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Ingénieur ITPE - independent mathematics, physics and infrastructure foundations study materials
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Ingénieur ITPE - independent mathematics, physics and infrastructure foundations practice questions

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Question 1Inner products, spectral reasoning and matrix structure

In the PCSI linear algebra curriculum, consider the 3x3 real matrix A = [[1, 2, 1], [0, 3, 1], [2, 1, 4]]. What is the determinant of A?

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

Explanation

Expanding along the first column or using Sarrus' rule: det(A) = 1*(3*4 - 1*1) - 0 + 2*(2*1 - 1*3) = 1*(12 - 1) + 2*(2 - 3) = 11 - 2 = 9. Determinant calculation is a foundational skill in the ITPE concours interne PCSI mathematics paper. Because det(A) != 0, the matrix A is invertible.

Question 2Inner products, spectral reasoning and matrix structure

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

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

Explanation

The characteristic polynomial of A is P(lambda) = det(A - 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. Since A has distinct eigenvalues, it is diagonalizable over R.

Question 3Inner products, spectral reasoning and matrix structure

Let E = R_3[X] be the vector space of real polynomials of degree at most 3 (dim E = 4). Consider the linear endomorphism u: E -> E defined by u(P) = P'' - 2*P'. What is the rank of u?

  1. 3 Correct answer.
  2. 2
  3. 4
  4. 1

Explanation

To determine dim(ker(u)), solve P'' - 2*P' = 0. The only polynomial solutions are constant polynomials P(X) = C, since any polynomial of degree d >= 1 would have deg(P') = d - 1 and deg(P'') = d - 2, meaning -2*P' cannot be cancelled by P''. Thus ker(u) = R_0[X] and dim(ker(u)) = 1. By the rank-nullity theorem (theoreme du rang), dim(E) = dim(ker(u)) + rg(u), so 4 = 1 + rg(u), which yields rg(u) = 3.

Question 4Inner products, spectral reasoning and matrix structure

Let A = [[1, 1], [0, 1]] in M_2(R). What is the matrix power A^n for any positive integer n?

  1. [[1, n], [0, 1]] Correct answer.
  2. [[1, 2^n - 1], [0, 1]]
  3. [[1, n^2], [0, 1]]
  4. [[n, n], [0, n]]

Explanation

We can decompose A as A = I_2 + N, where N = [[0, 1], [0, 0]]. Since N^2 = 0 (N is nilpotent of index 2) and I_2 commutes with N, the binomial theorem gives A^n = (I_2 + N)^n = I_2 + n*N = [[1, n], [0, 1]]. This can also be verified directly by induction.

Question 5Inner products, spectral reasoning and matrix structure

Let f: M_2(R) -> M_2(R) be the linear map defined by f(M) = M - M^T, where M^T is the transpose of M. What is the rank (dimension of the image) of f?

  1. 1 Correct answer.
  2. 3
  3. 2
  4. 4

Explanation

A matrix M is in ker(f) if and only if M = M^T, which is the subspace S_2(R) of symmetric 2x2 matrices. The dimension of S_2(R) is n*(n + 1) / 2 = 2*3 / 2 = 3. Since dim(M_2(R)) = 4, the rank-nullity theorem gives rg(f) = dim(M_2(R)) - dim(ker(f)) = 4 - 3 = 1. The image consists of skew-symmetric matrices A_2(R), which have dimension n*(n - 1) / 2 = 1.

Question 6Inner products, spectral reasoning and matrix structure

For which real values of parameter a is the 3x3 matrix M = [[1, a, 0], [0, 1, a], [a, 0, 1]] invertible?

  1. a != -1 Correct answer.
  2. a != 1
  3. a != 0
  4. a not in {-1, 1}

Explanation

Compute the determinant of M by expansion along the first column: det(M) = 1*(1*1 - a*0) + a*(a*a - 1*0) = 1 + a^3. A matrix is invertible if and only if its determinant is non-zero. In the real field R, 1 + a^3 = 0 has the single real solution a = -1. Therefore, M is invertible if and only if a != -1.

Question 7Inner products, spectral reasoning and matrix structure

In the complex plane C, what are the three roots of the equation z^3 = -8*i?

