges (Riemann series with $\alpha = 2 > 1$), the series \sum u_n converges normally (convergence normale) on all of $\mathbb{R}$. Normal convergence implies uniform convergence; since each $u_n$ is continuous and bounded, the sum $S(x)$ is continuous and bounded on $\mathbb{R}$.

Question 7Advanced analysis, convergence and multivariable reasoning

Let $f: \mathbb{R} \to \mathbb{R}$ be the $2\pi$-periodic function defined on $(-\pi, \pi]$ by $f(t) = t$. By computing its Fourier series and applying Parseval's identity, what is the exact sum of $\sum_{n=1}^\infty \frac{1}{n^2}$?

  1. $\frac{\pi^2}{12}$
  2. $\frac{\pi^2}{8}$
  3. $\frac{\pi^2}{6}$ Correct answer.
  4. $\frac{\pi^2}{4}$

Explanation

Because $f$ is odd, its Fourier cosine coefficients vanish: $a_n = 0$. The sine coefficients are $b_n = \frac{2}{\pi} \int_0^\pi t \sin(nt) \, dt = \frac{2}{\pi} \left( \left[-\frac{t\cos(nt)}{n}\right]_0^\pi + \frac{1}{n} \int_0^\pi \cos(nt) \, dt \right) = -\frac{2(-1)^n}{n} = \frac{2(-1)^{n+1}}{n}$. Parseval's identity states: $\frac{a_0^2}{4} + \frac{1}{2} \sum_{n=1}^\infty (a_n^2 + b_n^2) = \frac{1}{2\pi} \int_{-\pi}^\pi |f(t)|^2 \, dt$. Here $\frac{1}{2\pi} \int_{-\pi}^\pi t^2 \, dt = \frac{1}{\pi} \frac{\pi^3}{3} = \frac{\pi^2}{3}$. Thus, $\frac{1}{2} \sum_{n=1}^\infty \frac{4}{n^2} = 2 \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{3} \implies \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$.

Question 8Advanced analysis, convergence and multivariable reasoning

For $f \in L^1(\mathbb{R})$, define the Fourier transform by $\mathcal{F}(f)(\xi) = \hat{f}(\xi) = \int_{-\infty}^{+\infty} f(t) e^{-i \xi t} \, dt$. For $a > 0$, define the scaled function $g(t) = f(at)$. What is the Fourier transform $\hat{g}(\xi)$?

  1. $\hat{g}(\xi) = a \hat{f}(a\xi)$
  2. $\hat{g}(\xi) = \hat{f}\left(\frac{\xi}{a}\right)$
  3. $\hat{g}(\xi) = \frac{1}{a} \hat{f}(a\xi)$
  4. $\hat{g}(\xi) = \frac{1}{a} \hat{f}\left(\frac{\xi}{a}\right)$ Correct answer.

Explanation

By definition, $\hat{g}(\xi) = \int_{-\infty}^{+\infty} f(at) e^{-i \xi t} \, dt$. Performing the substitution $u = at$ gives $t = u/a$ and $dt = du/a$. Since $a > 0$, the integration limits remain $-\infty$ to $+\infty$. Thus, $\hat{g}(\xi) = \int_{-\infty}^{+\infty} f(u) e^{-i \xi (u/a)} \frac{du}{a} = \frac{1}{a} \int_{-\infty}^{+\infty} f(u) e^{-i (\xi/a) u} \, du = \frac{1}{a} \hat{f}\left(\frac{\xi}{a}\right)$.

Question 9Advanced analysis, convergence and multivariable reasoning

Let $f(x) = e^{-x^2 / 2}$ for $x \in \mathbb{R}$. What is the Fourier transform $\hat{f}(\xi) = \int_{-\infty}^{+\infty} e^{-x^2 / 2} e^{-i \xi x} \, dx$?

  1. $\hat{f}(\xi) = \sqrt{2\pi} e^{-\xi^2 / 2}$ Correct answer.
  2. $\hat{f}(\xi) = e^{-\xi^2 / 2}$
  3. $\hat{f}(\xi) = \frac{1}{\sqrt{2\pi}} e^{-\xi^2 / 2}$
  4. $\hat{f}(\xi) = \sqrt{2\pi} e^{-\xi^2}$

Explanation

Differentiating under the integral sign gives $\hat{f}'(\xi) = \int_{-\infty}^{+\infty} (-ix) e^{-x^2/2} e^{-i\xi x} \, dx$. Integrating by parts with $u = e^{-i\xi x}$ and $v' = -x e^{-x^2/2}$ yields $\hat{f}'(\xi) = [e^{-x^2/2} e^{-i\xi x}]_{-\infty}^{+\infty} - \int_{-\infty}^{+\infty} (-i\xi) e^{-x^2/2} e^{-i\xi x} \, dx = -\xi \hat{f}(\xi)$. Solving this linear differential equation gives $\hat{f}(\xi) = \hat{f}(0) e^{-\xi^2/2}$. The initial value is the standard Gaussian integral $\hat{f}(0) = \int_{-\infty}^{+\infty} e^{-x^2/2} \, dx = \sqrt{2\pi}$. Therefore, $\hat{f}(\xi) = \sqrt{2\pi} e^{-\xi^2/2}$.

Question 10Advanced analysis, convergence and multivariable reasoning

Let $f \in C^0([0, 1], \mathbb{R})$. Applying the Cauchy-Schwarz inequality to $f$ and the constant function $g(t) = 1$ in $L^2([0, 1])$ gives $\left( \int_0^1 f(t) \, dt \right)^2 \le \int_0^1 (f(t))^2 \, dt$. Under what condition does equality hold?

  1. $f$ is a constant function on $[0, 1]$ Correct answer.
  2. $f$ is an odd function centered at $1/2$
  3. $\int_0^1 f(t) \, dt = 0$
  4. $f(0) = f(1) = 0$

Explanation

In an inner product space, equality in Cauchy-Schwarz |\langle f, g \rangle| = \|f\| \|g\| occurs if and only if $f$ and $g$ are linearly dependent. Here $g(t) = 1$ is the constant function 1. Therefore, $f$ must be collinear to 1, meaning $f(t) = c$ for some constant $c \in \mathbb{R}$ on $[0, 1]$. In that case, both sides equal $c^2$.

Question 11Advanced analysis, convergence and multivariable reasoning

Let $\omega = P \, dx + Q \, dy + R \, dz$ be a differential 1-form of class $C^1$ on an open subset $U \subset \mathbb{R}^3$. According to the Poincaré Lemma (lemme de Poincaré), under what topological condition on $U$ is the closure condition $d\omega = 0$ (equivalent to $\text{curl}(\vec{V}) = \vec{0}$) both necessary and sufficient for $\omega$ to be exact (i.e., $\omega = df$ for some scalar potential $f$)?

  1. $U$ is compact and bounded
  2. $U$ is any connected open set
  3. $U$ is star-shaped (étoilé) or simply connected Correct answer.
  4. $U$ has an empty boundary $\partial U = \emptyset$

Explanation

By Schwarz's theorem, any exact form is closed ($d(df) = 0$), so $d\omega = 0$ is always necessary. By the Poincaré Lemma, on a star-shaped open set (or more generally any simply connected open set in $\mathbb{R}^3$), every closed differential form of degree 1 is exact. On domains with non-trivial fundamental group $\pi_1(U) \ne 0$ (such as $\mathbb{R}^3 \setminus \{z\text{-axis}\}$), closed forms need not be exact (e.g., the vortex form $\frac{-y dx + x dy}{x^2 + y^2}$).

