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Concours ATS - independent subject practice study materials
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Concours ATS - independent subject practice practice questions

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Question 1Advanced analysis, convergence and multivariable reasoning

What is the unique solution to the first-order linear differential equation y' + 2y = 4x satisfying the initial condition y(0) = 1?

  1. y(x) = 2 e^(-2x) + 2x - 1 Correct answer.
  2. y(x) = e^(-2x) + 2x
  3. y(x) = 3 e^(-2x) + 2x - 2
  4. y(x) = 2 e^(2x) - 2x - 1

Explanation

The homogeneous equation y' + 2y = 0 has general solution y_h(x) = C e^(-2x). A particular solution of the form y_p(x) = ax + b gives y_p' = a; substituting yields a + 2(ax + b) = 2ax + (a + 2b) = 4x, so 2a = 4 (a = 2) and a + 2b = 0 (b = -1). Thus y(x) = C e^(-2x) + 2x - 1. Applying y(0) = C - 1 = 1 yields C = 2, giving y(x) = 2 e^(-2x) + 2x - 1.

Question 2Advanced analysis, convergence and multivariable reasoning

What is the general real solution to the homogeneous second-order linear differential equation y'' + 4y' + 13y = 0?

  1. y(x) = e^(2x) [A cos(3x) + B sin(3x)]
  2. y(x) = e^(-2x) [A cos(3x) + B sin(3x)] Correct answer.
  3. y(x) = A e^(-2x) + B e^(-3x)
  4. y(x) = e^(-4x) [A cos(13x) + B sin(13x)]

Explanation

The characteristic equation is r^2 + 4r + 13 = 0. Its discriminant is Delta = 4^2 - 4(1)(13) = 16 - 52 = -36 = (6i)^2. The complex conjugate roots are r = (-4 +- 6i)/2 = -2 +- 3i. Therefore, the general real solution is y(x) = e^(-2x) [A cos(3x) + B sin(3x)], where A and B are real arbitrary constants.

Question 3Advanced analysis, convergence and multivariable reasoning

Which function represents a particular solution y_p(x) to the resonant differential equation y'' + 9y = 6 cos(3x)?

  1. y_p(x) = cos(3x)
  2. y_p(x) = x cos(3x)
  3. y_p(x) = x sin(3x) Correct answer.
  4. y_p(x) = 2x sin(3x)

Explanation

The characteristic roots of y'' + 9y = 0 are r = +-3i. Because the excitation frequency omega = 3 matches the natural frequency, resonance occurs. We seek a particular solution of the form y_p(x) = x(A cos(3x) + B sin(3x)). Differentiating twice gives y_p'' + 9y_p = -6A sin(3x) + 6B cos(3x). Setting this equal to 6 cos(3x) yields -6A = 0 (A = 0) and 6B = 6 (B = 1), so y_p(x) = x sin(3x).

Question 4Advanced analysis, convergence and multivariable reasoning

What is the Taylor expansion (développement limité) of f(x) = ln(1 + x) / (1 + x) at x = 0 to order 3?

  1. f(x) = x - (3/2) x^2 + (11/6) x^3 + o(x^3) Correct answer.
  2. f(x) = x - (1/2) x^2 + (1/3) x^3 + o(x^3)
  3. f(x) = x - x^2 + x^3 + o(x^3)
  4. f(x) = x - (3/2) x^2 + (7/6) x^3 + o(x^3)

Explanation

We use the standard expansions at x = 0: ln(1 + x) = x - x^2/2 + x^3/3 + o(x^3) and 1/(1 + x) = 1 - x + x^2 - x^3 + o(x^3). Multiplying the polynomials: f(x) = (x - x^2/2 + x^3/3)(1 - x + x^2) + o(x^3). Degree 1: x. Degree 2: -x^2 - x^2/2 = -(3/2) x^2. Degree 3: x^3 + x^3/2 + x^3/3 = (1 + 1/2 + 1/3) x^3 = (11/6) x^3. Hence f(x) = x - (3/2) x^2 + (11/6) x^3 + o(x^3).

Question 5Advanced analysis, convergence and multivariable reasoning

What is the radius of convergence R of the real power series sum_{n=1}^infty (3^n / n^2) x^(2n)?

  1. R = 1/3
  2. R = 1 / sqrt(3) Correct answer.
  3. R = sqrt(3)
  4. R = 3

Explanation

Let u_n = (3^n / n^2) x^(2n). Applying d'Alembert's ratio test: |u_{n+1} / u_n| = [3^(n+1) / (n+1)^2 * x^(2n+2)] / [3^n / n^2 * x^(2n)] = 3 * (n / (n+1))^2 * x^2. As n -> infty, this ratio approaches 3 x^2. For absolute convergence, we require 3 x^2 < 1, which implies x^2 < 1/3, or |x| < 1 / sqrt(3). Thus, the radius of convergence is R = 1 / sqrt(3) = sqrt(3)/3.

Question 6Advanced analysis, convergence and multivariable reasoning

For |x| < 1, what is the closed-form sum of the power series S(x) = sum_{n=1}^infty n x^n?

  1. S(x) = 1 / (1 - x)^2
  2. S(x) = x / (1 - x)
  3. S(x) = x / (1 - x)^2 Correct answer.
  4. S(x) = (1 + x) / (1 - x)^3

Explanation

Starting from the standard geometric series sum_{n=0}^infty x^n = 1 / (1 - x) for |x| < 1, we differentiate term-by-term with respect to x: sum_{n=1}^infty n x^(n-1) = d/dx [1 / (1 - x)] = 1 / (1 - x)^2. Multiplying both sides by x yields S(x) = sum_{n=1}^infty n x^n = x / (1 - x)^2.

Question 7Advanced analysis, convergence and multivariable reasoning

What is the value of the double integral I = iint_D (2x + y) dx dy over the triangular region D bounded by x = 0, y = 0, and x + y = 1?

