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Concours X-ENS-ESPCI - independent subject practice study materials
Concours X-ENS-ESPCI - independent subject practice study resources

Concours X-ENS-ESPCI - independent subject practice practice questions

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Question 1Advanced analysis, convergence and multivariable reasoning

Consider the polynomial P(X) = X^4 + 1. Over which of the following fields is P(X) irreducible?

  1. Over the field of real numbers R
  2. Over the finite field F_3 = Z/3Z
  3. Over the field of rational numbers Q Correct answer.
  4. Over the finite field F_2 = Z/2Z

Explanation

Over Q, P(X) = X^4 + 1 is the 8th cyclotomic polynomial Φ_8(X), which is irreducible. Alternatively, substituting X = Y + 1 yields P(Y + 1) = Y^4 + 4Y^3 + 6Y^2 + 4Y + 2; applying Eisenstein's criterion with the prime p = 2 (which divides 4, 6, 4, 2 but 2^2 = 4 does not divide 2) proves irreducibility over Q. Over R, P(X) = (X^2 - X√2 + 1)(X^2 + X√2 + 1) factors into real quadratics. Over F_3, X^4 + 1 = (X^2 + X + 2)(X^2 + 2X + 2). Over F_2, X^4 + 1 = (X + 1)^4.

Question 2Advanced analysis, convergence and multivariable reasoning

Using Burnside's Lemma, determine the number of distinct colorings of the 6 vertices of a regular planar hexagon using at most 2 colors (e.g. black and white), up to rotations and reflections belonging to the dihedral group D_6 of order 12.

  1. 9
  2. 13 Correct answer.
  3. 16
  4. 24

Explanation

Burnside's lemma states |X/G| = (1/|G|) ∑_{g ∈ G} |X^g|. Here |G| = 12. The elements of D_6 act on vertices as follows: 1 identity element fixes all 2^6 = 64 colorings; 2 rotations by ±π/3 (cycle structure 6) fix 2^1 = 2 colorings each; 2 rotations by ±2π/3 (two 3-cycles) fix 2^2 = 4 colorings each; 1 rotation by π (three 2-cycles) fixes 2^3 = 8 colorings; 3 reflections across axes passing through opposite vertices (two fixed points, two 2-cycles, cycle structure 1^2 2^2) fix 2^4 = 16 colorings each; 3 reflections across axes passing through midpoints of opposite edges (three 2-cycles, cycle structure 2^3) fix 2^3 = 8 colorings each. Summing fixed points: 64 + 2(2) + 2(4) + 8 + 3(16) + 3(8) = 64 + 4 + 8 + 8 + 48 + 24 = 156. Dividing by |G| = 12 gives 156 / 12 = 13 orbits.

Question 3Advanced analysis, convergence and multivariable reasoning

Let S_4 be the symmetric group on 4 elements. What are the successive derived subgroups (derived series) D^1(S_4) = [S_4, S_4] and D^2(S_4) = [D^1(S_4), D^1(S_4)], and what does this establish regarding the solvability of S_4?

  1. D^1(S_4) = A_4 and D^2(S_4) = V_4 (Klein 4-group); since D^3(S_4) = {id}, S_4 is solvable. Correct answer.
  2. D^1(S_4) = A_4 and D^2(S_4) = A_4; since the series stabilizes at a non-trivial subgroup, S_4 is not solvable.
  3. D^1(S_4) = V_4 and D^2(S_4) = {id}; S_4 is nilpotent of class 2.
  4. D^1(S_4) = Z/2Z and D^2(S_4) = {id}; S_4 is a p-group.

Explanation

The commutator subgroup of S_n for n ≥ 2 is the alternating group A_n, so D^1(S_4) = A_4. The derived subgroup of A_4 is the Klein four-group V_4 = {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, which is abelian. Because V_4 is abelian, its commutator subgroup D^3(S_4) = [V_4, V_4] = {id}. Because the derived series terminates at the trivial subgroup in finite steps, S_4 is a solvable group.

Question 4Advanced analysis, convergence and multivariable reasoning

According to the Arzelà-Ascoli theorem, what conditions are necessary and sufficient for a subset F ⊂ C([0, 1], R) (equipped with the uniform norm ‖f‖_∞ = sup_{x ∈ [0, 1]} |f(x)|) to be relatively compact?

  1. F is closed and bounded in the L^1 norm.
  2. F consists of infinitely differentiable functions with uniformly bounded derivatives.
  3. F is uniformly bounded and equicontinuous. Correct answer.
  4. F is pointwise bounded and closed under uniform convergence.

Explanation

The Arzelà-Ascoli theorem states that for a compact metric space X (here [0, 1]), a subset F of C(X, R) is relatively compact (its closure is compact) if and only if F is equicontinuous (for every ε > 0, there exists δ > 0 such that |x - y| < δ implies |f(x) - f(y)| < ε for all f ∈ F) and pointwise bounded (which, together with equicontinuity on a compact space, is equivalent to uniform boundedness).

Question 5Advanced analysis, convergence and multivariable reasoning

Let X be a complete metric space. According to the Baire Category Theorem, what can be asserted regarding a countable intersection ⋂_{n=1}^∞ U_n of open and dense subsets U_n ⊂ X?

  1. The intersection may be empty if the sets U_n are non-compact.
  2. The intersection is guaranteed to be compact.
  3. The intersection is open and dense in X.
  4. The intersection is dense in X (a dense G_δ set). Correct answer.

Explanation

The Baire Category Theorem asserts that in any complete metric space (or locally compact Hausdorff space), the intersection of countably many dense open sets is dense in X. A countable intersection of open sets is called a G_δ set. Although each U_n is open, a countable intersection of open sets is generally not open (e.g., ⋂ (-1/n, 1/n) = {0} in R).

Question 6Advanced analysis, convergence and multivariable reasoning

Let (X, d) be a complete metric space and T: X → X a k-contraction mapping with Lipschitz constant k ∈ [0, 1). Given an initial point x_0 ∈ X and sequence x_{n+1} = T(x_n), let x* be the unique fixed point. What is the standard a priori error bound on the distance d(x_n, x*)?

  1. d(x_n, x*) ≤ (k^n / (1 + k)) d(x_0, x_1)
  2. d(x_n, x*) ≤ (k^n / (1 - k)) d(x_0, x_1) Correct answer.
  3. d(x_n, x*) ≤ k^n d(x_0, x_1)
  4. d(x_n, x*) ≤ (1 - k)^n d(x_0, x_1)

Explanation

For any m > n, by the triangle inequality: d(x_n, x_m) ≤ ∑_{j=n}^{m-1} d(x_j, x_{j+1}). Since d(x_j, x_{j+1}) ≤ k^j d(x_0, x_1), we have d(x_n, x_m) ≤ d(x_0, x_1) ∑_{j=n}^{m-1} k^j = k^n d(x_0, x_1) ∑_{i=0}^{m-n-1} k^i. Taking the limit as m → ∞ gives the a priori bound d(x_n, x*) ≤ (k^n / (1 - k)) d(x_0, x_1).

Question 7Advanced analysis, convergence and multivariable reasoning

Consider the topologist's sine curve T = {(x, sin(1/x)) ∈ R^2 : x ∈ (0, 1]} ∪ {(0, y) ∈ R^2 : y ∈ [-1, 1]}. Which of the following topological statements correctly describes T?

  1. T is path-connected and compact.
  2. T is locally connected and simply connected.
  3. T is connected and compact, but not path-connected. Correct answer.
  4. T is disconnected because the segment {(0, y) : y ∈ [-1, 1]} is closed and open in T.