Question 12Advanced analysis, convergence and multivariable reasoning

Consider the equation $F(x, y) = x^2 + y + e^{xy} - 1 = 0$ in a neighborhood of $(0, 0)$. By the Implicit Function Theorem (théorème des fonctions implicites), this equation defines a unique $C^1$ function $y = \phi(x)$ such that $\phi(0) = 0$. What is the first derivative $\phi'(0)$?

  1. $-1$
  2. $1$
  3. $-2$
  4. $0$ Correct answer.

Explanation

We check the conditions of the implicit function theorem: $F(0, 0) = 0 + 0 + 1 - 1 = 0$. The partial derivatives are $\frac{\partial F}{\partial x} = 2x + y e^{xy}$ and $\frac{\partial F}{\partial y} = 1 + x e^{xy}$. At $(0, 0)$, $\frac{\partial F}{\partial x}(0, 0) = 0$ and $\frac{\partial F}{\partial y}(0, 0) = 1 \ne 0$. Because $\frac{\partial F}{\partial y}(0, 0) \ne 0$, the implicit function $\phi$ exists and its derivative is given by $\phi'(0) = -\frac{\frac{\partial F}{\partial x}(0, 0)}{\frac{\partial F}{\partial y}(0, 0)} = -\frac{0}{1} = 0$.

Question 13Advanced analysis, convergence and multivariable reasoning

According to Dini's First Theorem (premier théorème de Dini), under what exact hypotheses does the pointwise convergence of a sequence of continuous real-valued functions $(f_n)_{n \ge 1}$ to a continuous limit function $f$ on a topological space $X$ imply uniform convergence?

  1. $X$ is bounded, each $f_n$ is Lipschitz, and $(f_n)$ is equicontinuous
  2. $X$ is compact, and for every $x \in X$, the numerical sequence $(f_n(x))_{n \ge 1}$ is monotonic (either non-decreasing for all $x$, or non-increasing for all $x$) Correct answer.
  3. $X$ is complete, and the functions $f_n$ are uniformly bounded on $X$
  4. $X = [a, b]$, and the derivative sequence $(f_n')$ converges uniformly on $[a, b]$

Explanation

Dini's theorem states that if $X$ is a compact topological space, $(f_n)$ is a sequence of continuous real-valued functions converging pointwise to a continuous function $f$, and the sequence is monotonic at every point (for instance, $f_n(x) \le f_{n+1}(x)$ for all $n$ and all $x \in X$), then the convergence $f_n \to f$ is uniform on $X$. The proof relies on Cantor's intersection theorem or the finite subcover property applied to open sets $O_n = \{x \in X \mid f(x) - f_n(x) < \varepsilon\}$.

Question 14Advanced analysis, convergence and multivariable reasoning

Using the Lebesgue Dominated Convergence Theorem (théorème de convergence dominée), what is the limit $L = \lim_{n \to \infty} \int_0^{+\infty} \frac{n \sin(x/n)}{x(1 + x^2)} \, dx$?

  1. $0$
  2. $1$
  3. $\frac{\pi}{2}$ Correct answer.
  4. $\pi$

Explanation

Let $f_n(x) = \frac{n \sin(x/n)}{x(1 + x^2)}$ for $x > 0$. For any fixed $x > 0$, as $n \to \infty$, $n \sin(x/n) = x \frac{\sin(x/n)}{x/n} \to x$, so the pointwise limit is $f(x) = \frac{x}{x(1 + x^2)} = \frac{1}{1 + x^2}$. Furthermore, using the standard inequality $|\sin(u)| \le |u|$ for all $u \ge 0$, we have $|n \sin(x/n)| \le n(x/n) = x$, which provides the domination: $|f_n(x)| \le \frac{x}{x(1 + x^2)} = \frac{1}{1 + x^2} = g(x)$. The dominating function $g(x) = \frac{1}{1 + x^2}$ is continuous and integrable on $[0, +\infty)$ because $\int_0^{+\infty} \frac{dx}{1 + x^2} = [\arctan(x)]_0^{+\infty} = \frac{\pi}{2} < +\infty$. By the Dominated Convergence Theorem, $L = \int_0^{+\infty} f(x) \, dx = \int_0^{+\infty} \frac{dx}{1 + x^2} = \frac{\pi}{2}$.

Question 15Inner products, spectral reasoning and matrix structure

In a real pre-Hilbert space $(E, \langle \cdot, \cdot \rangle)$ with induced norm \|\cdot\|, let $x, y \in E$ be non-zero vectors. Under what exact condition does the Cauchy-Schwarz equality |\langle x, y \rangle| = \|x\| \|y\| hold?

  1. The vectors $x$ and $y$ are linearly dependent (collinear) Correct answer.
  2. The vectors $x$ and $y$ are orthogonal, satisfying $\langle x, y \rangle = 0$
  3. The vectors $x$ and $y$ have identical Euclidean norms, \|x\| = \|y\|
  4. The sum vector satisfies \|x + y\|^2 = \|x\|^2 + \|y\|^2

Explanation

In any real pre-Hilbert space, the quadratic polynomial $P(t) = \|tx + y\|^2 = t^2 \|x\|^2 + 2t\langle x, y \rangle + \|y\|^2$ is non-negative for all $t \in \mathbb{R}$. Its discriminant $\Delta = 4(\langle x, y \rangle^2 - \|x\|^2 \|y\|^2)$ must be non-positive, yielding the Cauchy-Schwarz inequality. The equality |\langle x, y \rangle| = \|x\| \|y\| occurs if and only if $\Delta = 0$, which corresponds to the existence of a real root $t_0$ such that \|t_0 x + y\| = 0$, proving that $x$ and $y$ are linearly dependent.

Question 16Inner products, spectral reasoning and matrix structure

Let $A \in S_n^{++}(\mathbb{R})$ be a real symmetric positive-definite matrix and let $B \in S_n(\mathbb{R})$ be a real symmetric matrix. According to the simultaneous reduction theorem (théorème de réduction simultanée), which of the following statements is unconditionally true?

  1. There exists an orthogonal matrix $Q \in O_n(\mathbb{R})$ such that both $Q^T A Q$ and $Q^T B Q$ are diagonal
  2. There exists an invertible matrix $P \in GL_n(\mathbb{R})$ such that $P^T A P = I_n$ and $P^T B P$ is diagonal Correct answer.
  3. The matrices $A$ and $B$ must commute ($AB = BA$) for simultaneous congruence to hold
  4. There exists an invertible matrix $P \in GL_n(\mathbb{R})$ such that $P^{-1} A P$ and $P^{-1} B P$ are both the identity matrix

Explanation

Because $A$ is symmetric positive-definite, it defines an inner product on $\mathbb{R}^n$. By the spectral theorem applied to the self-adjoint endomorphism associated with the quadratic form of $B$ relative to this inner product, there exists a basis orthonormal for $A$ and orthogonal for $B$. In matrix terms, there exists $P \in GL_n(\mathbb{R})$ such that $P^T A P = I_n$ and $P^T B P = D$ where $D$ is real diagonal.

Question 17Inner products, spectral reasoning and matrix structure

In the pre-Hilbert space $E = C^0([-1, 1], \mathbb{R})$ equipped with the inner product $\langle f, g \rangle = \int_{-1}^1 f(t)g(t) \, dt$, what is the orthogonal projection of the polynomial function $h(t) = t^2$ onto the subspace $F = \mathbb{R}_1[X]$ of polynomial functions of degree at most 1?