  1. I = 1/6
  2. I = 1/3
  3. I = 1
  4. I = 1/2 Correct answer.

Explanation

The domain is described by 0 <= x <= 1 and 0 <= y <= 1 - x. Integrating with respect to y first: int_0^(1-x) (2x + y) dy = [2xy + y^2/2]_0^(1-x) = 2x(1 - x) + (1 - x)^2 / 2 = 2x - 2x^2 + (1 - 2x + x^2)/2 = 1/2 + x - (3/2) x^2. Now integrate with respect to x: int_0^1 [1/2 + x - (3/2) x^2] dx = [x/2 + x^2/2 - x^3/2]_0^1 = 1/2 + 1/2 - 1/2 = 1/2.

Question 8Advanced analysis, convergence and multivariable reasoning

What is the value of the double integral I = iint_D sqrt(x^2 + y^2) dx dy where D is the circular disk x^2 + y^2 <= 9?

  1. I = 18 pi Correct answer.
  2. I = 9 pi
  3. I = 27 pi
  4. I = 54 pi

Explanation

Switching to polar coordinates x = r cos(theta), y = r sin(theta) with Jacobian r, the region D is defined by 0 <= r <= 3 and 0 <= theta <= 2 pi. The integrand becomes sqrt(x^2 + y^2) = r, so the integral is I = int_0^(2 pi) d theta int_0^3 r * r dr = 2 pi * [r^3 / 3]_0^3 = 2 pi * (27 / 3) = 18 pi.

Question 9Advanced analysis, convergence and multivariable reasoning

For the scalar field f(x, y, z) = x^2 y - 3 y z^2 + 2 x z, what is the gradient vector grad(f) evaluated at the point P(1, 2, -1)?

  1. grad(f)(P) = (4, -2, 10)
  2. grad(f)(P) = (2, -2, 14) Correct answer.
  3. grad(f)(P) = (2, 1, -10)
  4. grad(f)(P) = (-2, -2, 14)

Explanation

The partial derivatives are: df/dx = 2xy + 2z; df/dy = x^2 - 3z^2; df/dz = -6yz + 2x. Evaluating at P(1, 2, -1): df/dx = 2(1)(2) + 2(-1) = 4 - 2 = 2; df/dy = 1^2 - 3(-1)^2 = 1 - 3 = -2; df/dz = -6(2)(-1) + 2(1) = 12 + 2 = 14. Therefore, grad(f)(P) = (2, -2, 14).

Question 10Advanced analysis, convergence and multivariable reasoning

What is the divergence of the vector field F = (x^2 y) i + (y^2 z) j + (z^2 x) k evaluated at the point (1, 1, 1)?

  1. div(F) = 2
  2. div(F) = 3
  3. div(F) = 6 Correct answer.
  4. div(F) = 0

Explanation

The divergence of a vector field F = (F_x, F_y, F_z) is div(F) = dF_x/dx + dF_y/dy + dF_z/dz. Differentiating each component: d(x^2 y)/dx = 2xy, d(y^2 z)/dy = 2yz, and d(z^2 x)/dz = 2zx. Thus div(F) = 2(xy + yz + zx). At the point (1, 1, 1), div(F) = 2(1 + 1 + 1) = 2(3) = 6.

Question 11Advanced analysis, convergence and multivariable reasoning

What is the curl (rotationnel) of the vector field V = (y z) i + (x z) j + (x y) k?

  1. curl(V) = (x, y, z)
  2. curl(V) = 2(i + j + k)
  3. curl(V) = (y - z) i + (z - x) j + (x - y) k
  4. curl(V) = 0 Correct answer.

Explanation

The components of curl(V) = rot(V) are: (rot V)_x = d(xy)/dy - d(xz)/dz = x - x = 0; (rot V)_y = d(yz)/dz - d(xy)/dx = y - y = 0; (rot V)_z = d(xz)/dx - d(yz)/dy = z - z = 0. Because all components vanish identically, curl(V) = 0. This vector field is conservative, deriving from the scalar potential phi(x, y, z) = x y z.

Question 12Advanced analysis, convergence and multivariable reasoning

Using Green's theorem, what is the line integral oint_C (-y dx + x dy) along the circle C defined by x^2 + y^2 = 4, oriented counterclockwise?

  1. 8 pi Correct answer.
  2. 4 pi
  3. 16 pi
  4. 0

Explanation

By Green's theorem in the plane, oint_C (P dx + Q dy) = iint_D (dQ/dx - dP/dy) dx dy. Here P = -y and Q = x, so dQ/dx = 1 and dP/dy = -1, yielding dQ/dx - dP/dy = 1 - (-1) = 2. The integral becomes 2 * iint_D dx dy = 2 * Area(D). For a circle of radius R = 2, the area is pi R^2 = 4 pi. Hence the circulation is 2 * (4 pi) = 8 pi.

Question 13Advanced analysis, convergence and multivariable reasoning

For what value of parameter k does the 3x3 linear system have infinitely many solutions? x + 2y + z = 1 2x + k y + 2z = 2 3x + 6y + 4z = 3

  1. k = 2
  2. k = 4 Correct answer.
  3. k = 6
  4. There is no such value of k

Explanation

Observe the first two equations: the left side of the second equation is 2x + ky + 2z, and its right side is 2. Comparing with twice the first equation, 2(x + 2y + z) = 2x + 4y + 2z = 2. If k = 4, the second equation is completely redundant with the first (row 2 = 2 * row 1). Subtracting 3 * (eq 1) from eq 3 gives: (3-3)x + (6-6)y + (4-3)z = 3 - 3 => z = 0. Then eq 1 becomes x + 2y = 1, which yields a one-parameter line of solutions (x = 1 - 2y, y in R, z = 0). Thus k = 4 yields infinitely many solutions.

Question 14Advanced analysis, convergence and multivariable reasoning

What is the solution to the separable differential equation dy/dx = 2x (1 + y^2) satisfying the initial condition y(0) = 0?

  1. y(x) = e^(x^2) - 1
  2. y(x) = sin(x^2)
  3. y(x) = x^2 / (1 - x^2)
  4. y(x) = tan(x^2) Correct answer.