Explanation

The graph G = {(x, sin(1/x)) : x ∈ (0, 1]} is the continuous image of the interval (0, 1], so it is connected. The topologist's sine curve T is the closure of G in R^2: T = Cl(G). The closure of any connected set is connected, so T is connected. Furthermore, T is closed and bounded in R^2, hence compact by the Heine-Borel theorem. However, any continuous path γ: [0, 1] → T starting on the vertical segment and entering the curve must oscillate infinitely often, which violates continuity at t = 0; thus T is not path-connected.

Question 8Advanced analysis, convergence and multivariable reasoning

For p ∈ (1, ∞), let ℓ^p denote the Banach space of sequences x = (x_n)_{n≥1} with norm ‖x‖_p = (∑_{n=1}^∞ |x_n|^p)^{1/p}. Which of the following statements regarding the topological dual (ℓ^p)* and reflexivity is correct?

  1. (ℓ^p)* is isometrically isomorphic to ℓ^p, meaning ℓ^p is a Hilbert space for all p ∈ (1, ∞).
  2. (ℓ^p)* is isometrically isomorphic to ℓ^q where 1/p + 1/q = 1, and ℓ^p is reflexive. Correct answer.
  3. (ℓ^p)* is isomorphic to ℓ^∞, and ℓ^p is non-reflexive.
  4. ℓ^p is reflexive if and only if p = 2.

Explanation

By the Riesz representation theorem for sequence spaces (ℓ^p duality via Hölder's inequality), for any 1 < p < ∞ the dual space (ℓ^p)* is isometrically isomorphic to ℓ^q where the conjugate exponent satisfies 1/p + 1/q = 1. Since 1 < q < ∞, the bidual (ℓ^p)** ≅ (ℓ^q)* ≅ ℓ^p, meaning the canonical injection J: ℓ^p → (ℓ^p)** is surjective. Thus ℓ^p is reflexive for all p ∈ (1, ∞).

Question 9Advanced analysis, convergence and multivariable reasoning

Let u(x, y) = x^3 - 3xy^2 be the real part of an entire holomorphic function f(z) = u(x, y) + i v(x, y) on C. Using the Cauchy-Riemann equations, what is the harmonic conjugate v(x, y) satisfying v(0, 0) = 0?

  1. v(x, y) = 3x^2 y + y^3
  2. v(x, y) = x^3 + 3xy^2
  3. v(x, y) = 3x^2 y - y^3 Correct answer.
  4. v(x, y) = -3x^2 y + y^3

Explanation

The Cauchy-Riemann equations require ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. We compute ∂u/∂x = 3x^2 - 3y^2 and ∂u/∂y = -6xy. Integrating ∂v/∂y = 3x^2 - 3y^2 with respect to y gives v(x, y) = 3x^2 y - y^3 + g(x). Then ∂v/∂x = 6xy + g'(x). Setting this equal to -∂u/∂y = -(-6xy) = 6xy yields g'(x) = 0, so g(x) = C. With v(0, 0) = 0, we find C = 0 and v(x, y) = 3x^2 y - y^3. Note that this corresponds to f(z) = z^3 = (x + i y)^3 = (x^3 - 3xy^2) + i(3x^2 y - y^3).

Question 10Advanced analysis, convergence and multivariable reasoning

Evaluate the contour integral ∮_{|z|=2} (e^(3z) / (z - 1)^3) dz oriented counterclockwise, using Cauchy's integral formula for derivatives.

  1. 3 π i e^3
  2. 6 π i e^3
  3. 18 π i e^3
  4. 9 π i e^3 Correct answer.

Explanation

Cauchy's integral formula for higher derivatives states f^(n)(z_0) = (n! / (2π i)) ∮_γ (f(z) / (z - z_0)^(n+1)) dz. Here f(z) = e^(3z), z_0 = 1, and the exponent in the denominator is 3 = n + 1, so n = 2. The second derivative of f is f'(z) = 3 e^(3z) and f''(z) = 9 e^(3z). Evaluating at z_0 = 1 gives f''(1) = 9 e^3. Therefore, ∮_{|z|=2} (e^(3z) / (z - 1)^3) dz = (2π i / 2!) f''(1) = π i (9 e^3) = 9 π i e^3.

Question 11Advanced analysis, convergence and multivariable reasoning

Using the residue theorem, evaluate the improper real integral I = ∫_{-∞}^∞ (1 / (x^4 + 1)) dx.

  1. π / √2 Correct answer.
  2. π √2
  3. π / 2
  4. 2 π / √2

Explanation

Consider the upper half-plane contour consisting of [-R, R] and the semicircular arc Γ_R: z = R e^(i θ) (0 ≤ θ ≤ π). The integrand f(z) = 1/(z^4 + 1) has poles where z^4 = -1 = e^(i π). The poles in the upper half-plane are z_1 = e^(i π/4) and z_2 = e^(i 3π/4). At each simple pole z_k, Res(f, z_k) = 1 / (4 z_k^3) = -z_k / 4 (since z_k^4 = -1). Summing the residues: Res(f, z_1) + Res(f, z_2) = -(1/4)(e^(i π/4) + e^(i 3π/4)) = -(1/4)[(√2/2 + i √2/2) + (-√2/2 + i √2/2)] = -(1/4)(i √2) = -i √2 / 4. By the residue theorem, I = 2π i ∑ Res = 2π i (-i √2 / 4) = 2π (√2 / 4) = π √2 / 2 = π / √2.

Question 12Advanced analysis, convergence and multivariable reasoning

How many roots (counting multiplicities) does the polynomial P(z) = z^5 + 3z + 1 have inside the open unit disk D = {z ∈ C : |z| < 1}, according to Rouché's theorem?

  1. 0
  2. 1 Correct answer.
  3. 3
  4. 5

Explanation

On the boundary circle |z| = 1, let f(z) = 3z and g(z) = z^5 + 1. We estimate: |f(z)| = 3|z| = 3. By the triangle inequality, |g(z)| = |z^5 + 1| ≤ |z|^5 + 1 = 1 + 1 = 2. Since |f(z)| = 3 > 2 ≥ |g(z)| on |z| = 1, Rouché's theorem implies that P(z) = f(z) + g(z) has the same number of roots inside |z| < 1 as f(z) = 3z. The function f(z) has exactly one root inside |z| < 1 (namely z = 0). Therefore, P(z) has exactly 1 root in |z| < 1.

Question 13Advanced analysis, convergence and multivariable reasoning

Consider the function f(z) = 1 / (z(z - 2)). In the annulus A = {z ∈ C : 0 < |z| < 2}, what is the Laurent series expansion of f(z) around z = 0, and what is the residue Res(f, 0)?

  1. f(z) = ∑_{n=0}^∞ (z^n / 2^(n+1)), and Res(f, 0) = 1/2
  2. f(z) = -(1/z) - ∑_{n=0}^∞ (z^n / 2^n), and Res(f, 0) = -1
  3. f(z) = -(1 / (2z)) - ∑_{n=0}^∞ (z^n / 2^(n+2)), and Res(f, 0) = -1/2 Correct answer.
  4. f(z) = (1 / (2z)) + ∑_{n=0}^∞ (z^n / 2^(n+2)), and Res(f, 0) = 1/2

Explanation

We write f(z) = (1/z) · (1 / (z - 2)) = -(1 / (2z)) · (1 / (1 - z/2)). In the domain 0 < |z| < 2, we have |z/2| < 1, so the geometric series converges: 1 / (1 - z/2) = ∑_{n=0}^∞ (z/2)^n = ∑_{n=0}^∞ (z^n / 2^n). Multiplying by -(1/(2z)) gives f(z) = -∑_{n=0}^∞ (z^(n-1) / 2^(n+1)) = -(1 / (2z)) - ∑_{k=0}^∞ (z^k / 2^(k+2)). The residue is the coefficient of 1/z (corresponding to n = 0), which is -1/2.