  1. $p_F(h)(t) = 0$
  2. $p_F(h)(t) = t$
  3. $p_F(h)(t) = \frac{1}{3}$ Correct answer.
  4. $p_F(h)(t) = \frac{1}{2} + t$

Explanation

The subspace $F$ has orthogonal basis $\{e_0, e_1\}$ where $e_0(t) = 1$ and $e_1(t) = t$ because $\langle 1, t \rangle = \int_{-1}^1 t \, dt = 0$. The orthogonal projection is $p_F(h) = \frac{\langle h, 1 \rangle}{\|1\|^2} 1 + \frac{\langle h, t \rangle}{\|t\|^2} t$. We compute $\langle h, 1 \rangle = \int_{-1}^1 t^2 \, dt = \frac{2}{3}$, $\|1\|^2 = \int_{-1}^1 1 \, dt = 2$, and $\langle h, t \rangle = \int_{-1}^1 t^3 \, dt = 0$ by parity. Thus, $p_F(h)(t) = \frac{2/3}{2} \cdot 1 + 0 \cdot t = \frac{1}{3}$.

Question 18Inner products, spectral reasoning and matrix structure

Let $H$ be a real or complex Hilbert space and let $u \in \mathcal{L}(H)$ be a continuous linear operator with adjoint $u^*$. Which relation correctly characterizes the orthogonal complement of the image of $u$?

  1. $(\text{Im}\, u)^\perp = \text{Im}(u^*)$
  2. $(\text{Im}\, u)^\perp = \ker(u)$
  3. $(\text{Im}\, u)^\perp = (\ker(u^*))^\perp$
  4. $(\text{Im}\, u)^\perp = \ker(u^*)$ Correct answer.

Explanation

For any $y \in H$, $y \in (\text{Im}\, u)^\perp \iff \forall x \in H, \langle u(x), y \rangle = 0 \iff \forall x \in H, \langle x, u^*(y) \rangle = 0 \iff u^*(y) = 0 \iff y \in \ker(u^*)$. Hence, $(\text{Im}\, u)^\perp = \ker(u^*)$ holds in any Hilbert space.

Question 19Inner products, spectral reasoning and matrix structure

Let $A \in \mathcal{M}_n(\mathbb{R})$ ($n \ge 2$) be a matrix of rank 1. What is the characteristic polynomial $\chi_A(X) = \det(X I_n - A)$?

  1. $\chi_A(X) = X^n - \text{Tr}(A) X^{n-1}$ Correct answer.
  2. $\chi_A(X) = (X - \text{Tr}(A))^n$
  3. $\chi_A(X) = X^n - \det(A)$
  4. $\chi_A(X) = X^{n-1}(X - 1)$

Explanation

By the rank-nullity theorem, $\dim \ker(A) = n - \text{rk}(A) = n - 1$. Therefore, 0 is an eigenvalue of $A$ with geometric multiplicity at least $n - 1$, so $X^{n-1}$ divides $\chi_A(X)$. Hence $\chi_A(X) = X^{n-1}(X - \lambda)$ where $\lambda$ is the remaining eigenvalue. Since the trace of $A$ is the sum of its eigenvalues (counted with multiplicity), $\text{Tr}(A) = 0 + \dots + 0 + \lambda = \lambda$. Thus $\chi_A(X) = X^{n-1}(X - \text{Tr}(A)) = X^n - \text{Tr}(A) X^{n-1}$.

Question 20Inner products, spectral reasoning and matrix structure

Let $D_N(t) = 1 + 2\sum_{n=1}^N \cos(nt) = \frac{\sin((N + 1/2)t)}{\sin(t/2)}$ be the Dirichlet kernel on $[-\pi, \pi]$. According to Dirichlet's theorem, if a $2\pi$-periodic function $f: \mathbb{R} \to \mathbb{R}$ is piecewise $C^1$ (continûment différentiable par morceaux), what is the limit of its Fourier partial sum $S_N(f)(t_0) = \frac{1}{2\pi} \int_{-\pi}^\pi f(t_0 - u) D_N(u) \, du$ as $N \to \infty$ at any point $t_0$?

  1. $\lim_{N \to \infty} S_N(f)(t_0) = \sup_{t \in [-\pi, \pi]} f(t)$
  2. $\lim_{N \to \infty} S_N(f)(t_0) = \frac{f(t_0^+) + f(t_0^-)}{2}$ Correct answer.
  3. $\lim_{N \to \infty} S_N(f)(t_0) = f(t_0)$ only if $f$ is infinitely differentiable at $t_0$
  4. $\lim_{N \to \infty} S_N(f)(t_0) = 0$ at every jump discontinuity

Explanation

Dirichlet's theorem is a cornerstone of CPGE Fourier analysis: for any $2\pi$-periodic function $f$ that is piecewise $C^1$ on $\mathbb{R}$, the Fourier series converges at every point $t_0 \in \mathbb{R}$ to the arithmetic mean of its left and right limits, $\frac{f(t_0^+) + f(t_0^-)}{2}$. At points of continuity, this limit matches $f(t_0)$ exactly. The proof relies on writing $S_N(f)(t_0) - \frac{f(t_0^+) + f(t_0^-)}{2} = \frac{1}{\pi} \int_0^\pi \frac{f(t_0+u) + f(t_0-u) - f(t_0^+) - f(t_0^-)}{2\sin(u/2)} \sin((N+1/2)u) \, du$ and applying the Riemann-Lebesgue lemma.

Question 21Inner products, spectral reasoning and matrix structure

Let $H$ be an infinite-dimensional separable real Hilbert space, and let $T: H \to H$ be a compact self-adjoint linear operator. According to the Hilbert-Schmidt spectral theorem, which statement regarding the spectrum $\text{Sp}(T)$ and eigenspaces of $T$ is strictly true?

  1. The spectrum of $T$ contains only continuous spectrum, and $T$ has no eigenvectors
  2. The operator $T$ has an uncountable set of eigenvalues with infinite-dimensional eigenspaces
  3. The eigenvalues of $T$ are real, each eigenspace associated with a non-zero eigenvalue is finite-dimensional, non-zero eigenvalues form a finite set or a sequence tending to 0, and $H$ admits a Hilbert basis of eigenvectors of $T$ Correct answer.
  4. The spectral radius satisfies $\rho(T) = 0$ because $T$ is compact

Explanation

The spectral theorem for compact self-adjoint operators states that all eigenvalues are real, eigenspaces $E_\lambda(T) = \ker(T - \lambda I)$ for $\lambda \ne 0$ are finite-dimensional (by Riesz's theorem on locally compact normed spaces), and the non-zero eigenvalues can be ordered as a sequence $(\lambda_n)$ with $\lim_{n \to \infty} \lambda_n = 0$. Furthermore, $H$ is the orthogonal direct sum of $\ker(T)$ and the eigenspaces $E_{\lambda_n}$, providing an orthonormal Hilbert basis of eigenvectors.

Question 22Advanced probability, moments and conditional models

Let $X$ and $Y$ be independent discrete random variables following Poisson distributions: $X \sim \mathcal{P}(\lambda_1)$ and $Y \sim \mathcal{P}(\lambda_2)$ with $\lambda_1, \lambda_2 > 0$. What is the distribution and variance of their sum $Z = X + Y$?