  1. 2*i, -sqrt(3) - i, and sqrt(3) - i Correct answer.
  2. -2*i, sqrt(3) + i, and -sqrt(3) + i
  3. 2, -1 - i*sqrt(3), and -1 + i*sqrt(3)
  4. 2*exp(i*pi/3), 2*exp(i*pi), and 2*exp(i*5*pi/3)

Explanation

Write -8*i in polar form: -8*i = 8*exp(-i*pi/2) = 8*exp(i*3*pi/2). The cube roots are z_k = 2*exp(i*(3*pi/2 + 2*k*pi)/3) = 2*exp(i*(pi/2 + 2*k*pi/3)) for k in {0, 1, 2}. For k = 0: z_0 = 2*exp(i*pi/2) = 2*i. For k = 1: z_1 = 2*exp(i*7*pi/6) = 2*(-sqrt(3)/2 - i/2) = -sqrt(3) - i. For k = 2: z_2 = 2*exp(i*11*pi/6) = 2*(sqrt(3)/2 - i/2) = sqrt(3) - i.

Question 8Inner products, spectral reasoning and matrix structure

In Euclidean space R^3 equipped with the canonical dot product, what is the distance from the point M(3, 4, 1) to the plane P defined by x + 2*y + 2*z = 0?

  1. 13/3 Correct answer.
  2. 11/3
  3. 5
  4. 4

Explanation

The distance from a point M(x_0, y_0, z_0) to a plane ax + by + cz + d = 0 is given by d(M, P) = |a*x_0 + b*y_0 + c*z_0 + d| / sqrt(a^2 + b^2 + c^2). Here, the normal vector is n = (1, 2, 2) with norm ||n|| = sqrt(1^2 + 2^2 + 2^2) = sqrt(1 + 4 + 4) = 3. The numerator is |1*3 + 2*4 + 2*1 + 0| = |3 + 8 + 2| = 13. Thus d(M, P) = 13/3.

Question 9Inner products, spectral reasoning and matrix structure

Let M be the block upper-triangular matrix M = [[A, B], [0, C]], where A = [[2, 1], [0, 3]], C = [[4, 5], [1, 2]], and B is any 2x2 real matrix. What is det(M)?

  1. 18 Correct answer.
  2. 36
  3. 12
  4. 24

Explanation

For a block triangular matrix with square diagonal blocks, the determinant is the product of the determinants of the diagonal blocks: det(M) = det(A)*det(C), regardless of block B. Here, det(A) = 2*3 - 1*0 = 6, and det(C) = 4*2 - 5*1 = 8 - 5 = 3. Therefore, det(M) = 6*3 = 18.

Question 10Inner products, spectral reasoning and matrix structure

Let A = [[3, 1], [2, 4]] in M_2(R). Using the Cayley-Hamilton theorem, what is the inverse matrix A^(-1) expressed as a linear combination of the identity I_2 and A?

  1. (7/10)*I_2 - (1/10)*A Correct answer.
  2. (1/10)*A - (7/10)*I_2
  3. (4/10)*I_2 - (3/10)*A
  4. (1/10)*I_2 + (7/10)*A

Explanation

The characteristic polynomial of A is chi_A(X) = X^2 - Tr(A)*X + det(A). Here Tr(A) = 3 + 4 = 7 and det(A) = 3*4 - 1*2 = 10, so chi_A(X) = X^2 - 7*X + 10. By the Cayley-Hamilton theorem, A^2 - 7*A + 10*I_2 = 0. Rearranging gives 10*I_2 = 7*A - A^2 = A*(7*I_2 - A). Multiplying by A^(-1) and dividing by 10 yields A^(-1) = (7/10)*I_2 - (1/10)*A.

Question 11Inner products, spectral reasoning and matrix structure

Consider the 2D linear first-order differential system X'(t) = A*X(t) where A = [[0, 1], [-4, 0]]. What is the nature of the equilibrium point (0, 0)?

  1. A stable center (centre) with periodic sinusoidal trajectories Correct answer.
  2. An unstable saddle point (col)
  3. An asymptotically stable sink (noeud attractif)
  4. An unstable spiral source (foyer repulsif)

Explanation

The characteristic polynomial of A is det(A - lambda*I) = lambda^2 + 4 = 0, giving pure imaginary eigenvalues lambda = +/- 2*i. In linear ODE phase plane analysis, purely imaginary eigenvalues with zero real part correspond to a center (centre). The trajectories in phase space are concentric ellipses traversed periodically with period T = 2*pi / 2 = pi.

Question 12Inner products, spectral reasoning and matrix structure

In R^3, what is the orthogonal projection matrix P onto the line directed by the unit vector u = (1/3)*(1, 2, 2)^T?