Question 14Advanced analysis, convergence and multivariable reasoning

Which of the following Möbius transformations maps the upper half-plane H = {z ∈ C : Im(z) > 0} conformally onto the open unit disk D = {w ∈ C : |w| < 1} such that the point z_0 = i is mapped to the origin w = 0?

  1. f(z) = (z + i) / (z - i)
  2. f(z) = (z - 1) / (z + 1)
  3. f(z) = (z - i) / (z + 1)
  4. f(z) = (z - i) / (z + i) Correct answer.

Explanation

Any conformal mapping of the upper half-plane onto the unit disk has the form f(z) = e^(i θ) (z - z_0) / (z - z̄_0) with Im(z_0) > 0. To map z_0 = i to 0, the numerator must be z - i. The denominator must be z - ī = z - (-i) = z + i so that on the real axis (z = x ∈ R), |f(x)| = |x - i| / |x + i| = 1, ensuring the real boundary maps to the unit circle. Setting θ = 0 gives f(z) = (z - i) / (z + i).

Question 15Advanced analysis, convergence and multivariable reasoning

Consider the circular helix parameterized by r(t) = (a cos t, a sin t, b t) with a > 0 and b > 0. What are the Frenet-Serret curvature κ and torsion τ of this curve?

  1. κ = a / (a^2 + b^2) and τ = b / (a^2 + b^2) Correct answer.
  2. κ = a / √(a^2 + b^2) and τ = b / √(a^2 + b^2)
  3. κ = 1 / a and τ = 1 / b
  4. κ = b / (a^2 + b^2) and τ = a / (a^2 + b^2)

Explanation

The velocity vector is r'(t) = (-a sin t, a cos t, b), with speed v = ‖r'(t)‖ = √(a^2 + b^2). The unit tangent is T(t) = r'(t)/v. The derivative with respect to arc length s is dT/ds = (1/v) dT/dt = (1/v^2) (-a cos t, -a sin t, 0). The curvature is κ = ‖dT/ds‖ = a / v^2 = a / (a^2 + b^2). The normal vector is N = (-cos t, -sin t, 0), and the binormal is B = T × N = (b sin t, -b cos t, a) / v. Computing dB/ds = (1/v) dB/dt = (b cos t, b sin t, 0) / v^2 = -(b/v^2) N. Since dB/ds = -τ N, we find τ = b / v^2 = b / (a^2 + b^2).

Question 16Advanced analysis, convergence and multivariable reasoning

For a right circular cylinder of radius R > 0 in R^3 parameterized by r(u, v) = (R cos u, R sin u, v), what are the principal curvatures κ_1, κ_2 and the Gaussian curvature K?

  1. κ_1 = 1/R, κ_2 = 1/R, and K = 1/R^2
  2. κ_1 = 1/R, κ_2 = 0, and K = 0 Correct answer.
  3. κ_1 = 1/R, κ_2 = -1/R, and K = -1/R^2
  4. κ_1 = 0, κ_2 = 0, and K = 0

Explanation

The tangent vectors are r_u = (-R sin u, R cos u, 0) and r_v = (0, 0, 1). The unit normal is n = (cos u, sin u, 0). The coefficients of the first fundamental form are E = R^2, F = 0, G = 1. The second derivatives are r_{uu} = (-R cos u, -R sin u, 0), r_{uv} = 0, r_{vv} = 0. The second fundamental form coefficients are L = r_{uu} · n = -R, M = 0, N = 0. The principal curvatures are the eigenvalues of the shape operator: κ_1 = -L/E = R/R^2 = 1/R (along the circular cross-section) and κ_2 = -N/G = 0 (along the straight rulings). The Gaussian curvature is K = κ_1 · κ_2 = (1/R) · 0 = 0. This reflects the developable nature of the cylinder (Gauss's Theorema Egregium).

Question 17Advanced analysis, convergence and multivariable reasoning

Using Green's theorem (the Green-Riemann formula), evaluate the line integral I = ∮_C (y^2 dx + 3xy dy), where C is the boundary of the unit square [0, 1] × [0, 1] oriented counterclockwise.

  1. 0
  2. 1/4
  3. 1/2 Correct answer.
  4. 1

Explanation

By Green's theorem, ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x - ∂P/∂y) dx dy, where D = [0, 1] × [0, 1]. Here P(x, y) = y^2 and Q(x, y) = 3xy. We compute ∂Q/∂x = 3y and ∂P/∂y = 2y. Therefore, ∂Q/∂x - ∂P/∂y = 3y - 2y = y. Integrating over the square: I = ∫_0^1 dx ∫_0^1 y dy = (1) · [y^2 / 2]_0^1 = 1/2.

Question 18Advanced analysis, convergence and multivariable reasoning

Let ω be a smooth differential 1-form on R^3 that is nowhere vanishing. According to the Frobenius theorem, what is the necessary and sufficient condition for the hyperplane distribution ker(ω) to be completely integrable (i.e. tangent to a foliation of surfaces)?

  1. dω = 0
  2. ω ∧ dω ≠ 0
  3. *dω = ω
  4. ω ∧ dω = 0 Correct answer.

Explanation

The Frobenius integrability theorem for a co-dimension 1 distribution ker(ω) states that the distribution is involutive (and hence completely integrable by Frobenius) if and only if the differential 3-form ω ∧ dω = 0. In vector notation, with ω = u dx + v dy + w dz and curl V = curl(u, v, w), this condition corresponds to V · curl(V) = 0.

Question 19Advanced analysis, convergence and multivariable reasoning

Let (S_n)_{n≥0} be a symmetric simple random walk on Z with S_0 = 0 and step probabilities P(X_i = 1) = P(X_i = -1) = 1/2. Let T = inf{n ≥ 0 : S_n = a or S_n = -b} for positive integers a, b > 0. By applying the Optional Stopping Theorem to the martingale M_n = S_n^2 - n, what is the expected stopping time E[T]?

  1. E[T] = a + b
  2. E[T] = a · b Correct answer.
  3. E[T] = (a^2 + b^2) / 2
  4. E[T] = a^2 b^2

Explanation

First, applying the Optional Stopping Theorem to the martingale S_n gives E[S_T] = S_0 = 0. Since S_T takes values a with probability p and -b with probability 1 - p, we have p a - (1 - p) b = 0, so p = b / (a + b). Second, M_n = S_n^2 - n is a martingale. Applying the Optional Stopping Theorem to M_{T ∧ n} and using dominated/monotone convergence yields E[M_T] = E[S_T^2 - T] = M_0 = 0, so E[T] = E[S_T^2]. Computing E[S_T^2] = p a^2 + (1 - p) b^2 = (b / (a + b)) a^2 + (a / (a + b)) b^2 = (a^2 b + a b^2) / (a + b) = a b (a + b) / (a + b) = a · b.

Question 20Advanced analysis, convergence and multivariable reasoning

For f ∈ L^1(R) ∩ L^2(R), let f̂(ξ) = ∫_{-∞}^∞ f(x) e^(-i ξ x) dx. According to the Plancherel identity, what is the relation between ‖f̂‖_2 and ‖f‖_2 under this convention?

  1. ‖f̂‖_2 = ‖f‖_2
  2. ‖f̂‖_2 = (1 / √(2π)) ‖f‖_2
  3. ‖f̂‖_2 = √(2π) ‖f‖_2 Correct answer.
  4. ‖f̂‖_2 = 2π ‖f‖_2

Explanation

With the non-unitary angular frequency convention f̂(ξ) = ∫_{-∞}^∞ f(x) e^(-i ξ x) dx, the inverse Fourier transform is f(x) = (1 / (2π)) ∫_{-∞}^∞ f̂(ξ) e^(i ξ x) dξ. By Parseval's theorem, ∫_{-∞}^∞ |f̂(ξ)|^2 dξ = 2π ∫_{-∞}^∞ |f(x)|^2 dx. Taking square roots yields ‖f̂‖_2 = √(2π) ‖f‖_2.