Explanation

Separating variables: dy / (1 + y^2) = 2x dx. Integrating both sides gives arctan(y) = x^2 + C. Using the initial condition y(0) = 0: arctan(0) = 0^2 + C, so C = 0. Taking the tangent of both sides yields y(x) = tan(x^2).

Question 15Advanced analysis, convergence and multivariable reasoning

For what real values of parameter p does the improper integral int_1^infty (ln x) / x^p dx converge?

  1. p > 1 Correct answer.
  2. p >= 1
  3. p > 0
  4. p > 2

Explanation

Let x = e^u, so dx = e^u du. The limits transform from x in [1, infty) to u in [0, infty). The integral becomes int_0^infty u e^(-p u) e^u du = int_0^infty u e^(-(p - 1)u) du. This is a standard Laplace-type integral that converges at infinity if and only if the exponential decay factor is strictly positive, i.e., p - 1 > 0, which means p > 1. At p = 1, int_1^infty (ln x)/x dx = [ (ln x)^2 / 2 ]_1^infty = infty (diverges).

Question 16Advanced analysis, convergence and multivariable reasoning

What is the numerical value of the improper integral int_0^infty x e^(-3x) dx?

  1. 1/3
  2. 1/9 Correct answer.
  3. 1/6
  4. 2/9

Explanation

We evaluate by integration by parts: set u = x (so du = dx) and dv = e^(-3x) dx (so v = -1/3 e^(-3x)). Then int_0^infty x e^(-3x) dx = [-x/3 e^(-3x)]_0^infty - int_0^infty (-1/3 e^(-3x)) dx. By standard growth comparison, lim_{x -> infty} x e^(-3x) = 0, so the boundary term evaluates to 0 - 0 = 0. The remaining integral is 1/3 int_0^infty e^(-3x) dx = 1/3 [-1/3 e^(-3x)]_0^infty = 1/3 (0 - (-1/3)) = 1/9.

Question 17Advanced analysis, convergence and multivariable reasoning

For the 2pi-periodic function defined by f(x) = x on (-pi, pi), what is the Fourier coefficient b_1 in the Fourier series expansion f(x) = sum_{n=1}^infty b_n sin(nx)?

  1. b_1 = 1
  2. b_1 = pi
  3. b_1 = 0
  4. b_1 = 2 Correct answer.

Explanation

Since f(x) is an odd function, its cosine coefficients vanish (a_n = 0). The sine coefficients are given by b_n = (2/pi) int_0^pi x sin(nx) dx. Integrating by parts with u = x and dv = sin(nx) dx: b_n = (2/pi) [ -x cos(nx)/n ]_0^pi + (2/pi n) int_0^pi cos(nx) dx = (2/pi) [ -pi cos(n pi)/n ] + 0 = -2 (-1)^n / n = 2 (-1)^(n+1) / n. For n = 1: b_1 = 2 (-1)^2 / 1 = 2.

Question 18Advanced analysis, convergence and multivariable reasoning

Applying Parseval's identity to the Fourier series of f(x) = x on (-pi, pi) (where b_n = 2(-1)^(n+1)/n), what is the celebrated sum of the Basel problem sum_{n=1}^infty 1/n^2?

  1. pi^2 / 6 Correct answer.
  2. pi^2 / 8
  3. pi^2 / 12
  4. pi / 4

Explanation

Parseval's identity for an odd 2pi-periodic function states: (1/pi) int_{-pi}^pi |f(x)|^2 dx = sum_{n=1}^infty b_n^2. The left-hand side is (2/pi) int_0^pi x^2 dx = (2/pi) [x^3/3]_0^pi = (2/pi)(pi^3/3) = 2 pi^2 / 3. The right-hand side is sum_{n=1}^infty [2(-1)^(n+1)/n]^2 = sum_{n=1}^infty 4 / n^2 = 4 sum_{n=1}^infty 1/n^2. Equating both sides gives 4 sum_{n=1}^infty 1/n^2 = 2 pi^2 / 3, which simplifies to sum_{n=1}^infty 1/n^2 = (2 pi^2 / 3) / 4 = pi^2 / 6.

Question 19Advanced analysis, convergence and multivariable reasoning

For the multivariable function f(x, y) = x^3 + y^3 - 3xy, what is the classification of the critical point (1, 1)?

  1. Saddle point (point col)
  2. Strict local minimum Correct answer.
  3. Strict local maximum
  4. Non-isolated critical point

Explanation

We compute the partial derivatives: df/dx = 3x^2 - 3y and df/dy = 3y^2 - 3x. Setting both to zero yields y = x^2 and x = y^2 = x^4, giving critical points (0, 0) and (1, 1). The second derivatives are: r = d^2f/dx^2 = 6x, t = d^2f/dy^2 = 6y, and s = d^2f/dxdy = -3. At (1, 1): r = 6, t = 6, and s = -3. The Hessian determinant is Delta = rt - s^2 = (6)(6) - (-3)^2 = 36 - 9 = 27 > 0. Because Delta > 0 and r = 6 > 0, the Hessian matrix is positive definite, confirming that (1, 1) is a strict local minimum.

Question 20Advanced analysis, convergence and multivariable reasoning

What is the directional derivative of f(x, y) = x^2 + 3xy - y^2 at point P(1, 2) in the direction of vector u = (3, 4)?

  1. 20
  2. 8
  3. 4 Correct answer.
  4. -4

Explanation

The directional derivative along a vector u is given by D_u f(P) = grad(f)(P) . e, where e = u / ||u|| is the unit vector. First, ||u|| = sqrt(3^2 + 4^2) = 5, so e = (3/5, 4/5). Next, grad(f) = (2x + 3y, 3x - 2y). At P(1, 2): df/dx = 2(1) + 3(2) = 8, and df/dy = 3(1) - 2(2) = -1, so grad(f)(P) = (8, -1). Taking the dot product: D_u f = 8 * (3/5) + (-1) * (4/5) = 24/5 - 4/5 = 20/5 = 4.