  1. (1/9)*[[1, 2, 2], [2, 4, 4], [2, 4, 4]] Correct answer.
  2. (1/3)*[[1, 2, 2], [2, 4, 4], [2, 4, 4]]
  3. [[1, 0, 0], [0, 2, 0], [0, 0, 2]]
  4. (1/9)*[[1, 0, 0], [0, 4, 0], [0, 0, 4]]

Explanation

The orthogonal projection matrix onto a line directed by a unit vector u is given by the outer product P = u * u^T. Here u = (1/3)*(1, 2, 2)^T, so P = (1/9)*[[1*1, 1*2, 1*2], [2*1, 2*2, 2*2], [2*1, 2*2, 2*2]] = (1/9)*[[1, 2, 2], [2, 4, 4], [2, 4, 4]]. Notice that P is symmetric, Tr(P) = (1 + 4 + 4)/9 = 1 = rg(P), and P^2 = P.

Question 13Advanced analysis, convergence and multivariable reasoning

Solve the first-order linear differential equation y' + (2*x / (1 + x^2))*y = 1 with initial condition y(0) = 0. What is y(1)?

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

Explanation

The integrating factor is exp(integral(2*x / (1 + x^2) dx)) = exp(ln(1 + x^2)) = 1 + x^2. Multiplying the ODE by this factor yields ((1 + x^2)*y)' = 1 + x^2. Integrating both sides gives (1 + x^2)*y = x + x^3 / 3 + C. Using y(0) = 0 gives C = 0, so y(x) = (x + x^3 / 3) / (1 + x^2). Evaluating at x = 1: y(1) = (1 + 1/3) / (1 + 1) = (4/3) / 2 = 2/3.

Question 14Advanced analysis, convergence and multivariable reasoning

Consider the second-order Cauchy problem y'' + 4*y' + 4*y = 0 with initial conditions y(0) = 1 and y'(0) = 1. What is the solution y(x)?

  1. (1 + 3*x)*exp(-2*x) Correct answer.
  2. (1 + x)*exp(-2*x)
  3. exp(-2*x) + 3*exp(2*x)
  4. cos(2*x) + sin(2*x)

Explanation

The characteristic equation is r^2 + 4*r + 4 = (r + 2)^2 = 0, which has a double real root r = -2. The general solution is y(x) = (A + B*x)*exp(-2*x). Applying y(0) = 1 gives A = 1. The derivative is y'(x) = B*exp(-2*x) - 2*(A + B*x)*exp(-2*x). Evaluating at x = 0 gives y'(0) = B - 2*A = B - 2 = 1, so B = 3. Hence y(x) = (1 + 3*x)*exp(-2*x).

Question 15Advanced analysis, convergence and multivariable reasoning

What is the Taylor expansion (developpement limite) of f(x) = ln(1 + x)*cos(x) around x = 0 up to order 3?

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

Explanation

At x = 0, ln(1 + x) = x - x^2 / 2 + x^3 / 3 + o(x^3) and cos(x) = 1 - x^2 / 2 + o(x^3). Multiplying the two expansions: f(x) = (x - x^2 / 2 + x^3 / 3)*(1 - x^2 / 2) + o(x^3) = x - x^3 / 2 - x^2 / 2 + x^3 / 3 + o(x^3) = x - x^2 / 2 + (-1/2 + 1/3)*x^3 + o(x^3) = x - x^2 / 2 - x^3 / 6 + o(x^3).

Question 16Advanced analysis, convergence and multivariable reasoning

What is the exact sum of the convergent numerical series sum_{n=1}^infty (n / 2^n)?

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

Explanation

For |x| < 1, the geometric series sum_{n=0}^infty x^n = 1 / (1 - x). Differentiating both sides gives sum_{n=1}^infty n*x^(n-1) = 1 / (1 - x)^2. Multiplying by x yields sum_{n=1}^infty n*x^n = x / (1 - x)^2. Substituting x = 1/2 gives (1/2) / (1 - 1/2)^2 = (1/2) / (1/4) = 2.

Question 17Advanced analysis, convergence and multivariable reasoning

Evaluate the improper integral I = integral_0^{+infty} x*exp(-3*x) dx.

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

Explanation

Using integration by parts, set u = x and v' = exp(-3*x), so u' = 1 and v = -exp(-3*x) / 3. Then I = [-x*exp(-3*x) / 3]_0^{+infty} + (1/3)*integral_0^{+infty} exp(-3*x) dx. The boundary term vanishes at both limits. The remaining integral is (1/3)*[-exp(-3*x) / 3]_0^{+infty} = (1/3)*(0 - (-1/3)) = 1/9.