  1. $Z \sim \mathcal{P}(\lambda_1 \lambda_2)$ with $\text{Var}(Z) = \lambda_1 \lambda_2$
  2. $Z \sim \mathcal{P}(\lambda_1 + \lambda_2)$ with $\text{Var}(Z) = \lambda_1 + \lambda_2$ Correct answer.
  3. $Z \sim \mathcal{P}(\lambda_1 + \lambda_2)$ with $\text{Var}(Z) = (\lambda_1 + \lambda_2)^2$
  4. $Z$ is not Poisson-distributed, but $\text{Var}(Z) = \lambda_1 + \lambda_2$

Explanation

The probability generating function of a Poisson variable $X \sim \mathcal{P}(\lambda)$ is $G_X(t) = \mathbb{E}[t^X] = e^{\lambda(t-1)}$. By independence, $G_Z(t) = G_X(t) G_Y(t) = e^{\lambda_1(t-1)} e^{\lambda_2(t-1)} = e^{(\lambda_1 + \lambda_2)(t-1)}$, which is the generating function of $\mathcal{P}(\lambda_1 + \lambda_2)$. For any Poisson distribution $\mathcal{P}(\lambda)$, the variance equals the parameter, so $\text{Var}(Z) = \lambda_1 + \lambda_2$.

Question 23Advanced probability, moments and conditional models

Let $(X_n)_{n \ge 1}$ be a sequence of independent and identically distributed random variables with mean $\mu = 10$ and variance $\sigma^2 = 4$. Let $\bar{X}_{100} = \frac{1}{100} \sum_{i=1}^{100} X_i$ denote the empirical mean of 100 observations. Using the Central Limit Theorem and the standard normal table value $\Phi(1.50) \approx 0.9332$, what is the approximate probability $P(\bar{X}_{100} > 10.30)$?

  1. $0.9332$
  2. $0.5000$
  3. $0.1587$
  4. $0.0668$ Correct answer.

Explanation

The empirical mean $\bar{X}_n$ has expectation $\mathbb{E}[\bar{X}_{100}] = \mu = 10$ and standard deviation $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{\sqrt{4}}{\sqrt{100}} = \frac{2}{10} = 0.20$. Standardizing the variable gives: $Z = \frac{\bar{X}_{100} - 10}{0.20}$. The condition $\bar{X}_{100} > 10.30$ translates to $Z > \frac{10.30 - 10}{0.20} = \frac{0.30}{0.20} = 1.50$. By the Central Limit Theorem, $P(Z > 1.50) \approx 1 - \Phi(1.50) = 1 - 0.9332 = 0.0668$.

Question 24Advanced probability, moments and conditional models

Let $X$ and $Y$ be two real random variables with variances $\text{Var}(X) = 9$, $\text{Var}(Y) = 16$, and covariance $\text{Cov}(X, Y) = -6$. What is the linear correlation coefficient $\rho(X, Y)$ and the variance of the linear combination $W = 2X - Y$?

  1. $\rho(X, Y) = -0.25$ and $\text{Var}(W) = 28$
  2. $\rho(X, Y) = -0.50$ and $\text{Var}(W) = 52$
  3. $\rho(X, Y) = 0.50$ and $\text{Var}(W) = 76$
  4. $\rho(X, Y) = -0.50$ and $\text{Var}(W) = 76$ Correct answer.

Explanation

The correlation coefficient is $\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} = \frac{-6}{\sqrt{9} \sqrt{16}} = \frac{-6}{3 \times 4} = \frac{-6}{12} = -0.50$. Using bilinear properties of variance: $\text{Var}(2X - Y) = 2^2 \text{Var}(X) + (-1)^2 \text{Var}(Y) + 2(2)(-1) \text{Cov}(X, Y) = 4(9) + 1(16) - 4(-6) = 36 + 16 + 24 = 76$.

Question 25Advanced probability, moments and conditional models

Let $X$ be a random variable with finite expectation $\mathbb{E}[X] = 50$ and standard deviation $\sigma = 5$. Using the Bienaymé-Chebyshev inequality (inégalité de Bienaymé-Tchebychev), what is the tightest theoretical upper bound for the probability $P(|X - 50| \ge 15)$?

  1. $\frac{1}{9}$ Correct answer.
  2. $\frac{1}{3}$
  3. $\frac{5}{15} = \frac{1}{3}$
  4. $\frac{1}{25}$

Explanation

The Bienaymé-Chebyshev inequality states that for any $\varepsilon > 0$, $P(|X - \mathbb{E}[X]| \ge \varepsilon) \le \frac{\text{Var}(X)}{\varepsilon^2}$. Here $\mathbb{E}[X] = 50$, $\varepsilon = 15$, and $\text{Var}(X) = \sigma^2 = 5^2 = 25$. Therefore, $P(|X - 50| \ge 15) \le \frac{25}{15^2} = \frac{25}{225} = \frac{1}{9}$. Equivalently, writing $\varepsilon = k\sigma$ with $k = 15/5 = 3$, the bound is $1/k^2 = 1/9$.

Question 26Advanced probability, moments and conditional models

Let $X \sim \mathcal{N}(\mu, \sigma^2)$ be a real Gaussian random variable with mean $\mu \in \mathbb{R}$ and variance $\sigma^2 > 0$. What is the characteristic function $\phi_X(t) = \mathbb{E}[e^{itX}]$ for $t \in \mathbb{R}$?

  1. $\phi_X(t) = e^{it\mu - \frac{1}{2}\sigma^2 t^2}$ Correct answer.
  2. $\phi_X(t) = e^{it\mu - \sigma^2 t^2}$
  3. $\phi_X(t) = e^{it\mu + \frac{1}{2}\sigma^2 t^2}$
  4. $\phi_X(t) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{it\mu - \frac{1}{2}\sigma^2 t^2}$

Explanation

For a standard normal variable $Z \sim \mathcal{N}(0, 1)$, its characteristic function is $\phi_Z(t) = \int_{-\infty}^{+\infty} \frac{1}{\sqrt{2\pi}} e^{itz - z^2/2} \, dz = e^{-t^2/2}$ (by completing the square or solving $\phi'(t) = -t\phi(t)$). Since $X = \mu + \sigma Z$, we have $\phi_X(t) = \mathbb{E}[e^{it(\mu + \sigma Z)}] = e^{it\mu} \mathbb{E}[e^{i(t\sigma) Z}] = e^{it\mu} e^{-(t\sigma)^2 / 2} = e^{it\mu - \frac{1}{2}\sigma^2 t^2}$.

Question 27Analytical mechanics, oscillation and model reduction

In the Navier-Stokes momentum equation for an incompressible fluid $\rho \left( \frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \vec{\nabla})\vec{v} \right) = -\vec{\nabla} p + \eta \Delta \vec{v} + \vec{f}_{\text{vol}}$, what physical process is represented by the non-linear term $(\vec{v} \cdot \vec{\nabla})\vec{v}$?

  1. The rate of kinetic energy dissipation due to dynamic shear viscosity
  2. The local unsteady acceleration measured at a fixed point in space
  3. The convective acceleration experienced by a fluid particle as it travels through a spatially non-uniform velocity field Correct answer.
  4. The barotropic pressure gradient driving the flow

Explanation

In the Eulerian description of continuum mechanics, the material (particle) derivative is $\frac{D\vec{v}}{Dt} = \frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \vec{\nabla})\vec{v}$. The first term $\frac{\partial \vec{v}}{\partial t}$ is the local acceleration at a fixed spatial coordinate, while $(\vec{v} \cdot \vec{\nabla})\vec{v}$ is the convective acceleration. It accounts for the change in velocity of a fluid parcel moving from one point to another in a velocity field with spatial gradients (e.g. fluid speeding up when entering a nozzle even in steady flow where $\frac{\partial \vec{v}}{\partial t} = 0$).