Question 21Advanced analysis, convergence and multivariable reasoning

Given the unilateral Laplace transform L{f(t)} = F(s), what is the Laplace transform L{f''(t)} if f(0) = 2 and f'(0) = -3?

  1. s^2 F(s) - 2s - 3
  2. s^2 F(s) + 3s - 2
  3. s^2 F(s) - 3s + 2
  4. s^2 F(s) - 2s + 3 Correct answer.

Explanation

By the operational property for the Laplace transform of higher-order derivatives: L{f''(t)} = s^2 F(s) - s f(0) - f'(0). Substituting the given initial values f(0) = 2 and f'(0) = -3 yields: L{f''(t)} = s^2 F(s) - s(2) - (-3) = s^2 F(s) - 2s + 3.

Question 22Advanced analysis, convergence and multivariable reasoning

What is the inverse Laplace transform f(t) = L^(-1){ 6 / (s^2 + 4s + 13) } for t >= 0?

  1. f(t) = 2 e^(-2t) sin(3t) Correct answer.
  2. f(t) = 6 e^(-2t) cos(3t)
  3. f(t) = 2 e^(2t) sin(3t)
  4. f(t) = 3 e^(-2t) sin(2t)

Explanation

Completing the square in the denominator: s^2 + 4s + 13 = (s + 2)^2 + 9 = (s + 2)^2 + 3^2. Recall that L{e^(-a t) sin(omega t)} = omega / [(s + a)^2 + omega^2]. Here a = 2 and omega = 3. We rewrite the numerator as 6 = 2 * 3: F(s) = 2 * [3 / ((s + 2)^2 + 3^2)]. Taking the inverse transform immediately yields f(t) = 2 e^(-2t) sin(3t).

Question 23Inner products, spectral reasoning and matrix structure

What is the determinant of the 3x3 matrix A = [[2, 1, 3], [0, -1, 4], [1, 2, 0]]?

  1. -15
  2. 7
  3. 0
  4. -9 Correct answer.

Explanation

Expanding along the first row: det(A) = 2 * [(-1)(0) - (4)(2)] - 1 * [(0)(0) - (4)(1)] + 3 * [(0)(2) - (-1)(1)]. Evaluating each 2x2 minor: 2 * (-8) - 1 * (-4) + 3 * (1) = -16 + 4 + 3 = -9.

Question 24Inner products, spectral reasoning and matrix structure

What are the eigenvalues of the real matrix M = [[4, 2], [1, 3]]?

  1. lambda_1 = 2 and lambda_2 = 5 Correct answer.
  2. lambda_1 = 1 and lambda_2 = 6
  3. lambda_1 = 3 and lambda_2 = 4
  4. lambda_1 = -2 and lambda_2 = -5

Explanation

The characteristic polynomial is det(M - lambda I) = (4 - lambda)(3 - lambda) - (2)(1) = lambda^2 - 7 lambda + 12 - 2 = lambda^2 - 7 lambda + 10. Factoring gives (lambda - 2)(lambda - 5) = 0, yielding eigenvalues lambda_1 = 2 and lambda_2 = 5. Note that Tr(M) = 4 + 3 = 7 = 2 + 5 and det(M) = 10 = 2 * 5.

Question 25Inner products, spectral reasoning and matrix structure

For the matrix M = [[4, 2], [1, 3]], which vector is an eigenvector corresponding to the eigenvalue lambda = 5?

  1. v = (1, -1)^T
  2. v = (2, 1)^T Correct answer.
  3. v = (1, 2)^T
  4. v = (-1, 2)^T

Explanation

An eigenvector satisfies (M - 5I) v = 0. We set up [[4 - 5, 2], [1, 3 - 5]] [x, y]^T = [[-1, 2], [1, -2]] [x, y]^T = [0, 0]^T. Both rows give -x + 2y = 0, so x = 2y. Choosing y = 1 yields v = (2, 1)^T. Testing: M v = [[4, 2], [1, 3]] [2, 1]^T = [10, 5]^T = 5 [2, 1]^T.

Question 26Inner products, spectral reasoning and matrix structure

Let A = P D P^(-1) where D = diag(1, 2) and P = [[1, 1], [0, 1]]. What is the matrix power A^4?

  1. [[1, 16], [0, 16]]
  2. [[1, 8], [0, 16]]
  3. [[1, 15], [0, 16]] Correct answer.
  4. [[4, 30], [0, 32]]

Explanation

Since A = P D P^(-1), we have A^4 = P D^4 P^(-1). Here D^4 = diag(1^4, 2^4) = diag(1, 16). The inverse of P is P^(-1) = [[1, -1], [0, 1]]. Computing the product: P D^4 = [[1, 1], [0, 1]] [[1, 0], [0, 16]] = [[1, 16], [0, 16]]. Then A^4 = [[1, 16], [0, 16]] [[1, -1], [0, 1]] = [[1*1 + 16*0, 1*(-1) + 16*1], [0*1 + 16*0, 0*(-1) + 16*1]] = [[1, 15], [0, 16]].

Question 27Inner products, spectral reasoning and matrix structure

Let A be a 4x4 real matrix such that rank(A) = 3. What are the values of det(A) and det(2A)?

  1. det(A) = 3 and det(2A) = 48
  2. det(A) = 0 and det(2A) = 8
  3. det(A) = 1 and det(2A) = 16
  4. det(A) = 0 and det(2A) = 0 Correct answer.

Explanation

A square n x n matrix is invertible if and only if rank(A) = n. For a 4x4 matrix with rank(A) = 3 < 4, the columns are linearly dependent and A is not invertible, which implies det(A) = 0. Furthermore, by the scalar multiplication property of determinants for an n x n matrix, det(c A) = c^n det(A). Here det(2A) = 2^4 det(A) = 16 * 0 = 0.

Question 28Inner products, spectral reasoning and matrix structure

In R^3 with the standard Euclidean inner product, apply the Gram-Schmidt process to vectors u_1 = (1, 1, 0) and u_2 = (1, 0, 1). What is the orthogonal vector v_2 before normalization?