Question 18Advanced analysis, convergence and multivariable reasoning

Compute the definite integral I = integral_0^{pi/2} x*sin(x) dx.

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

Explanation

Integrate by parts: let u = x, dv = sin(x) dx, so du = dx and v = -cos(x). Then I = [-x*cos(x)]_0^{pi/2} + integral_0^{pi/2} cos(x) dx. The boundary term evaluates to -(pi/2)*cos(pi/2) - (-0) = 0. The remaining integral is [sin(x)]_0^{pi/2} = sin(pi/2) - sin(0) = 1 - 0 = 1.

Question 19Advanced analysis, convergence and multivariable reasoning

Compute the limit L = lim_{x -> 0} (exp(x) - cos(x) - sin(x)) / x^2.

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

Explanation

Use standard Taylor expansions near 0 up to order 2: exp(x) = 1 + x + x^2 / 2 + o(x^2), cos(x) = 1 - x^2 / 2 + o(x^2), and sin(x) = x + o(x^2). Substituting into the numerator: exp(x) - cos(x) - sin(x) = (1 + x + x^2 / 2) - (1 - x^2 / 2) - x + o(x^2) = (x^2 / 2 + x^2 / 2) + o(x^2) = x^2 + o(x^2). Dividing by x^2 gives lim_{x -> 0} (x^2 + o(x^2)) / x^2 = 1.

Question 20Advanced analysis, convergence and multivariable reasoning

Consider the real sequence defined by u_0 = 1 and u_{n+1} = sqrt(2 + u_n) for all n >= 0. What is the limit of (u_n) as n -> +infty?

  1. 2 Correct answer.
  2. sqrt(2)
  3. 1 + sqrt(2)
  4. +infty

Explanation

The function f(x) = sqrt(2 + x) is continuous and strictly increasing on [0, +infty[. Solving the fixed-point equation L = f(L) gives L = sqrt(2 + L) <=> L^2 - L - 2 = 0 <=> (L - 2)(L + 1) = 0. Since u_n >= 0 for all n, L must be non-negative, so L = 2. By induction, 1 <= u_n <= u_{n+1} <= 2; being bounded and monotonic, (u_n) converges to L = 2.

Question 21Advanced analysis, convergence and multivariable reasoning

A terrain height model is defined by z = h(x, y) = x^2 + 3*x*y + 2*y^2. What is the directional derivative of h at the point (1, 1) in the direction of the unit vector u = (3/5, 4/5)?

  1. 8.6 Correct answer.
  2. 7.2
  3. 9.4
  4. 6.0

Explanation

The gradient of h is grad(h) = (dh/dx, dh/dy) = (2*x + 3*y, 3*x + 4*y). At (1, 1), grad(h)(1, 1) = (2*1 + 3*1, 3*1 + 4*1) = (5, 7). The directional derivative along the unit vector u = (3/5, 4/5) is the dot product grad(h) . u = 5*(3/5) + 7*(4/5) = 3 + 28/5 = 15/5 + 28/5 = 43/5 = 8.6.

Question 22Advanced analysis, convergence and multivariable reasoning

Evaluate the improper integral I = integral_0^{+infty} (x / (1 + x^2)^2) dx.

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

Explanation

Perform the substitution u = 1 + x^2, which gives du = 2*x dx, or x dx = du / 2. The integration limits transform as: x = 0 -> u = 1, and x -> +infty -> u -> +infty. Then I = (1/2)*integral_1^{+infty} (1 / u^2) du = (1/2)*[-1/u]_1^{+infty} = (1/2)*(0 - (-1)) = 1/2.

Question 23Advanced analysis, convergence and multivariable reasoning

An undamped harmonic oscillator satisfies the differential equation y'' + omega_0^2*y = F_0*cos(omega_0*t). What is the particular solution y_p(t) satisfying y_p(0) = 0?