Question 28Analytical mechanics, oscillation and model reduction

Taking the curl of the Navier-Stokes equation for a homogeneous, incompressible Newtonian fluid with kinematic viscosity $\nu = \eta/\rho$ subjected to conservative body forces ($\vec{f}_{\text{vol}} = -\vec{\nabla} U$), what is the exact vorticity transport equation for $\vec{\omega} = \vec{\nabla} \times \vec{v}$?

  1. $\frac{\partial \vec{\omega}}{\partial t} = \nu \Delta \vec{\omega}$
  2. $\frac{\partial \vec{\omega}}{\partial t} + (\vec{v} \cdot \vec{\nabla})\vec{\omega} = -\vec{\nabla} p + \nu \Delta \vec{\omega}$
  3. $\frac{\partial \vec{\omega}}{\partial t} + (\vec{\omega} \cdot \vec{\nabla})\vec{v} = \nu \vec{\nabla}(\vec{\nabla} \cdot \vec{\omega})$
  4. $\frac{\partial \vec{\omega}}{\partial t} + (\vec{v} \cdot \vec{\nabla})\vec{\omega} = (\vec{\omega} \cdot \vec{\nabla})\vec{v} + \nu \Delta \vec{\omega}$ Correct answer.

Explanation

We use the vector identity $(\vec{v} \cdot \vec{\nabla})\vec{v} = \vec{\nabla}\left(\frac{v^2}{2}\right) + (\vec{\nabla} \times \vec{v}) \times \vec{v} = \vec{\nabla}\left(\frac{v^2}{2}\right) + \vec{\omega} \times \vec{v}$. Taking the curl eliminates all gradient terms: $\vec{\nabla} \times \vec{\nabla}(p/\rho + v^2/2 + U/\rho) = \vec{0}$. Using the curl of a cross product with $\vec{\nabla} \cdot \vec{v} = 0$ and $\vec{\nabla} \cdot \vec{\omega} = 0$: $\vec{\nabla} \times (\vec{\omega} \times \vec{v}) = (\vec{v} \cdot \vec{\nabla})\vec{\omega} - (\vec{\omega} \cdot \vec{\nabla})\vec{v}$. Since $\vec{\nabla} \times (\Delta \vec{v}) = \Delta \vec{\omega}$, we obtain $\frac{D\vec{\omega}}{Dt} = \frac{\partial \vec{\omega}}{\partial t} + (\vec{v} \cdot \vec{\nabla})\vec{\omega} = (\vec{\omega} \cdot \vec{\nabla})\vec{v} + \nu \Delta \vec{\omega}$. The term $(\vec{\omega} \cdot \vec{\nabla})\vec{v}$ represents vortex stretching and tilting.

Question 29Analytical mechanics, oscillation and model reduction

For a one-dimensional quantum harmonic oscillator of mass $m$ and classical angular frequency $\omega$, governed by the Hamiltonian $\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}^2$, what are the ground state energy $E_0$ and the product of uncertainties $\Delta x \Delta p$ in the ground state $|0\rangle$?

  1. $E_0 = 0$ and $\Delta x \Delta p = 0$
  2. $E_0 = \hbar \omega$ and $\Delta x \Delta p = \hbar$
  3. $E_0 = \frac{1}{2}\hbar \omega$ and $\Delta x \Delta p = \hbar$
  4. $E_0 = \frac{1}{2}\hbar \omega$ and $\Delta x \Delta p = \frac{\hbar}{2}$ Correct answer.

Explanation

The eigenvalues of the 1D quantum harmonic oscillator are $E_n = \left(n + \frac{1}{2}\right)\hbar\omega$ for $n = 0, 1, 2, \dots$. The ground state energy is the zero-point energy $E_0 = \frac{1}{2}\hbar\omega$. In the ground state $|0\rangle$, the position wave function is a Gaussian $\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-\frac{m\omega}{2\hbar}x^2}$. For this Gaussian state, $\langle x \rangle = \langle p \rangle = 0$, $(\Delta x)^2 = \langle x^2 \rangle = \frac{\hbar}{2m\omega}$, and $(\Delta p)^2 = \langle p^2 \rangle = \frac{m\omega\hbar}{2}$. Therefore, $\Delta x \Delta p = \sqrt{\frac{\hbar}{2m\omega} \cdot \frac{m\omega\hbar}{2}} = \frac{\hbar}{2}$, which exactly saturates the minimum bound of the Heisenberg uncertainty principle.

Question 30Analytical mechanics, oscillation and model reduction

For an incompressible, inviscid fluid in irrotational motion (écoulement potentiel), the velocity field is given by $\vec{v} = \vec{\nabla} \Phi$, where $\Phi(\vec{r}, t)$ is the velocity potential. Under uniform gravity $\vec{g} = -g\vec{e}_z$, what is the unsteady Bernoulli equation (relation de Bernoulli instationnaire) valid throughout the entire fluid domain?

  1. $\frac{\partial \Phi}{\partial t} + \frac{1}{2} v^2 + \frac{p}{\rho} + gz = C(t)$ Correct answer.
  2. $\frac{1}{2} v^2 + \frac{p}{\rho} + gz = \text{constant along each streamline only}$
  3. $\frac{\partial \Phi}{\partial t} + v^2 + \frac{p}{\rho} - gz = 0$
  4. $\Delta \Phi + \frac{p}{\rho} + gz = C(t)$

Explanation

Starting from Euler's momentum equation $\frac{\partial \vec{v}}{\partial t} + (\vec{v} \cdot \vec{\nabla})\vec{v} = -\frac{1}{\rho}\vec{\nabla} p - g\vec{e}_z$ and using irrotationality $\vec{\nabla} \times \vec{v} = \vec{0}$, we have $(\vec{v} \cdot \vec{\nabla})\vec{v} = \vec{\nabla}\left(\frac{v^2}{2}\right)$. Writing $\vec{v} = \vec{\nabla}\Phi$ gives $\frac{\partial \vec{v}}{\partial t} = \vec{\nabla}\left(\frac{\partial \Phi}{\partial t}\right)$. Factoring the spatial gradient across all terms: $\vec{\nabla}\left( \frac{\partial \Phi}{\partial t} + \frac{1}{2} v^2 + \frac{p}{\rho} + gz \right) = \vec{0}$. Integrating across space yields $\frac{\partial \Phi}{\partial t} + \frac{1}{2} v^2 + \frac{p}{\rho} + gz = C(t)$. Because $\Phi$ can absorb any arbitrary function of time via gauge shift $\Phi \mapsto \Phi + \int C(t)dt$, this relation holds globally throughout the fluid, not just along streamlines.

Question 31Analytical mechanics, oscillation and model reduction

A point projectile is launched with speed v_0 at angle α above horizontal and lands at the same height under uniform gravity g, with no drag. What is its horizontal range for this given angle?

  1. X_{max} = \frac{v_0^2 \sin(2\alpha)}{g} Correct answer.
  2. X_{max} = \frac{v_0^2 \cos(2\alpha)}{g}
  3. X_{max} = \frac{2 v_0 \sin(\alpha)}{g}
  4. X_{max} = \frac{v_0^2 \sin^2(\alpha)}{2g}

Explanation

Parametric equations: $x(t) = v_0 \cos(\alpha) t$, $y(t) = -\frac{1}{2}g t^2 + v_0 \sin(\alpha) t$. Flight time when landing ($y = 0$): $t_{flight} = \frac{2 v_0 \sin(\alpha)}{g}$. Substituting into $x(t)$: $X_{max} = v_0 \cos(\alpha) \left(\frac{2 v_0 \sin(\alpha)}{g}\right) = \frac{v_0^2 (2 \sin\alpha \cos\alpha)}{g} = \frac{v_0^2 \sin(2\alpha)}{g}$. (Maximum range occurs at $\alpha = 45^\circ$).