  1. v_2 = (0, -1, 1)
  2. v_2 = (1, -1, 1)
  3. v_2 = (1/2, -1/2, 1) Correct answer.
  4. v_2 = (-1/2, 1/2, 1)

Explanation

In the Gram-Schmidt process, we set v_1 = u_1 = (1, 1, 0). The second orthogonal vector is v_2 = u_2 - proj_{v_1}(u_2) = u_2 - [(u_2 . v_1) / ||v_1||^2] v_1. The dot product is u_2 . v_1 = 1*1 + 0*1 + 1*0 = 1, and ||v_1||^2 = 1^2 + 1^2 + 0^2 = 2. Thus proj_{v_1}(u_2) = (1/2) (1, 1, 0) = (1/2, 1/2, 0). Subtracting: v_2 = (1, 0, 1) - (1/2, 1/2, 0) = (1/2, -1/2, 1). We can verify orthogonality: v_2 . v_1 = (1/2)(1) + (-1/2)(1) + (1)(0) = 0.

Question 29Inner products, spectral reasoning and matrix structure

Using the Cayley-Hamilton theorem for the matrix A = [[1, 2], [3, 4]], how can the inverse matrix A^(-1) be expressed as a linear combination of A and the identity matrix I?

  1. A^(-1) = A - 5 I
  2. A^(-1) = (1/2)(5 I + A)
  3. A^(-1) = (1/2)(A - 5 I) Correct answer.
  4. A^(-1) = -(1/2)(A - 4 I)

Explanation

The characteristic polynomial of A is chi(lambda) = lambda^2 - Tr(A) lambda + det(A). Here Tr(A) = 1 + 4 = 5 and det(A) = 1*4 - 2*3 = -2, so chi(lambda) = lambda^2 - 5 lambda - 2. By the Cayley-Hamilton theorem, A satisfies its own characteristic equation: A^2 - 5 A - 2 I = 0. Rearranging: A^2 - 5 A = 2 I, or A (A - 5 I) = 2 I. Multiplying on the left by A^(-1) gives A - 5 I = 2 A^(-1), so A^(-1) = (1/2)(A - 5 I).

Question 30Inner products, spectral reasoning and matrix structure

Let f: R^5 -> R^3 be a linear map represented by a 3x5 real matrix of rank 3. According to the rank-nullity theorem (théorème du rang), what is the dimension of the kernel, dim(ker(f))?

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

Explanation

The rank-nullity theorem states that for any linear map f defined on a finite-dimensional vector space E, dim(E) = dim(ker(f)) + rank(f). Here E = R^5, so dim(E) = 5. The rank is given as rank(f) = 3. Therefore, dim(ker(f)) = dim(E) - rank(f) = 5 - 3 = 2.

Question 31Advanced probability, moments and conditional models

A workshop has two machines. Machine A produces 60% of the parts and 3% of its output is defective; machine B produces the remaining 40% and 5% of its output is defective. A part drawn at random is found to be defective. What is the probability that it came from machine A?

  1. 9/19 ≈ 0.474 Correct answer.
  2. 3/100 = 0.030
  3. 3/5 = 0.600
  4. 10/19 ≈ 0.526

Explanation

Total probability of a defective part: $P(D) = P(A)P(D|A) + P(B)P(D|B) = 0.60 \times 0.03 + 0.40 \times 0.05 = 0.018 + 0.020 = 0.038$. Bayes' rule then gives $P(A|D) = \frac{P(A)P(D|A)}{P(D)} = \frac{0.018}{0.038} = \frac{18}{38} = \frac{9}{19} \approx 0.474$.

Question 32Advanced probability, moments and conditional models

Let X follow a geometric distribution with parameter p in ]0, 1[ on N* (P(X = k) = p (1 - p)^(k - 1) for k >= 1). What are the expectation E(X) and the memoryless property of X?

  1. E(X) = 1/p^2 and P(X > s + t | X > s) = P(X > t)
  2. E(X) = (1 - p)/p and P(X = s + t | X > s) = P(X = t)
  3. E(X) = 1/p and P(X > s + t | X > s) = P(X > t) for all integers s, t >= 1 Correct answer.
  4. E(X) = p and P(X > s + t) = P(X > s) P(X > t)

Explanation

For a geometric distribution supported on N*, the expectation is E(X) = 1/p. The memoryless property states that P(X > s + t | X > s) = P(X > s + t) / P(X > s) = (1 - p)^(s + t) / (1 - p)^s = (1 - p)^t = P(X > t) for all integers s, t in N*.

Question 33Advanced probability, moments and conditional models

Let X follow a Poisson distribution with parameter lambda = 3. What is the conditional probability P(X = 2 | X >= 1)?

  1. (9 / 2) e^(-3)
  2. (9 / 2) / (1 - e^(-3))
  3. (9 / 2) e^(-3) / (1 - 4 e^(-3))
  4. (9 / 2) e^(-3) / (1 - e^(-3)) Correct answer.

Explanation

By Bayes' formula, P(X = 2 | X >= 1) = P(X = 2 and X >= 1) / P(X >= 1) = P(X = 2) / (1 - P(X = 0)). For a Poisson variable with parameter 3, P(X = 2) = (3^2 / 2!) e^(-3) = (9/2) e^(-3) and P(X = 0) = e^(-3). Thus P(X = 2 | X >= 1) = (9/2) e^(-3) / (1 - e^(-3)).

Question 34Advanced probability, moments and conditional models

What is the probability generating function (fonction génératrice) G_X(t) = E(t^X) of a random variable X following the Binomial distribution B(n, p)?

  1. G_X(t) = (1 - p + p t)^n Correct answer.
  2. G_X(t) = (p + (1 - p) t)^n
  3. G_X(t) = e^(n p (t - 1))
  4. G_X(t) = (p t)^n

Explanation

By definition, G_X(t) = sum_{k=0}^n (n choose k) p^k (1 - p)^(n - k) t^k = sum_{k=0}^n (n choose k) (p t)^k (1 - p)^(n - k). Applying the binomial theorem directly yields ((1 - p) + p t)^n.