  1. (F_0 / (2*omega_0))*t*sin(omega_0*t) Correct answer.
  2. (F_0 / omega_0^2)*cos(omega_0*t)
  3. -(F_0 / (2*omega_0))*t*cos(omega_0*t)
  4. (F_0 / (2*omega_0^2))*t^2*sin(omega_0*t)

Explanation

Because the forcing frequency omega equals the natural frequency omega_0, pure resonance occurs. The standard trial solution A*cos(omega_0*t) + B*sin(omega_0*t) solves the homogeneous equation. We try y_p(t) = t*(A*cos(omega_0*t) + B*sin(omega_0*t)). Differentiating twice and substituting gives y_p''(t) + omega_0^2*y_p(t) = -2*omega_0*A*sin(omega_0*t) + 2*omega_0*B*cos(omega_0*t). Equating with F_0*cos(omega_0*t) yields A = 0 and B = F_0 / (2*omega_0), so y_p(t) = (F_0 / (2*omega_0))*t*sin(omega_0*t).

Question 24Advanced analysis, convergence and multivariable reasoning

What is the radius of convergence R of the real power series sum_{n=0}^infty ((2^n / (n + 1))*x^n)?

  1. 1/2 Correct answer.
  2. 2
  3. 1
  4. +infty

Explanation

Let a_n = 2^n / (n + 1). By d'Alembert's ratio test: |a_{n+1} / a_n| = (2^(n+1) / (n + 2)) * ((n + 1) / 2^n) = 2 * ((n + 1) / (n + 2)). As n -> +infty, this ratio approaches 2. By Cauchy-Hadamard / d'Alembert criterion, the radius of convergence is R = 1 / L = 1/2.

Question 25Advanced analysis, convergence and multivariable reasoning

Evaluate the double integral I = iint_D (x + 2*y) dx dy over the rectangular domain D = [0, 1] x [0, 2].

  1. 5 Correct answer.
  2. 4
  3. 6
  4. 3

Explanation

By Fubini's theorem, I = integral_0^1 dx integral_0^2 (x + 2*y) dy. The inner integral evaluates to [x*y + y^2]_0^2 = 2*x + 4. Integrating with respect to x over [0, 1] gives integral_0^1 (2*x + 4) dx = [x^2 + 4*x]_0^1 = 1 + 4 = 5.

Question 26Advanced analysis, convergence and multivariable reasoning

Compute the limit L = lim_{x -> +infty} x*(sqrt(x^2 + 4) - x).

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

Explanation

Multiply and divide by the conjugate expression: sqrt(x^2 + 4) - x = (x^2 + 4 - x^2) / (sqrt(x^2 + 4) + x) = 4 / (sqrt(x^2 + 4) + x). Then x*(sqrt(x^2 + 4) - x) = 4*x / (x*(sqrt(1 + 4/x^2) + 1)) = 4 / (sqrt(1 + 4/x^2) + 1). As x -> +infty, 4/x^2 -> 0, so the limit is 4 / (1 + 1) = 2.

Question 27Advanced analysis, convergence and multivariable reasoning

Consider the function f(x, y) = x^3 + y^3 - 3*x*y. What is the nature of the critical point (1, 1)?

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

Explanation

The gradient is grad(f) = (3*x^2 - 3*y, 3*y^2 - 3*x). Setting grad(f) = (0, 0) yields y = x^2 and x = y^2 = x^4, giving real critical points (0, 0) and (1, 1). The Hessian matrix is H(x, y) = [[6*x, -3], [-3, 6*y]]. At (1, 1), H(1, 1) = [[6, -3], [-3, 6]]. The determinant is det(H) = 36 - 9 = 27 > 0, and the top-left entry is d^2f/dx^2 = 6 > 0. By the second derivative test, (1, 1) is a strict local minimum.

Question 28Advanced analysis, convergence and multivariable reasoning

Using Euler's formulas, what is the linearization of sin^3(x)?

  1. (3*sin(x) - sin(3*x)) / 4 Correct answer.
  2. (3*cos(x) + cos(3*x)) / 4
  3. (sin(3*x) + 3*sin(x)) / 4
  4. (sin(x) - sin(3*x)) / 2

Explanation

Using sin(x) = (exp(i*x) - exp(-i*x)) / (2*i), we have sin^3(x) = (exp(i*x) - exp(-i*x))^3 / (-8*i). Expanding via the binomial theorem: (exp(3*i*x) - 3*exp(i*x) + 3*exp(-i*x) - exp(-3*i*x)) / (-8*i) = [(exp(3*i*x) - exp(-3*i*x)) - 3*(exp(i*x) - exp(-i*x))] / (-8*i) = [2*i*sin(3*x) - 6*i*sin(x)] / (-8*i) = (3*sin(x) - sin(3*x)) / 4.