Question 32Analytical mechanics, oscillation and model reduction

A simple pendulum of length $L = 0.981\text{ m}$ swings with small angular oscillations in a gravitational field $g = 9.81\text{ m/s}^2$. What is its period of oscillation $T_0$?

  1. 1.99 s (~2.0 s, T_0 = 2π √(L/g)) Correct answer.
  2. 1.00 s
  3. 6.28 s
  4. 0.31 s

Explanation

For small oscillations ($\sin\theta \approx \theta$), the period is $T_0 = 2\pi \sqrt{\frac{L}{g}} = 2\pi \sqrt{\frac{0.981}{9.81}} = 2\pi \sqrt{0.1} = 2\pi \times 0.316227 \approx 1.9869\text{ s} \approx 2.0\text{ seconds}$.

Question 33Analytical mechanics, oscillation and model reduction

What is the moment of inertia $J_{\Delta}$ of a solid homogeneous cylinder of mass $M$ and radius $R$ about its central longitudinal axis of symmetry?

  1. J_{\Delta} = \frac{1}{2} M R^2 Correct answer.
  2. J_{\Delta} = M R^2
  3. J_{\Delta} = \frac{2}{5} M R^2
  4. J_{\Delta} = \frac{1}{12} M R^2

Explanation

Integrating concentric thin cylindrical shells: $J_{\Delta} = \int_0^R r^2 \, dm = \int_0^R r^2 (\rho 2\pi r L \, dr) = 2\pi \rho L \left[ \frac{r^4}{4} \right]_0^R = \frac{1}{2} (\rho \pi R^2 L) R^2 = \frac{1}{2} M R^2$.

Question 34Analytical mechanics, oscillation and model reduction

According to Bernoulli's equation for steady, incompressible, frictionless streamline fluid flow with density $\rho$, what quantity remains constant along a streamline?

  1. P + \frac{1}{2} \rho v^2 + \rho g z = \text{constant} Correct answer.
  2. P + \rho v + g z = \text{constant}
  3. \frac{P}{\rho} + v^2 + g z = \text{constant}
  4. P · v · z = \text{constant}

Explanation

Bernoulli's theorem represents conservation of mechanical energy per unit volume for an inviscid, incompressible fluid in steady flow: Static pressure ($P$) + Dynamic pressure ($\frac{1}{2}\rho v^2$) + Hydrostatic potential pressure ($\rho g z$) = constant along any streamline.

Question 35Analytical mechanics, oscillation and model reduction

A venturi tube with inlet cross-section $S_1 = 100\text{ cm}^2$ and throat constriction $S_2 = 25\text{ cm}^2$ carries water ($\rho = 1000\text{ kg/m}^3$). If the measured pressure difference is $\Delta P = P_1 - P_2 = 30\text{ kPa}$ ($30000\text{ Pa}$), what is the volume flow rate $Q$?

  1. 20.0 L/s (0.020 m³/s) Correct answer.
  2. 40.0 L/s
  3. 10.0 L/s
  4. 80.0 L/s

Explanation

By continuity: $v_1 S_1 = v_2 S_2 \implies v_2 = v_1 \frac{S_1}{S_2} = 4 v_1$. From Bernoulli: $P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2) = \frac{1}{2}\rho(16 v_1^2 - v_1^2) = \frac{15}{2}\rho v_1^2$. Thus, $30000 = 7.5 \times 1000 \times v_1^2 = 7500 v_1^2 \implies v_1^2 = \frac{30000}{7500} = 4.0 \implies v_1 = 2.0\text{ m/s}$. Volume flow rate is $Q = S_1 \cdot v_1 = (100 \times 10^{-4}\text{ m}^2) \times 2.0\text{ m/s} = 0.020\text{ m}^3/\text{s} = 20.0\text{ L/s}$.

Question 36Analytical mechanics, oscillation and model reduction

A mass $m = 0.5\text{ kg}$ on a horizontal frictionless table is attached to an ideal spring with spring constant $k = 50\text{ N/m}$. The mass is displaced by $x_0 = 0.10\text{ m}$ from equilibrium and released from rest. What is the maximum velocity $v_{max}$ of the mass?

  1. 1.0 m/s (v_max = x_0 · √(k/m) = 0.10 · √(50/0.5) = 0.10 · 10 = 1.0 m/s) Correct answer.
  2. 10.0 m/s
  3. 0.10 m/s
  4. 5.0 m/s

Explanation

By conservation of mechanical energy: $\frac{1}{2} k x_0^2 = \frac{1}{2} m v_{max}^2 \implies v_{max} = x_0 \sqrt{\frac{k}{m}} = 0.10 \sqrt{\frac{50}{0.5}} = 0.10 \sqrt{100} = 0.10 \times 10 = 1.0\text{ m/s}$.

Question 37Analytical mechanics, oscillation and model reduction

What is the escape velocity $v_{esc}$ from the surface of a spherical planet of mass $M$ and radius $R$?

  1. v_{esc} = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g_0 R} Correct answer.
  2. v_{esc} = \sqrt{\frac{G M}{R}}
  3. v_{esc} = \frac{2 G M}{R}
  4. v_{esc} = \sqrt{\frac{G M}{2 R}}

Explanation

To escape the gravitational potential well to infinity with zero residual kinetic energy: $E_m = E_k + E_p = \frac{1}{2} m v_{esc}^2 - \frac{G M m}{R} = 0 \implies \frac{1}{2} v_{esc}^2 = \frac{G M}{R} \implies v_{esc} = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g_0 R}$. (For Earth, $v_{esc} \approx 11.2\text{ km/s}$).

Question 38Analytical mechanics, oscillation and model reduction

What is the Reynolds number ($Re$) of fluid flow with density $\rho = 1000\text{ kg/m}^3$, dynamic viscosity $\mu = 10^{-3}\text{ Pa}\cdot\text{s}$, velocity $v = 0.5\text{ m/s}$ in a pipe of diameter $D = 0.04\text{ m}$ ($4\text{ cm}$), and what is the flow regime?

  1. Re = 20,000 (Turbulent flow / régime turbulent car Re > 4000) Correct answer.
  2. Re = 2,000 (Laminar flow)
  3. Re = 200 (Creeping flow)
  4. Re = 200,000

Explanation

Reynolds number is $Re = \frac{\rho \cdot v \cdot D}{\mu} = \frac{1000 \times 0.5 \times 0.04}{10^{-3}} = \frac{20}{10^{-3}} = 20000$. Since $Re > 4000$, the flow is fully turbulent (laminar flow occurs for $Re < 2300$).

Question 39Analytical mechanics, oscillation and model reduction

In the presence of quadratic fluid drag force $\vec{F}_d = -\frac{1}{2} C_d \rho A v^2 \vec{u}_v$, what is the terminal falling velocity $v_t$ (vitesse limite de chute) of a spherical body of mass $m$ falling in air under gravity $g$?