Question 21Inner products, spectral reasoning and matrix structure

Let A ∈ M_3(R) be a real 3 × 3 matrix with characteristic polynomial χ_A(X) = (X - 2)^2 (X + 1). If dim(ker(A - 2 I_3)) = 1, what is the minimal polynomial μ_A(X) of A?

  1. μ_A(X) = (X - 2)(X + 1)
  2. μ_A(X) = (X - 2)^2 (X + 1) Correct answer.
  3. μ_A(X) = (X - 2)(X + 1)^2
  4. μ_A(X) = (X - 2)^3

Explanation

The minimal polynomial μ_A(X) shares the same irreducible factors as the characteristic polynomial χ_A(X), so μ_A(X) must be either (X - 2)(X + 1) or (X - 2)^2 (X + 1). The matrix is diagonalizable over C if and only if its minimal polynomial is split with simple roots. Since the geometric multiplicity dim(ker(A - 2 I_3)) = 1 is strictly less than the algebraic multiplicity 2, the eigenspace corresponding to eigenvalue 2 is not full, meaning A is not diagonalizable. Therefore, the multiplicity of the root (X - 2) in μ_A(X) must be 2, yielding μ_A(X) = (X - 2)^2 (X + 1).

Question 22Inner products, spectral reasoning and matrix structure

In the vector space E = R_2[X] of real polynomials of degree at most 2, consider the evaluation linear forms f_0(P) = P(-1), f_1(P) = P(0), and f_2(P) = P(1). The family B* = (f_0, f_1, f_2) forms a basis of the dual space E*. What is the polynomial P_1(X) belonging to the dual basis (biorthogonal basis) B = (P_0, P_1, P_2) of E such that f_i(P_j) = δ_{ij}?

  1. P_1(X) = (1/2)(X^2 - X)
  2. P_1(X) = (1/2)(X^2 + X)
  3. P_1(X) = X^2 - 1
  4. P_1(X) = 1 - X^2 Correct answer.

Explanation

The basis polynomials P_j are the Lagrange interpolation polynomials for nodes x_0 = -1, x_1 = 0, x_2 = 1. For j = 1, P_1(X) must satisfy P_1(-1) = 0, P_1(0) = 1, and P_1(1) = 0. The roots are -1 and 1, so P_1(X) = c(X - (-1))(X - 1) = c(X^2 - 1). Evaluating at X = 0 gives P_1(0) = -c = 1, hence c = -1 and P_1(X) = 1 - X^2.

Question 23Inner products, spectral reasoning and matrix structure

Let E be a finite-dimensional complex Hilbert space with inner product ⟨·, ·⟩, and let u ∈ L(E) be a self-adjoint operator with eigenvalues λ_1 ≤ λ_2 ≤ ... ≤ λ_n. According to the Courant-Fischer min-max theorem, how is the k-th eigenvalue λ_k characterized?

  1. λ_k = min_{dim(V) = k} max_{x ∈ V, x ≠ 0} (⟨u(x), x⟩ / ⟨x, x⟩) Correct answer.
  2. λ_k = max_{dim(V) = k} max_{x ∈ V, x ≠ 0} (⟨u(x), x⟩ / ⟨x, x⟩)
  3. λ_k = min_{dim(V) = n - k} min_{x ∈ V, x ≠ 0} (⟨u(x), x⟩ / ⟨x, x⟩)
  4. λ_k = (1/k) ∑_{j=1}^k max_{x ⊥ v_j} ⟨u(x), x⟩

Explanation

The Courant-Fischer min-max theorem states that for a self-adjoint operator with ordered eigenvalues λ_1 ≤ λ_2 ≤ ... ≤ λ_n, the k-th eigenvalue is given by λ_k = min_{dim(V)=k} max_{x ∈ V, x ≠ 0} R_u(x) = max_{dim(W)=n-k+1} min_{x ∈ W, x ≠ 0} R_u(x), where R_u(x) = ⟨u(x), x⟩ / ‖x‖^2 is the Rayleigh quotient.

Question 24Inner products, spectral reasoning and matrix structure

In the Euclidean space R^3 equipped with the standard dot product, consider the subspace F = Vect(u_1, u_2) spanned by u_1 = (1, 0, 1) and u_2 = (1, 1, 0). What is the distance d(x, F) from the point x = (1, 2, 3) to F, computed using the Gram matrix G(u_1, u_2)?

  1. 1 / √3
  2. √2 / 3
  3. 4 / √3 Correct answer.
  4. 2 / √3

Explanation

The squared distance from x to F is d(x, F)^2 = det G(u_1, u_2, x) / det G(u_1, u_2). Alternatively, the unit normal to F is n = (u_1 × u_2) / ‖u_1 × u_2‖. We compute u_1 × u_2 = (1·0 - 1·1, 1·1 - 1·0, 1·1 - 0·1) = (-1, 1, 1). Its norm is √((-1)^2 + 1^2 + 1^2) = √3. The distance is the absolute value of the projection of x onto n: d(x, F) = |x · n| / ‖n‖ = |1(-1) + 2(1) + 3(1)| / √3 = |-1 + 2 + 3| / √3 = 4 / √3. Checking Gram determinants: det G(u_1, u_2) = (u_1 · u_1)(u_2 · u_2) - (u_1 · u_2)^2 = (2)(2) - (1)^2 = 3. det G(u_1, u_2, x) = [det(u_1, u_2, x)]^2 = (-1·1 + 1·2 + 1·3)^2 = 4^2 = 16. Thus d^2 = 16/3, so d = 4 / √3.

Question 25Inner products, spectral reasoning and matrix structure

Let J ∈ M_2(R) be the Jordan block J = [[2, 1], [0, 2]]. For any t ∈ R, what is the matrix exponential exp(t J)?

  1. [[e^(2t), e^t], [0, e^(2t)]]
  2. e^(2t) [[1, t], [0, 1]] Correct answer.
  3. [[e^(2t), t e^t], [0, e^(2t)]]
  4. e^(2t) [[1, 0], [t, 1]]

Explanation

We decompose J into J = 2 I_2 + N, where N = [[0, 1], [0, 0]]. Since 2 I_2 commutes with N, exp(t J) = exp(2t I_2) · exp(t N). We have exp(2t I_2) = e^(2t) I_2. Because N^2 = 0, the power series for exp(t N) terminates: exp(t N) = I_2 + t N = [[1, t], [0, 1]]. Multiplying by e^(2t) yields exp(t J) = e^(2t) [[1, t], [0, 1]] = [[e^(2t), t e^(2t)], [0, e^(2t)]].

Question 26Inner products, spectral reasoning and matrix structure

On the vector space E = C([0, 1], R), consider the norms ‖f‖_1 = ∫_0^1 |f(t)| dt and ‖f‖_∞ = sup_{t ∈ [0, 1]} |f(t)|. Which of the following statements demonstrates that these two norms are NOT equivalent?

  1. The sequence f_n(t) = t^n satisfies ‖f_n‖_1 = 1/(n + 1) → 0 while ‖f_n‖_∞ = 1 for all n ≥ 1. Correct answer.
  2. The sequence f_n(t) = 1/n satisfies ‖f_n‖_∞ → 0 while ‖f_n‖_1 → 1.
  3. The space (E, ‖·‖_∞) is not complete, whereas (E, ‖·‖_1) is a Banach space.
  4. There exists no constant C such that ‖f‖_1 ≤ C ‖f‖_∞ for all f ∈ E.