Question 35Advanced probability, moments and conditional models

A random variable X has expectation E(X) = 20 and variance V(X) = 4. Using the Bienaymé-Chebyshev inequality, what is the best guaranteed upper bound on P(|X - 20| >= 6)?

  1. 1/4
  2. 1/9 Correct answer.
  3. 1/3
  4. 2/3

Explanation

The Bienaymé-Chebyshev inequality states P(|X - E(X)| >= a) <= V(X) / a^2 for any a > 0. Setting E(X) = 20, V(X) = 4, and a = 6 gives P(|X - 20| >= 6) <= 4 / 6^2 = 4 / 36 = 1/9.

Question 36Analytical mechanics, oscillation and model reduction

A point projectile is launched from flat ground with an initial speed v_0 = 20 m/s at an angle alpha = 30° above the horizontal. Taking g = 10 m/s^2 and neglecting air resistance, what is the maximum height H reached by the projectile above ground level?

  1. 15.0 m
  2. 10.0 m
  3. 5.0 m Correct answer.
  4. 2.5 m

Explanation

The vertical component of the initial velocity is v_{0y} = v_0 sin(alpha) = 20 * sin(30°) = 20 * 0.5 = 10 m/s. At maximum height, the vertical velocity is zero: v_y(t) = v_{0y} - g t = 0, so t_{peak} = v_{0y} / g = 10 / 10 = 1.0 s. The maximum height is H = v_{0y} t_{peak} - (1/2) g t_{peak}^2 = 10(1.0) - 0.5(10)(1.0)^2 = 10 - 5 = 5.0 m. Equivalently, by Torricelli / energy conservation: H = v_{0y}^2 / (2g) = 100 / 20 = 5.0 m.

Question 37Analytical mechanics, oscillation and model reduction

A uniform solid cylinder of mass M = 4.0 kg and radius R = 0.50 m rotates about its fixed central axis (moment of inertia J = 1/2 M R^2). A constant tangential braking force F = 10 N is applied at its rim. According to the angular momentum theorem (théorème du moment cinétique), what is the magnitude of the cylinder's angular deceleration alpha?

  1. 2.5 rad/s^2
  2. 5.0 rad/s^2
  3. 20 rad/s^2
  4. 10 rad/s^2 Correct answer.

Explanation

First calculate the moment of inertia: J = (1/2) M R^2 = 0.5 * 4.0 * (0.50)^2 = 2.0 * 0.25 = 0.50 kg.m^2. The braking torque exerted by the tangential force is Gamma = F * R = 10 * 0.50 = 5.0 N.m. By the theorem of angular momentum for rotation about a fixed axis, J * alpha = Gamma, so alpha = Gamma / J = 5.0 / 0.50 = 10 rad/s^2.

Question 38Analytical mechanics, oscillation and model reduction

A damped harmonic oscillator consists of a mass m = 0.50 kg attached to a spring of stiffness k = 50 N/m and a viscous damper with damping coefficient lambda = 2.0 N.s/m. What is the pseudo-angular frequency omega_d of the resulting underdamped oscillations?

  1. omega_d = sqrt(96) rad/s (approx 9.80 rad/s) Correct answer.
  2. omega_d = 10 rad/s
  3. omega_d = sqrt(92) rad/s (approx 9.59 rad/s)
  4. omega_d = sqrt(104) rad/s (approx 10.2 rad/s)

Explanation

The differential equation of motion is m x'' + lambda x' + k x = 0, which can be written in canonical form x'' + 2 gamma x' + omega_0^2 x = 0. Here omega_0^2 = k / m = 50 / 0.50 = 100 rad^2/s^2 (so omega_0 = 10 rad/s), and 2 gamma = lambda / m = 2.0 / 0.50 = 4.0 s^(-1), which means gamma = 2.0 s^(-1). Because gamma < omega_0 (2 < 10), the motion is in the pseudoperiodic regime with pseudo-angular frequency omega_d = sqrt(omega_0^2 - gamma^2) = sqrt(100 - 2^2) = sqrt(100 - 4) = sqrt(96) approx 9.80 rad/s.

Question 39Analytical mechanics, oscillation and model reduction

A Carnot heat engine operates between a hot reservoir at T_h = 600 K and a cold reservoir at T_c = 300 K. If the engine absorbs Q_h = 2000 J of heat from the hot source during each cycle, what is the maximum mechanical work W_out delivered per cycle?

  1. W_out = 500 J
  2. W_out = 1500 J
  3. W_out = 2000 J
  4. W_out = 1000 J Correct answer.

Explanation

The maximum theoretical efficiency of any heat engine operating between two thermal reservoirs is given by the Carnot efficiency: eta_C = 1 - T_c / T_h = 1 - 300 / 600 = 0.50 (or 50%). By definition, efficiency is the ratio of net mechanical work delivered to the heat absorbed: eta_C = W_out / Q_h. Therefore, the work produced is W_out = eta_C * Q_h = 0.50 * 2000 J = 1000 J.

Question 40Analytical mechanics, oscillation and model reduction

A planar discrete mechanical system consists of three point masses located in the Cartesian xy-plane: m_1 = 1.0 kg at (0, 0), m_2 = 2.0 kg at (3.0 m, 0), and m_3 = 3.0 kg at (0, 4.0 m). What are the coordinates (x_G, y_G) of the center of mass G?

  1. (1.5 m, 2.0 m)
  2. (2.0 m, 1.0 m)
  3. (1.0 m, 1.33 m)
  4. (1.0 m, 2.0 m) Correct answer.

Explanation

The total mass of the system is M = m_1 + m_2 + m_3 = 1.0 + 2.0 + 3.0 = 6.0 kg. The x-coordinate of G is x_G = (m_1 x_1 + m_2 x_2 + m_3 x_3) / M = (1.0 * 0 + 2.0 * 3.0 + 3.0 * 0) / 6.0 = 6.0 / 6.0 = 1.0 m. The y-coordinate of G is y_G = (m_1 y_1 + m_2 y_2 + m_3 y_3) / M = (1.0 * 0 + 2.0 * 0 + 3.0 * 4.0) / 6.0 = 12.0 / 6.0 = 2.0 m. Thus G = (1.0 m, 2.0 m).