  1. v_t = \sqrt{\frac{2 m g}{C_d \rho A}} Correct answer.
  2. v_t = \frac{m g}{C_d \rho A}
  3. v_t = \sqrt{\frac{m g}{2 C_d \rho A}}
  4. v_t = \frac{2 m g}{C_d \rho A}

Explanation

At terminal velocity, downward weight equals upward aerodynamic drag force ($a = 0$): $m g = \frac{1}{2} C_d \rho A v_t^2 \implies v_t^2 = \frac{2 m g}{C_d \rho A} \implies v_t = \sqrt{\frac{2 m g}{C_d \rho A}}$.

Question 40Analytical mechanics, oscillation and model reduction

What is the angular momentum $\vec{L}_O$ of a point particle of mass $m = 2\text{ kg}$ located at position $\vec{r} = (1, 3, 0)\text{ m}$ moving with velocity $\vec{v} = (4, -2, 0)\text{ m/s}$ relative to the origin $O$?

  1. \vec{L}_O = (0, 0, -28) \text{ kg}\cdot\text{m}^2\text{/s} \quad (-28 \vec{k}) Correct answer.
  2. \vec{L}_O = (0, 0, +28) \text{ kg}\cdot\text{m}^2\text{/s}
  3. \vec{L}_O = (0, 0, -14) \text{ kg}\cdot\text{m}^2\text{/s}
  4. \vec{L}_O = (0, 0, 0)

Explanation

Angular momentum is $\vec{L}_O = \vec{r} \times \vec{p} = m (\vec{r} \times \vec{v})$. $\vec{r} \times \vec{v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 1 & 3 & 0 \\ 4 & -2 & 0 \end{vmatrix} = \vec{k} \big((1)(-2) - (3)(4)\big) = \vec{k}(-2 - 12) = -14 \vec{k}$. Multiplying by mass $m = 2\text{ kg}$: $\vec{L}_O = 2 \times (-14 \vec{k}) = -28 \vec{k} = (0, 0, -28)\text{ kg}\cdot\text{m}^2\text{/s}$.

Question 41Analytical mechanics, oscillation and model reduction

A uniform solid cylinder of mass $M$ and radius $R$ rotates about its central axis with angular velocity $\omega_0$. What is its total kinetic energy of rotation $E_k$?

  1. E_k = \frac{1}{4} M R^2 \omega_0^2 Correct answer.
  2. E_k = \frac{1}{2} M R^2 \omega_0^2
  3. E_k = M R^2 \omega_0^2
  4. E_k = \frac{1}{2} M \omega_0^2

Explanation

Rotational kinetic energy is $E_k = \frac{1}{2} J \omega_0^2$. Since $J = \frac{1}{2} M R^2$ for a solid cylinder: $E_k = \frac{1}{2} \left(\frac{1}{2} M R^2\right) \omega_0^2 = \frac{1}{4} M R^2 \omega_0^2$.

Question 42Analytical mechanics, oscillation and model reduction

In a damped harmonic oscillator governed by $\ddot{x} + 2\lambda \dot{x} + \omega_0^2 x = 0$ with underdamping ($\lambda < \omega_0$), what is the pseudo-frequency $\omega_p$ of the oscillations?

  1. \omega_p = \sqrt{\omega_0^2 - \lambda^2} Correct answer.
  2. \omega_p = \omega_0 + \lambda
  3. \omega_p = \sqrt{\omega_0^2 + \lambda^2}
  4. \omega_p = \omega_0 - \lambda

Explanation

Characteristic equation: $r^2 + 2\lambda r + \omega_0^2 = 0 \implies r = -\lambda \pm \sqrt{\lambda^2 - \omega_0^2} = -\lambda \pm i \sqrt{\omega_0^2 - \lambda^2}$. The pseudo-frequency of the decaying sinusoidal response $x(t) = X_0 e^{-\lambda t} \cos(\omega_p t + \phi)$ is $\omega_p = \sqrt{\omega_0^2 - \lambda^2}$.

Question 43Analytical mechanics, oscillation and model reduction

What is the logarithmic decrement $\delta = \ln\left(\frac{x(t)}{x(t + T_p)}\right)$ of the damped oscillator in terms of damping coefficient $\lambda$ and pseudo-period $T_p$?

  1. \delta = \lambda · T_p Correct answer.
  2. \delta = \frac{\lambda}{T_p}
  3. \delta = \frac{2\pi}{\lambda T_p}
  4. \delta = \lambda^2 T_p

Explanation

The amplitude envelope decays as $A(t) = X_0 e^{-\lambda t}$. The ratio of two successive peak amplitudes separated by one pseudo-period $T_p$ is $\frac{x(t)}{x(t + T_p)} = \frac{e^{-\lambda t}}{e^{-\lambda(t + T_p)}} = e^{\lambda T_p}$. Taking natural logarithm: $\delta = \ln(e^{\lambda T_p}) = \lambda T_p$.

Question 44Analytical mechanics, oscillation and model reduction

What is the center of mass $G(\bar{x}, \bar{y})$ of a uniform thin semicircular wire of radius $R$ positioned in the upper half-plane $y \ge 0$ centered at the origin?

  1. \bar{x} = 0 \quad \text{and} \quad \bar{y} = \frac{2R}{\pi} Correct answer.
  2. \bar{x} = 0 \quad \text{and} \quad \bar{y} = \frac{4R}{3\pi}
  3. \bar{x} = 0 \quad \text{and} \quad \bar{y} = \frac{R}{2}
  4. \bar{x} = 0 \quad \text{and} \quad \bar{y} = \frac{R}{\pi}

Explanation

By symmetry, $\bar{x} = 0$. For a 1D thin wire of length $L = \pi R$: $\bar{y} = \frac{1}{L} \int y \, dl = \frac{1}{\pi R} \int_0^\pi (R \sin\theta) (R \, d\theta) = \frac{R}{\pi} [-\cos\theta]_0^\pi = \frac{R}{\pi} (1 - (-1)) = \frac{2R}{\pi} \approx 0.637 R$.

Question 45Analytical mechanics, oscillation and model reduction

What is the center of mass $\bar{y}$ of a uniform solid 2D planar semicircular disk of radius $R$ ($y \ge 0$)?

  1. \bar{y} = \frac{4R}{3\pi} Correct answer.
  2. \bar{y} = \frac{2R}{\pi}
  3. \bar{y} = \frac{3R}{8}
  4. \bar{y} = \frac{R}{3}

Explanation

For a 2D solid planar semicircular lamina of area $A = \frac{\pi R^2}{2}$: $\bar{y} = \frac{1}{A} \iint y \, dA = \frac{2}{\pi R^2} \int_0^\pi \int_0^R (r \sin\theta) r \, dr \, d\theta = \frac{2}{\pi R^2} \left( \int_0^R r^2 \, dr \right) \left( \int_0^\pi \sin\theta \, d\theta \right) = \frac{2}{\pi R^2} \left(\frac{R^3}{3}\right) (2) = \frac{4R}{3\pi} \approx 0.424 R$.

Question 46Electromagnetic fields, thermal physics and wave models

Under proper Lorentz transformations between inertial reference frames, the electric field $\vec{E}$ and magnetic field $\vec{B}$ transform as components of the antisymmetric Faraday tensor $F^{\mu\nu}$. Which of the following pairs of scalar quantities represents the two fundamental Lorentz invariants of the electromagnetic field?