Explanation

Two norms ‖·‖_a and ‖·‖_b are equivalent if there exist constants c, C > 0 such that c ‖f‖_a ≤ ‖f‖_b ≤ C ‖f‖_a. While ‖f‖_1 ≤ ‖f‖_∞ is always true on [0, 1] (with C = 1), there is no constant c > 0 such that ‖f‖_∞ ≤ (1/c) ‖f‖_1. For f_n(t) = t^n, ‖f_n‖_1 = ∫_0^1 t^n dt = 1/(n + 1) → 0 as n → ∞, whereas ‖f_n‖_∞ = 1 for every n. Thus ‖f_n‖_∞ / ‖f_n‖_1 = n + 1 → ∞, proving the norms are not equivalent.

Question 27Inner products, spectral reasoning and matrix structure

Let H be a real Hilbert space and K ⊂ H a non-empty closed convex subset. Let P_K: H → K denote the orthogonal projection onto K. Which variational inequality uniquely characterizes P_K(x) for any x ∈ H?

  1. ⟨x - P_K(x), y - P_K(x)⟩ ≥ 0 for all y ∈ K
  2. ⟨x - P_K(x), y⟩ = 0 for all y ∈ K
  3. ‖x - P_K(x)‖^2 + ‖y - P_K(x)‖^2 = ‖x - y‖^2 for all y ∈ K
  4. ⟨x - P_K(x), y - P_K(x)⟩ ≤ 0 for all y ∈ K Correct answer.

Explanation

The projection p = P_K(x) minimizes the distance ‖x - y‖ over y ∈ K. For any y ∈ K and t ∈ (0, 1], convexity of K ensures that p + t(y - p) ∈ K. Expanding ‖x - (p + t(y - p))‖^2 ≥ ‖x - p‖^2 and taking t → 0^+ yields -2t ⟨x - p, y - p⟩ + t^2 ‖y - p‖^2 ≥ 0, which reduces to ⟨x - P_K(x), y - P_K(x)⟩ ≤ 0 for all y ∈ K. Geometrically, this signifies that the angle between x - P_K(x) and y - P_K(x) is obtuse (≥ π/2).

Question 28Inner products, spectral reasoning and matrix structure

Let H be an infinite-dimensional separable complex Hilbert space and T ∈ L(H) a compact self-adjoint operator. What is the structure of the spectrum σ(T) of T?

  1. σ(T) is a countable set of real eigenvalues with 0 as the only possible accumulation point, and 0 ∈ σ(T). Correct answer.
  2. σ(T) is an uncountable continuous interval [a, b] ⊂ R.
  3. σ(T) consists purely of isolated eigenvalues with finite multiplicities, and 0 ∉ σ(T).
  4. σ(T) is empty because compact operators on infinite-dimensional spaces have no bounded inverses.

Explanation

By the spectral theorem for compact self-adjoint operators (Hilbert-Schmidt theorem), the spectrum σ(T) is real, countable, and consists of eigenvalues whose eigenspaces corresponding to λ ≠ 0 are finite-dimensional. The only possible accumulation point of eigenvalues is 0. Furthermore, because H is infinite-dimensional and T is compact, T cannot be invertible (its image cannot be the entire space by Riesz's lemma), which implies 0 ∈ σ(T).

Question 29Finite groups, polynomials, fields and structural hypotheses

Let G be a finite group of order |G| = 100 = 2^2 × 5^2. According to the Sylow theorems, what is the number n_5 of Sylow 5-subgroups of G, and what does this imply about the structure of G?

  1. n_5 = 1, meaning G possesses a unique normal Sylow 5-subgroup of order 25. Correct answer.
  2. n_5 = 2, meaning G is isomorphic to the direct product Z/4Z × Z/25Z.
  3. n_5 = 4, meaning G has four conjugate Sylow 5-subgroups and cannot be abelian.
  4. n_5 = 5, meaning the Sylow 5-subgroups partition the non-identity elements of G.

Explanation

By the Sylow theorems, the number of Sylow 5-subgroups n_5 must satisfy n_5 ≡ 1 (mod 5) and n_5 must divide |G|/5^2 = 4. The divisors of 4 are 1, 2, and 4. The only divisor satisfying n_5 ≡ 1 (mod 5) is n_5 = 1. Since any Sylow p-subgroup is unique if and only if it is normal, G necessarily contains a unique normal Sylow 5-subgroup of order 25.

Question 30Finite groups, polynomials, fields and structural hypotheses

What is the degree [Q(√2 + √3) : Q] of the field extension Q(√2 + √3) over Q, and what is the minimal polynomial of α = √2 + √3 over Q?

  1. Degree 2, with minimal polynomial X^2 - 5
  2. Degree 4, with minimal polynomial X^4 - 10X^2 + 25
  3. Degree 2, with minimal polynomial X^2 - 2√6
  4. Degree 4, with minimal polynomial X^4 - 10X^2 + 1 Correct answer.

Explanation

Let α = √2 + √3. Squaring gives α^2 = 5 + 2√6, so α^2 - 5 = 2√6. Squaring again gives (α^2 - 5)^2 = 24, which expands to α^4 - 10α^2 + 25 = 24, so α^4 - 10α^2 + 1 = 0. The polynomial P(X) = X^4 - 10X^2 + 1 has roots ±√2 ± √3, none of which are rational. Furthermore, P(X) does not factor into rational quadratics (its quadratic factors over R are (X^2 - 2√2 X + 1) and (X^2 + 2√2 X + 1), which are not in Q[X]). Thus P(X) is irreducible over Q, and [Q(√2 + √3) : Q] = deg(P) = 4.

Question 31Advanced probability, moments and conditional models

Let (A_n)_{n≥1} be a sequence of events in a probability space (Ω, F, P). According to the second Borel-Cantelli lemma, what condition guarantees that P(lim sup_{n→∞} A_n) = 1 (i.e. infinitely many A_n occur almost surely)?

  1. The events A_n are mutually independent and ∑_{n=1}^∞ P(A_n) = ∞. Correct answer.
  2. The events A_n are pairwise disjoint and ∑_{n=1}^∞ P(A_n) = 1.
  3. ∑_{n=1}^∞ P(A_n) < ∞, with no independence assumption required.
  4. P(A_n) ≥ 1/2 for all n ≥ 1, regardless of dependence.

Explanation

The first Borel-Cantelli lemma states that if ∑ P(A_n) < ∞, then P(lim sup A_n) = 0 without any requirement of independence. The second Borel-Cantelli lemma provides the converse under independence: if the events (A_n)_{n≥1} are mutually independent and ∑_{n=1}^∞ P(A_n) = ∞, then P(lim sup A_n) = P(A_n infinitely often) = 1.

Question 32Advanced probability, moments and conditional models

What is the characteristic function φ_X(t) = E[e^(i t X)] of a standard normal random variable X ~ N(0, 1)?

  1. φ_X(t) = e^(-t^2)
  2. φ_X(t) = e^(-t^2 / 2) Correct answer.
  3. φ_X(t) = 1 / (1 + t^2)
  4. φ_X(t) = e^(i t - t^2 / 2)

Explanation

By definition, φ_X(t) = (1/√(2π)) ∫_{-∞}^∞ e^(i t x) e^(-x^2 / 2) dx. Completing the square in the exponent gives i t x - x^2 / 2 = -(x - i t)^2 / 2 - t^2 / 2. Shifting the contour of integration in the complex plane (valid by Cauchy's integral theorem since the Gaussian is entire and decays rapidly along horizontal strips) yields φ_X(t) = e^(-t^2 / 2) · (1/√(2π)) ∫_{-∞}^∞ e^(-u^2 / 2) du = e^(-t^2 / 2).