Question 41Analytical mechanics, oscillation and model reduction

A cart of mass m_1 = 3.0 kg moving along a horizontal rail at speed v_1 = 4.0 m/s collides with and latches onto a stationary cart of mass m_2 = 1.0 kg (completely inelastic collision). How much mechanical energy Delta E_k is dissipated into deformation and heat during the collision?

  1. Delta E_k = 6.0 J Correct answer.
  2. Delta E_k = 18.0 J
  3. Delta E_k = 24.0 J
  4. Delta E_k = 0 J

Explanation

Because no external horizontal forces act on the system, total linear momentum is conserved: m_1 v_1 = (m_1 + m_2) v_f. Substituting values: 3.0 * 4.0 = (3.0 + 1.0) v_f => 12.0 = 4.0 v_f => v_f = 3.0 m/s. The initial kinetic energy is E_{k,i} = (1/2) m_1 v_1^2 = 0.5 * 3.0 * (4.0)^2 = 0.5 * 3.0 * 16.0 = 24.0 J. The final kinetic energy is E_{k,f} = (1/2)(m_1 + m_2) v_f^2 = 0.5 * 4.0 * (3.0)^2 = 0.5 * 4.0 * 9.0 = 18.0 J. The dissipated mechanical energy is Delta E_k = E_{k,i} - E_{k,f} = 24.0 - 18.0 = 6.0 J.

Question 42Analytical mechanics, oscillation and model reduction

In the Northern Hemisphere, a projectile moves horizontally due North with relative velocity v_r. Due to the Earth's rotation vector Omega (which points upwards out of the ground at latitude lambda > 0), what is the direction of the fictitious Coriolis inertial force F_cor = -2 m (Omega x v_r)?

  1. Towards the East (deflecting to the right of the trajectory) Correct answer.
  2. Towards the West (deflecting to the left of the trajectory)
  3. Vertically upwards towards the sky
  4. Vertically downwards towards the ground

Explanation

The rotation vector Omega has a positive vertical component Omega_z = Omega sin(lambda) pointing vertically upward. The relative velocity v_r points North. Applying the right-hand rule to the cross product: (Upwards) x (North) points West. Because the Coriolis force formula contains a negative sign, F_cor = -2 m (Omega x v_r) = -2 m (West) = +2 m (East). Therefore, the Coriolis force deflects the horizontal motion toward the East (to the right of the direction of motion in the Northern Hemisphere).

Question 43Analytical mechanics, oscillation and model reduction

A homogeneous wooden cube of edge length a = 0.20 m and density rho_{wood} = 600 kg/m^3 floats in calm freshwater of density rho_{water} = 1000 kg/m^3. What is the submerged draft (submerged depth) h_{sub} of the cube?

  1. h_{sub} = 0.08 m (8 cm)
  2. h_{sub} = 0.10 m (10 cm)
  3. h_{sub} = 0.16 m (16 cm)
  4. h_{sub} = 0.12 m (12 cm) Correct answer.

Explanation

According to Archimedes' principle, a floating body in static equilibrium experiences an upward buoyant force equal to its weight: F_b = P. Here P = rho_{wood} V_{total} g = rho_{wood} a^3 g, and F_b = rho_{water} V_{sub} g = rho_{water} a^2 h_{sub} g. Equating both expressions: rho_{water} a^2 h_{sub} = rho_{wood} a^3, which simplifies to h_{sub} = a * (rho_{wood} / rho_{water}). Substituting the numerical values: h_{sub} = 0.20 m * (600 / 1000) = 0.20 * 0.60 = 0.12 m = 12 cm.

Question 44Electromagnetic fields, thermal physics and wave models

A block of mass m = 2.0 kg slides from rest down a plane inclined at angle theta = 30° over a distance d = 5.0 m. The coefficient of kinetic friction between the block and the plane is mu = 0.20. Taking g = 10 m/s^2 and cos(30°) approx 0.866, what is the speed v of the block at the bottom of the incline?

  1. v = 7.07 m/s
  2. v approx 5.72 m/s Correct answer.
  3. v approx 4.25 m/s
  4. v = 10.0 m/s

Explanation

By the work-energy theorem (théorème de l'énergie cinétique): Delta E_k = W(P) + W(R_N) + W(f). The normal reaction does no work (W(R_N) = 0). The work done by gravity is W(P) = m g d sin(30°) = 2.0 * 10 * 5.0 * 0.50 = 50 J. The normal force is R_N = m g cos(30°) = 2.0 * 10 * 0.866 = 17.32 N. The friction force is f = mu R_N = 0.20 * 17.32 = 3.464 N, so its work is W(f) = -f * d = -3.464 * 5.0 = -17.32 J. The net work is W_net = 50 - 17.32 = 32.68 J. Equating to (1/2) m v^2: (1/2)(2.0) v^2 = v^2 = 32.68, so v = sqrt(32.68) approx 5.72 m/s.

Question 45Electromagnetic fields, thermal physics and wave models

Two identical blocks of liquid water, each of mass m = 1.0 kg and specific heat capacity c = 4184 J/(kg.K), at initial temperatures T_1 = 300 K and T_2 = 360 K, are placed in thermal contact inside an isolated calorimeter. What is the total entropy change Delta S of the isolated system upon reaching thermal equilibrium?