  1. $\|\vec{E}\|^2 + c^2 \|\vec{B}\|^2$ and $\|\vec{E} \times \vec{B}\|$
  2. $\|\vec{E}\|^2 - \|\vec{B}\|^2$ and $\vec{E} + c\vec{B}$
  3. $\|\vec{E}\|^2 \|\vec{B}\|^2$ and $(\vec{E} \cdot \vec{B})^2$
  4. $\|\vec{E}\|^2 - c^2 \|\vec{B}\|^2$ and $\vec{E} \cdot \vec{B}$ Correct answer.

Explanation

The electromagnetic field tensor contraction gives $F_{\mu\nu} F^{\mu\nu} = 2(c^2 \|\vec{B}\|^2 - \|\vec{E}\|^2)$, so $I_1 = \|\vec{E}\|^2 - c^2 \|\vec{B}\|^2$ is a Lorentz scalar invariant. Contracting with the Levi-Civita dual tensor $\tilde{F}^{\mu\nu} = \frac{1}{2} \varepsilon^{\mu\nu\rho\sigma} F_{\rho\sigma}$ yields $F_{\mu\nu} \tilde{F}^{\mu\nu} = -4c (\vec{E} \cdot \vec{B})$, proving that $I_2 = \vec{E} \cdot \vec{B}$ is a Lorentz pseudoscalar invariant. These invariants dictate that if $\vec{E} \perp \vec{B}$ in one frame, they remain perpendicular in all frames.

Question 47Electromagnetic fields, thermal physics and wave models

A non-relativistic point particle with electric charge $q$ undergoes accelerated motion with instantaneous acceleration $a(t)$ ($v \ll c$). According to Larmor's radiation formula (formule de Larmor), what is the total instantaneous power $P$ radiated as electromagnetic waves into all directions?

  1. $P = \frac{q^2 a^2}{6\pi \varepsilon_0 c^3}$ Correct answer.
  2. $P = \frac{q^2 a^2}{12\pi \varepsilon_0 c^2}$
  3. $P = \frac{q a^2}{4\pi \varepsilon_0 c^3}$
  4. $P = \frac{\mu_0 q^2 a}{6\pi c}$

Explanation

Integrating the radiated Poynting flux $\vec{\Pi}_{\text{rad}} = \frac{\mu_0 q^2 a^2 \sin^2 \theta}{16\pi^2 c r^2} \vec{e}_r$ over a sphere of radius $r$ at infinity gives $P = \int_0^\pi \frac{\mu_0 q^2 a^2 \sin^2 \theta}{16\pi^2 c} (2\pi \sin \theta) \, d\theta = \frac{\mu_0 q^2 a^2}{8\pi c} \int_0^\pi \sin^3 \theta \, d\theta = \frac{\mu_0 q^2 a^2}{8\pi c} \left(\frac{4}{3}\right) = \frac{\mu_0 q^2 a^2}{6\pi c}$. Substituting $\mu_0 = \frac{1}{\varepsilon_0 c^2}$ yields Larmor's celebrated formula: $P = \frac{q^2 a^2}{6\pi \varepsilon_0 c^3}$.

Question 48Electromagnetic fields, thermal physics and wave models

For an ideal center-fed thin half-wave dipole antenna (antenne demi-onde de longueur $L = \lambda/2$) radiating into free space, what is its theoretical radiation resistance $R_{\text{rad}}$ referred to the feed-point current antinode $I_0$?

  1. $R_{\text{rad}} \approx 36.6\ \Omega$
  2. $R_{\text{rad}} \approx 73.1\ \Omega$ Correct answer.
  3. $R_{\text{rad}} \approx 120\pi\ \Omega \approx 377\ \Omega$
  4. $R_{\text{rad}} \approx 50.0\ \Omega$

Explanation

For a half-wave dipole with sinusoidal current distribution $I(z) = I_0 \cos(kz)$, the total radiated power is obtained by integrating the Poynting flux: $P_{\text{rad}} = \frac{\eta_0 I_0^2}{4\pi} \int_0^\pi \frac{\cos^2\left(\frac{\pi}{2}\cos\theta\right)}{\sin\theta} \, d\theta = \frac{\eta_0 I_0^2}{4\pi} \text{Cin}(2\pi) \approx \frac{120\pi I_0^2}{4\pi} (1.2188) = 30 I_0^2 (1.2188) \approx 36.56 I_0^2$. Equating this to $\frac{1}{2} R_{\text{rad}} I_0^2$ gives $R_{\text{rad}} = 2 \times 36.56\ \Omega \approx 73.13\ \Omega$.

Question 49Electromagnetic fields, thermal physics and wave models

A semi-infinite flat plate is placed parallel to a uniform, steady incompressible laminar flow of free-stream velocity $U_0$ and kinematic viscosity $\nu$. According to Prandtl's laminar boundary layer theory (théorie de la couche limite de Prandtl), how does the boundary layer thickness $\delta(x)$ scale with the distance $x$ from the leading edge?

  1. $\delta(x) \propto x$
  2. $\delta(x) \propto \sqrt{\frac{\nu x}{U_0}} = \frac{x}{\sqrt{\text{Re}_x}}$ Correct answer.
  3. $\delta(x) \propto x^{4/5}$
  4. $\delta(x) \propto \frac{\nu}{U_0}$

Explanation

Inside the boundary layer, the convective inertial force scales as $\rho u \frac{\partial u}{\partial x} \sim \rho \frac{U_0^2}{x}$, while the viscous shear force scales as $\mu \frac{\partial^2 u}{\partial y^2} \sim \mu \frac{U_0}{\delta^2}$. Balancing inertia and viscosity yields $\rho \frac{U_0^2}{x} \sim \mu \frac{U_0}{\delta^2} \implies \delta^2 \sim \frac{\mu x}{\rho U_0} = \frac{\nu x}{U_0}$. Taking the square root gives the celebrated Blasius scaling $\delta(x) \approx 5.0 \sqrt{\frac{\nu x}{U_0}} = 5.0 \frac{x}{\sqrt{\text{Re}_x}}$, where $\text{Re}_x = \frac{U_0 x}{\nu}$ is the local Reynolds number.

Question 50Electromagnetic fields, thermal physics and wave models

In a positive uniaxial birefringent crystal (cristal uniaxe positif, where extraordinary and ordinary indices satisfy $n_e > n_o$), consider the ray velocity surfaces (surfaces des ondes) emitted by an isotropic point source located inside the crystal. What are the geometric shapes and relative positions of the ordinary and extraordinary wave surfaces?

  1. Both wavefronts are concentric spheres of radii $c/n_o$ and $c/n_e$
  2. The ordinary surface is an ellipsoid of revolution, and the extraordinary surface is a sphere enclosing it
  3. The ordinary surface is a sphere of radius $v_o = c/n_o$, and the extraordinary surface is an ellipsoid of revolution entirely enclosed within the sphere, touching it only at two points on the optic axis Correct answer.
  4. The ordinary surface is a sphere, and the extraordinary surface is an ellipsoid of revolution that lies entirely outside the sphere

Explanation

The ordinary wave propagates with phase velocity $v_o = c/n_o$ equally in all directions, so its wave surface is a sphere of radius $c/n_o$. The extraordinary wave velocity depends on angle $\theta$ relative to the optical axis: along the optical axis, $v_e = c/n_o = v_o$ (where the sphere and ellipsoid touch). Perpendicular to the optical axis, $v_e = c/n_e$. For a positive uniaxial crystal, $n_e > n_o \implies v_e(\pi/2) = c/n_e < c/n_o = v_o$. Therefore, the extraordinary ellipsoid of revolution has semi-minor axis $c/n_e$ and is situated inside the ordinary sphere.