Question 33Advanced probability, moments and conditional models

Let (X_n)_{n≥1} be i.i.d. real random variables with mean μ = 0, variance σ^2 > 0, and finite third absolute moment ρ = E[|X_1|^3] < ∞. Let F_n(x) be the cumulative distribution function of S_n / (σ √n). According to the Berry-Esseen theorem, what is the upper bound on sup_{x ∈ R} |F_n(x) - Φ(x)|, where Φ is the standard normal CDF?

  1. O(1/n)
  2. O(1/n^2)
  3. C · ρ / (σ^3 √n) Correct answer.
  4. C · ρ / (σ^2 n^(1/4))

Explanation

The Berry-Esseen theorem quantifies the rate of convergence in the Central Limit Theorem: there exists a universal constant C (with C < 0.4748) such that for all n ≥ 1, sup_{x ∈ R} |F_n(x) - Φ(x)| ≤ C · E[|X_1|^3] / (σ^3 √n). Thus the uniform approximation error decays at rate O(1/√n).

Question 34Advanced probability, moments and conditional models

In L^2(Ω, F, P), let G ⊂ F be a sub-σ-algebra. What is the geometric interpretation of the conditional expectation E[X | G] of a random variable X ∈ L^2(Ω, F, P)?

  1. It is the projection of X along G onto the subspace of constants.
  2. It is the maximum likelihood estimator of X conditioned on G.
  3. It is the conditional median minimizing E[|X - Y|] over Y ∈ L^1(G).
  4. It is the orthogonal projection of X onto the closed linear subspace L^2(Ω, G, P). Correct answer.

Explanation

In the Hilbert space L^2(Ω, F, P), the space L^2(Ω, G, P) of G-measurable square-integrable random variables forms a closed linear subspace. The conditional expectation E[X | G] is precisely the orthogonal projection of X onto L^2(Ω, G, P). This is equivalent to the defining property: E[(X - E[X | G]) · Y] = 0 for all Y ∈ L^2(G), meaning the error X - E[X | G] is orthogonal to every G-measurable test variable.

Question 35Advanced probability, moments and conditional models

Consider a discrete-time Markov chain on state space S = {1, 2} with transition probability matrix P = [[1 - α, α], [β, 1 - β]], where 0 < α, β < 1. What is the unique stationary distribution π = (π_1, π_2)?

  1. π = (β / (α + β), α / (α + β)) Correct answer.
  2. π = (α / (α + β), β / (α + β))
  3. π = (1/2, 1/2)
  4. π = (1 - α, 1 - β)

Explanation

The stationary distribution satisfies π P = π with π_1 + π_2 = 1. Expanding π_1 = π_1 (1 - α) + π_2 β gives α π_1 = β π_2. Substituting π_2 = 1 - π_1 yields α π_1 = β(1 - π_1) = β - β π_1, so (α + β) π_1 = β, which gives π_1 = β / (α + β). Then π_2 = 1 - π_1 = α / (α + β).

Question 36Analytical mechanics, oscillation and model reduction

For a one-dimensional quantum harmonic oscillator of frequency ω in thermal equilibrium at temperature T (with β = 1/(k_B T)), the discrete energy levels are E_n = (n + 1/2) ℏω (n ≥ 0). What is the canonical partition function Z and the average thermal energy ⟨E⟩?

  1. Z = 1 / (2 cosh(β ℏω / 2)) and ⟨E⟩ = (ℏω / 2) tanh(β ℏω / 2)
  2. Z = 1 / (2 sinh(β ℏω / 2)) and ⟨E⟩ = (ℏω / 2) coth(β ℏω / 2) Correct answer.
  3. Z = e^(-β ℏω) / (1 - e^(-β ℏω)) and ⟨E⟩ = ℏω / (e^(β ℏω) - 1)
  4. Z = 2 sinh(β ℏω / 2) and ⟨E⟩ = k_B T

Explanation

The canonical partition function is Z = ∑_{n=0}^∞ e^(-β (n + 1/2) ℏω) = e^(-β ℏω / 2) ∑_{n=0}^∞ (e^(-β ℏω))^n = e^(-β ℏω / 2) / (1 - e^(-β ℏω)). Multiplying numerator and denominator by e^(β ℏω / 2) gives Z = 1 / (e^(β ℏω / 2) - e^(-β ℏω / 2)) = 1 / (2 sinh(β ℏω / 2)). The average energy is ⟨E⟩ = -∂(ln Z)/∂β = ∂(ln(2 sinh(β ℏω / 2)))/∂β = (ℏω / 2) cosh(β ℏω / 2) / sinh(β ℏω / 2) = (ℏω / 2) coth(β ℏω / 2) = ℏω/2 + ℏω / (e^(β ℏω) - 1).

Question 37Analytical mechanics, oscillation and model reduction

Let L_x, L_y, L_z be the components of the orbital angular momentum operator L = r × p. What is the commutator [L_x, L_y], and what uncertainty relation does it imply for measurements of L_x and L_y?

  1. [L_x, L_y] = 0, implying ΔL_x ΔL_y = 0
  2. [L_x, L_y] = ℏ L_z, implying ΔL_x ΔL_y ≥ (ℏ/2) |⟨L_z⟩|
  3. [L_x, L_y] = i ℏ L_z, implying ΔL_x ΔL_y ≥ (ℏ/2) |⟨L_z⟩| Correct answer.
  4. [L_x, L_y] = -i ℏ L_z, implying ΔL_x ΔL_y ≥ ℏ |⟨L_z⟩|

Explanation

Using canonical commutators [r_j, p_k] = i ℏ δ_{jk}, we evaluate [L_x, L_y] = [y p_z - z p_y, z p_x - x p_z] = y p_x [p_z, z] + x p_y [z, p_z] = y p_x (-i ℏ) + x p_y (i ℏ) = i ℏ (x p_y - y p_x) = i ℏ L_z. By the Robertson-Schrödinger uncertainty relation ΔA ΔB ≥ (1/2) |⟨[A, B]⟩|, setting A = L_x and B = L_y gives ΔL_x ΔL_y ≥ (ℏ/2) |⟨L_z⟩|.

Question 38Analytical mechanics, oscillation and model reduction

For a 1D quantum harmonic oscillator, the annihilation operator is a = √(m ω / (2 ℏ)) (x + i p / (m ω)) and the creation operator is a^†. If |n⟩ denotes the normalized energy eigenstate with eigenvalue E_n = (n + 1/2) ℏω, what are the matrix elements ⟨n| x^2 |n⟩?

  1. ⟨n| x^2 |n⟩ = (ℏ / (2 m ω)) (2n)
  2. ⟨n| x^2 |n⟩ = ℏ / (m ω)
  3. ⟨n| x^2 |n⟩ = (ℏ / (m ω)) (n + 1)
  4. ⟨n| x^2 |n⟩ = (ℏ / (2 m ω)) (2n + 1) Correct answer.

Explanation

We express x in terms of ladder operators: x = √(ℏ / (2 m ω)) (a + a^†). Squaring this gives x^2 = (ℏ / (2 m ω)) (a^2 + a a^† + a^† a + (a^†)^2). When evaluating the diagonal matrix element ⟨n| x^2 |n⟩, the off-diagonal terms a^2 and (a^†)^2 have zero expectation value. For the remaining terms, a a^† |n⟩ = (n + 1) |n⟩ and a^† a |n⟩ = n |n⟩. Thus ⟨n| (a a^† + a^† a) |n⟩ = (n + 1) + n = 2n + 1. Therefore, ⟨n| x^2 |n⟩ = (ℏ / (2 m ω)) (2n + 1). Notice that by the virial theorem, ⟨V⟩ = (1/2) m ω^2 ⟨x^2⟩ = (1/2) ℏω (n + 1/2) = E_n / 2, confirming the result.