  1. Delta S approx +34.7 J/K Correct answer.
  2. Delta S = 0 J/K
  3. Delta S approx -34.7 J/K
  4. Delta S approx +145 J/K

Explanation

Because heat capacity is constant and the system is isolated, energy conservation yields the equilibrium temperature T_f = (T_1 + T_2) / 2 = (300 + 360) / 2 = 330 K. The total entropy variation is Delta S = Delta S_1 + Delta S_2 = m c ln(T_f / T_1) + m c ln(T_f / T_2) = m c ln(T_f^2 / (T_1 T_2)). Evaluating the ratio: T_f^2 / (T_1 T_2) = 330^2 / (300 * 360) = 108,900 / 108,000 = 1.00833. Using ln(1 + x) approx x: ln(1.00833) approx 0.00830. Thus Delta S = 1.0 * 4184 * 0.00830 approx +34.7 J/K. Because Delta S > 0, the spontaneous heat exchange is irreversible.

Question 46Electromagnetic fields, thermal physics and wave models

A series RLC circuit has resistance R = 100 Omega, inductance L = 0.25 H, and capacitance C = 1.0 muF (1.0 * 10^(-6) F). What is the quality factor Q of this resonant circuit?

  1. Q = 2.5
  2. Q = 5.0 Correct answer.
  3. Q = 10.0
  4. Q = 50.0

Explanation

For a series RLC circuit, the resonance angular frequency is omega_0 = 1 / sqrt(L C) = 1 / sqrt(0.25 * 10^(-6)) = 1 / (0.5 * 10^(-3)) = 2000 rad/s. The quality factor Q is given by Q = L omega_0 / R = (1 / R) sqrt(L / C). Substituting the values: Q = (0.25 * 2000) / 100 = 500 / 100 = 5.0. Alternatively, sqrt(L / C) = sqrt(0.25 / 10^(-6)) = sqrt(250,000) = 500 Omega, so Q = 500 / 100 = 5.0.

Question 47Electromagnetic fields, thermal physics and wave models

An air-core solenoid of length l = 0.50 m, cross-sectional area S = 10 cm^2 = 1.0 * 10^(-3) m^2, and N = 500 turns carries a steady current I = 2.0 A. Taking mu_0 = 4 pi * 10^(-7) H/m, what is the magnetic energy E_mag stored in the solenoid?

  1. E_mag approx 0.31 mJ
  2. E_mag approx 2.51 mJ
  3. E_mag approx 1.26 mJ Correct answer.
  4. E_mag approx 5.03 mJ

Explanation

First find the inductance: L = mu_0 N^2 S / l = (4 pi * 10^(-7) * 500^2 * 1.0 * 10^(-3)) / 0.50 = (4 pi * 10^(-7) * 250,000 * 10^(-3)) / 0.50 = (4 pi * 10^(-7) * 250) / 0.50 = (1000 pi * 10^(-7)) / 0.50 = 2 pi * 10^(-4) H approx 0.628 mH. The magnetic energy stored in an inductor is E_mag = (1/2) L I^2 = 0.5 * (2 pi * 10^(-4)) * (2.0)^2 = 4 pi * 10^(-4) J approx 1.257 * 10^(-3) J = 1.26 mJ.

Question 48Electromagnetic fields, thermal physics and wave models

A thin converging glass lens has a focal length f' = +20 cm. An object of height h = 2.0 cm is placed 30 cm in front of the optical center (algebraic position p = bar{OA} = -30 cm). What are the algebraic image position p' = bar{OA'} and the transverse magnification gamma = bar{A'B'} / bar{AB}?

  1. p' = +12 cm and gamma = -0.40
  2. p' = +30 cm and gamma = -1.0
  3. p' = -60 cm and gamma = +2.0
  4. p' = +60 cm and gamma = -2.0 Correct answer.

Explanation

Applying Descartes' thin lens conjugacy formula with origin at the optical center O: 1 / p' - 1 / p = 1 / f'. Substituting p = -30 cm and f' = +20 cm gives: 1 / p' - 1 / (-30) = 1 / 20, so 1 / p' = 1 / 20 - 1 / 30 = (3 - 2) / 60 = 1 / 60 cm^(-1). Therefore p' = +60 cm (a real image located 60 cm behind the lens). The transverse magnification is gamma = p' / p = (+60) / (-30) = -2.0 (an inverted image twice the height of the object, h' = -4.0 cm).

Question 49Electromagnetic fields, thermal physics and wave models

In a Young's double-slit experiment, two narrow parallel slits separated by a center-to-center distance a = 0.50 mm are illuminated by a monochromatic laser beam of wavelength lambda = 600 nm. The interference pattern is observed on a screen placed at a distance D = 2.0 m. What is the fringe spacing (interfrange) i?

  1. i = 2.4 mm Correct answer.
  2. i = 1.2 mm
  3. i = 4.8 mm
  4. i = 0.60 mm

Explanation

The optical path difference between the two coherent beams at position x on the screen is delta = a x / D. Constructive interference occurs when delta = p lambda (where p is an integer). The fringe spacing i is the distance between consecutive bright fringes: i = lambda D / a. Converting all units to meters: lambda = 600 * 10^(-9) m, D = 2.0 m, and a = 0.50 * 10^(-3) m. Thus i = (600 * 10^(-9) * 2.0) / (0.50 * 10^(-3)) = (1.20 * 10^(-6)) / (0.50 * 10^(-3)) = 2.40 * 10^(-3) m = 2.4 mm.

Question 50Electromagnetic fields, thermal physics and wave models

A scuba diver descends to a depth of h = 40 m in seawater of density rho = 1025 kg/m^3. Taking atmospheric surface pressure P_atm = 1.013 * 10^5 Pa (approx 1.01 bar) and g = 9.81 m/s^2, what is the absolute hydrostatic pressure P experienced by the diver?

  1. P approx 4.02 bar
  2. P approx 5.03 bar Correct answer.
  3. P approx 6.05 bar
  4. P approx 3.01 bar

Explanation

According to the fundamental equation of hydrostatics for an incompressible fluid: P = P_atm + rho g h. The gauge pressure contributed by the water column is Delta P = rho g h = 1025 kg/m^3 * 9.81 m/s^2 * 40 m = 402,210 Pa = 4.022 bar. Adding the atmospheric surface pressure: P = 1.013 bar + 4.022 bar = 5.035 bar approx 5.03 bar.