Question 39Analytical mechanics, oscillation and model reduction

A one-dimensional quantum harmonic oscillator of mass m and frequency ω is subjected to a small quartic perturbation H' = λ x^4 (with λ > 0). Using first-order time-independent non-degenerate perturbation theory, what is the energy correction E_0^(1) to the ground state |0⟩?

  1. E_0^(1) = (3/4) λ (ℏ / (m ω))
  2. E_0^(1) = (1/4) λ (ℏ / (m ω))^2
  3. E_0^(1) = (3/4) λ (ℏ / (m ω))^2 Correct answer.
  4. E_0^(1) = (3/2) λ (ℏ / (m ω))^2

Explanation

In first-order perturbation theory, E_0^(1) = ⟨0| H' |0⟩ = λ ⟨0| x^4 |0⟩. Writing x = √(ℏ / (2 m ω)) (a + a^†), we have x^4 = (ℏ / (2 m ω))^2 (a + a^†)^4. When expanding (a + a^†)^4 and evaluating the expectation value in the ground state |0⟩, only terms with an equal number of creation and annihilation operators contribute where annihilation operators do not act directly on |0⟩. The non-vanishing combinations acting on |0⟩ are a a a^† a^† |0⟩ and a a^† a a^† |0⟩. Specifically, (a + a^†)^2 |0⟩ = |0⟩ + √2 |2⟩. Then ‖(a + a^†)^2 |0⟩‖^2 = ⟨0| (a + a^†)^4 |0⟩ = 1^2 + (√2)^2 = 1 + 2 = 3. Thus ⟨0| x^4 |0⟩ = 3 (ℏ / (2 m ω))^2 = 3 (ℏ^2 / (4 m^2 ω^2)) = (3/4) (ℏ / (m ω))^2. Therefore, E_0^(1) = (3/4) λ (ℏ / (m ω))^2.

Question 40Analytical mechanics, oscillation and model reduction

The motion of an incompressible Newtonian fluid of density ρ and dynamic viscosity μ is governed by the Navier-Stokes equation: ρ(∂v/∂t + (v · ∇)v) = -∇p + μ ∇^2 v + ρ g. When non-dimensionalized using characteristic length L and velocity U, what dimensionless group governs the ratio of inertial forces to viscous forces?

  1. Froude number Fr = U / √(g L)
  2. Knudsen number Kn = λ / L
  3. Mach number Ma = U / c_s
  4. Reynolds number Re = ρ U L / μ Correct answer.

Explanation

Inertial forces scale as ρ (v · ∇)v ~ ρ U^2 / L, while viscous diffusion forces scale as μ ∇^2 v ~ μ U / L^2. The ratio of inertial to viscous forces is (ρ U^2 / L) / (μ U / L^2) = ρ U L / μ = U L / ν, which is the Reynolds number Re. High Re indicates inertia-dominated, potentially turbulent flow, whereas low Re (Re ≪ 1) corresponds to Stokes (creeping) flow.

Question 41Analytical mechanics, oscillation and model reduction

Deep-water surface gravity waves on an ideal fluid of infinite depth satisfy the dispersion relation ω^2 = g k, where g is gravitational acceleration and k is the wavenumber. What is the relation between the phase velocity v_φ and the group velocity v_g?

  1. v_g = 2 v_φ
  2. v_g = (1/2) v_φ Correct answer.
  3. v_g = v_φ
  4. v_g = (1/4) v_φ

Explanation

The dispersion relation is ω(k) = √(g k) = g^(1/2) k^(1/2). The phase velocity is v_φ = ω / k = √(g/k). The group velocity is the derivative of frequency with respect to wavenumber: v_g = dω/dk = (1/2) g^(1/2) k^(-1/2) = (1/2) √(g/k). Comparing the two expressions yields v_g = (1/2) v_φ. This means wave packets travel at half the speed of individual wave crests.

Question 42Analytical mechanics, oscillation and model reduction

According to the Euler-Bernoulli beam theory, the transverse deflection w(x) of a slender elastic beam with Young's modulus E and second moment of area I satisfies E I (d^4 w / dx^4) = q(x). For a cantilever beam of length L clamped at x = 0 (w(0) = 0, w'(0) = 0) and subjected to a transverse point load F at its free end x = L, what is the maximum tip deflection w(L)?

  1. w(L) = F L^3 / (8 E I)
  2. w(L) = F L^3 / (48 E I)
  3. w(L) = F L^3 / (3 E I) Correct answer.
  4. w(L) = F L^4 / (8 E I)

Explanation

The bending moment along the cantilever beam at distance x from the clamp is M(x) = F(L - x). By the Euler-Bernoulli moment-curvature equation, E I w''(x) = M(x) = F(L - x). Integrating once with boundary condition w'(0) = 0 yields E I w'(x) = F(L x - x^2 / 2). Integrating a second time with w(0) = 0 yields E I w(x) = F(L x^2 / 2 - x^3 / 6). Evaluating at the tip x = L: E I w(L) = F(L^3 / 2 - L^3 / 6) = F L^3 (3/6 - 1/6) = F L^3 / 3. Thus w(L) = F L^3 / (3 E I).

Question 43Analytical mechanics, oscillation and model reduction

Consider the nonlinear dynamical system of a damped pendulum governed by dx/dt = y and dy/dt = -sin(x) - γ y, with damping parameter γ > 0. What is the nature and stability of the fixed point at (x*, y*) = (π, 0) in the phase portrait?

  1. Stable spiral (focus)
  2. Stable node (sink)
  3. Center (neutral stability)
  4. Saddle point (unstable) Correct answer.

Explanation

The Jacobian matrix of the system is J(x, y) = [[0, 1], [-cos(x), -γ]]. Evaluating J at (π, 0): since cos(π) = -1, J(π, 0) = [[0, 1], [1, -γ]]. The characteristic equation is det(J - λ I) = λ(λ + γ) - 1 = λ^2 + γ λ - 1 = 0. The eigenvalues are λ_{1, 2} = (-γ ± √(γ^2 + 4)) / 2. Because √(γ^2 + 4) > γ for any real γ > 0, the discriminant is strictly positive, giving one strictly positive eigenvalue λ_1 > 0 and one strictly negative eigenvalue λ_2 < 0. By Hartman-Grobman theorem, a fixed point with eigenvalues of opposite real signs is an unstable hyperbolic saddle point.

Question 44Analytical mechanics, oscillation and model reduction

In a collisionless, cold, unmagnetized plasma with electron density n_e = 1.0 × 10^(18) m^(-3), what is the electron plasma frequency ω_p (Langmuir frequency), given electron mass m_e = 9.11 × 10^(-31) kg, charge e = 1.60 × 10^(-19) C, and vacuum permittivity ε_0 = 8.85 × 10^(-12) F/m?

  1. ω_p ≈ 5.6 × 10^7 rad/s
  2. ω_p ≈ 1.8 × 10^9 rad/s
  3. ω_p ≈ 5.6 × 10^(10) rad/s Correct answer.
  4. ω_p ≈ 3.2 × 10^(21) rad/s

Explanation

The Langmuir plasma frequency is ω_p = √(n_e e^2 / (ε_0 m_e)). Calculating the numerator: n_e e^2 = (1.0 × 10^(18)) × (1.60 × 10^(-19))^2 = 1.0 × 10^(18) × 2.56 × 10^(-38) = 2.56 × 10^(-20) C^2/m^3. Calculating the denominator: ε_0 m_e = (8.85 × 10^(-12)) × (9.11 × 10^(-31)) ≈ 8.06 × 10^(-42) F·kg/m. Dividing: ω_p^2 = (2.56 × 10^(-20)) / (8.06 × 10^(-42)) ≈ 3.175 × 10^(21) s^(-2). Taking the square root: ω_p = √(3.175 × 10^(21)) ≈ 5.63 × 10^(10) rad/s.

Question 45Analytical mechanics, oscillation and model reduction

An electron with rest mass m_e is accelerated to a relativistic kinetic energy equal to its rest energy: T = m_e c^2. What is its relativistic momentum p?

  1. p = m_e c
  2. p = 2 m_e c
  3. p = (1 / √3) m_e c
  4. p = √3 m_e c Correct answer.

Explanation

The total relativistic energy is E = T + m_e c^2. Given T = m_e c^2, we have E = 2 m_e c^2. The Lorentz factor is γ = E / (m_e c^2) = 2. Using the fundamental invariant relation E^2 = p^2 c^2 + m_e^2 c^4: (2 m_e c^2)^2 = p^2 c^2 + m_e^2 c^4, which expands to 4 m_e^2 c^4 = p^2 c^2 + m_e^2 c^4. Subtracting m_e^2 c^4 gives p^2 c^2 = 3 m_e^2 c^4, hence p = √3 m_e c.

Question 46Electromagnetic fields, thermal physics and wave models

In special relativistic covariant electrodynamics with metric signature (+, -, -, -), the four-potential is A^μ = (φ/c, A). What is the Lorenz gauge condition expressed in covariant four-divergence notation?

  1. ∂_μ A_ν - ∂_ν A_μ = 0
  2. ∂_0 A^0 = 0
  3. ∂^μ ∂_μ A^ν = 0
  4. ∂_μ A^μ = 0 Correct answer.

Explanation

In four-vector notation, ∂_μ = ( (1/c) ∂/∂t, ∇ ) and A^μ = ( φ/c, A ). The four-divergence is ∂_μ A^μ = (1/c) ∂(φ/c)/∂t + ∇ · A = (1/c^2) ∂φ/∂t + ∇ · A. Setting ∂_μ A^μ = 0 is precisely the Lorenz gauge condition (1/c^2) ∂φ/∂t + div A = 0.

Question 47Electromagnetic fields, thermal physics and wave models

An electron of charge q = -e = -1.6 × 10^(-19) C undergoes non-relativistic acceleration a = 3.0 × 10^17 m/s^2. Given ε_0 = 8.85 × 10^(-12) F/m and c = 3.0 × 10^8 m/s, what is the total power P radiated by the electron according to the Larmor formula?

  1. P ≈ 5.1 × 10^(-19) W Correct answer.
  2. P ≈ 1.7 × 10^(-36) W
  3. P ≈ 1.5 × 10^(-10) W
  4. P ≈ 9.6 × 10^(-18) W

Explanation

Larmor's formula states P = (q^2 a^2) / (6 π ε_0 c^3). Computing the numerator: q^2 = (1.6 × 10^(-19))^2 = 2.56 × 10^(-38) C^2; a^2 = (3.0 × 10^17)^2 = 9.0 × 10^34 m^2/s^4; q^2 a^2 = 2.304 × 10^(-3) C^2 m^2 / s^4. Denominator: c^3 = 2.7 × 10^25 m^3/s^3, so 6 π ε_0 c^3 = 6 × π × (8.85 × 10^(-12)) × (2.7 × 10^25) ≈ 4.50 × 10^15. Dividing: P = (2.304 × 10^(-3)) / (4.50 × 10^15) ≈ 5.1 × 10^(-19) W.

Question 48Electromagnetic fields, thermal physics and wave models

A high-frequency electromagnetic wave at f = 1.0 GHz (ω = 2π × 10^9 rad/s) impinges on a copper conductor with electrical conductivity σ = 5.8 × 10^7 S/m and magnetic permeability μ = μ_0 = 4π × 10^(-7) H/m. What is the characteristic skin depth δ of the electromagnetic field inside the copper?

  1. δ ≈ 21 μm
  2. δ ≈ 2.1 μm Correct answer.
  3. δ ≈ 0.21 μm
  4. δ ≈ 66 μm

Explanation

The skin depth is given by δ = √(2 / (μ σ ω)) = √(1 / (π f μ σ)). Substituting: π f = π × 10^9 s^(-1); μ = 4π × 10^(-7) H/m; σ = 5.8 × 10^7 S/m. The product in the denominator is π f μ σ = π × 10^9 × 4π × 10^(-7) × 5.8 × 10^7 = 4 π^2 × 5.8 × 10^9 ≈ 4 × 9.8696 × 5.8 × 10^9 ≈ 2.29 × 10^11 m^(-2). Taking the reciprocal: 1 / (2.29 × 10^11) ≈ 4.37 × 10^(-12) m^2. Taking the square root: δ = √(4.37 × 10^(-12)) ≈ 2.09 × 10^(-6) m = 2.1 μm.

Question 49Electromagnetic fields, thermal physics and wave models

For the fundamental transverse electric mode TE_10 propagating along a hollow rectangular metallic waveguide of width a (along x) and height b < a (along y), what is the relationship between the phase velocity v_φ, the group velocity v_g, and the speed of light in vacuum c?

  1. v_φ < c and v_g > c, with v_φ / v_g = c^2
  2. v_φ = v_g = c
  3. v_φ > c and v_g < c, with v_φ · v_g = c^2 Correct answer.
  4. v_φ · v_g = c^2 / 2

Explanation

The dispersion relation for TE_10 is ω^2 = c^2 k_z^2 + ω_c^2, where ω_c = c π / a is the cutoff angular frequency. Differentiating with respect to k_z gives 2ω dω/dk_z = 2 c^2 k_z, so v_g = dω/dk_z = c^2 k_z / ω. The phase velocity is v_φ = ω / k_z. Multiplying them yields v_φ · v_g = (ω / k_z) · (c^2 k_z / ω) = c^2. For propagating frequencies ω > ω_c, k_z = (1/c) √(ω^2 - ω_c^2) < ω/c, which implies v_φ = ω/k_z > c and v_g = c^2 / v_φ < c, fully consistent with relativistic causality (energy travels at v_g < c).

Question 50Electromagnetic fields, thermal physics and wave models

Under an arbitrary smooth gauge transformation parameterized by scalar function χ(r, t), the electromagnetic potentials transform as A' = A + ∇χ and φ' = φ - ∂χ/∂t. Why do the physical electric field E and magnetic field B remain invariant?

  1. Because ∇ · (∇χ) = 0 and ∂^2 χ / ∂t^2 = 0 for any scalar function.
  2. Because the Lorentz force equation contains only the potentials, not the fields.
  3. Because the Lagrangian of the electromagnetic field is identically zero under all gauge transformations.
  4. Because curl(grad χ) = 0 and the mixed time and space derivatives cancel in E = -∇φ - ∂A/∂t. Correct answer.

Explanation

The magnetic field is B' = curl A' = curl(A + ∇χ) = curl A + curl(∇χ) = B because the curl of any gradient is identically zero. The electric field is E' = -∇φ' - ∂A'/∂t = -∇(φ - ∂χ/∂t) - ∂(A + ∇χ)/∂t = -∇φ + ∇(∂χ/∂t) - ∂A/∂t - ∂(∇χ)/∂t. By Schwarz's theorem on mixed partial derivatives, ∇(∂χ/∂t) = ∂(∇χ)/∂t, so the two terms cancel exactly, leaving E' = -∇φ - ∂A/∂t = E.