Question 29Advanced analysis, convergence and multivariable reasoning

Consider the Bertrand series sum_{n=2}^infty (1 / (n * (ln(n))^beta)). For which values of the real parameter beta does this series converge?

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

Explanation

The function f(t) = 1 / (t * (ln(t))^beta) is positive and decreasing on [2, +infty[. By the integral test (comparaison serie-integrale), the series converges if and only if integral_2^{+infty} dt / (t * (ln(t))^beta) converges. With substitution u = ln(t), du = dt / t, the integral becomes integral_{ln(2)}^{+infty} du / u^beta. This is a standard Riemann integral which converges if and only if beta > 1. At beta = 1, it diverges logarithmically.

Question 30Advanced probability, moments and conditional models

Structural inspection on a highway viaduct detects an average of lambda = 2 micro-cracks per 100 meters according to a Poisson process. What is the probability of detecting exactly 1 micro-crack on a 100-meter section?

  1. 2*exp(-2) Correct answer.
  2. exp(-2)
  3. 4*exp(-2)
  4. 1 - exp(-2)

Explanation

For a Poisson distribution X ~ P(lambda), the probability mass function is P(X = k) = (lambda^k * exp(-lambda)) / k!. With lambda = 2 and k = 1, P(X = 1) = (2^1 * exp(-2)) / 1! = 2*exp(-2) approx 0.2707 (27.1%).

Question 31Advanced probability, moments and conditional models

A civil engineering precast plant produces concrete foundation blocks. Each block independently has a probability p = 0.10 of having a structural flaw. In a random sample of n = 10 blocks, what is the probability that at least one block has a flaw?

  1. 1 - (0.9)^10 Correct answer.
  2. (0.1)^10
  3. 10*(0.1)*(0.9)^9
  4. (0.9)^10

Explanation

Let X ~ B(10, 0.1) be the number of defective blocks. The probability of having at least one defective block is the complement of having zero defects: P(X >= 1) = 1 - P(X = 0) = 1 - (1 - p)^n = 1 - (0.9)^10 approx 1 - 0.3487 = 0.6513 (65.1%).

Question 32Advanced probability, moments and conditional models

For S=sum from n=1 to infinity of (-1)^(n-1)/n^2, use the alternating-series upper bound |S-S_N| <= 1/(N+1)^2. What smallest N makes this particular bound at most 0.01? The question asks about the bound, not the actual error.

  1. 9 Correct answer.
  2. 10
  3. 8
  4. 99

Explanation

The series satisfies the Leibniz test for alternating series (terms alternate in sign, |a_n| = 1/n^2 decreases monotonically to 0). The remainder theorem states that |R_N| = |S - S_N| <= a_{N+1} = 1 / (N + 1)^2. To ensure |R_N| <= 10^(-2) = 1/100, we require 1 / (N + 1)^2 <= 1/100 <=> (N + 1)^2 >= 100 <=> N + 1 >= 10 <=> N >= 9. The smallest integer is N = 9.

Question 33Advanced probability, moments and conditional models

In non-destructive testing of bridge prestressing cables, 5% of cables suffer severe corrosion (event C). An ultrasonic detection method has a sensitivity P(+ | C) = 0.90 and a false positive rate P(+ | not C) = 0.02. If a tested cable tests positive (+), what is the probability P(C | +) that it actually suffers severe corrosion?

  1. approx 70.3% Correct answer.
  2. approx 90.0%
  3. approx 50.0%
  4. approx 23.8%

Explanation

By Bayes' theorem, P(C | +) = (P(+ | C)*P(C)) / P(+). By the law of total probability, P(+) = P(+ | C)*P(C) + P(+ | not C)*P(not C) = 0.90*0.05 + 0.02*(1 - 0.05) = 0.045 + 0.02*0.95 = 0.045 + 0.019 = 0.064. Thus P(C | +) = 0.045 / 0.064 = 45 / 64 approx 0.7031 (70.3%).

Question 34Advanced probability, moments and conditional models

Let X and Y be two independent random variables with Var(X) = 2 and Var(Y) = 5. What is the variance of the linear combination Z = 3*X - 2*Y?

  1. 38 Correct answer.
  2. -2
  3. 26
  4. 14

Explanation

For independent random variables X and Y, Var(a*X + b*Y) = a^2*Var(X) + b^2*Var(Y). Here a = 3 and b = -2. Thus Var(Z) = 3^2*Var(X) + (-2)^2*Var(Y) = 9*2 + 4*5 = 18 + 20 = 38.

Question 35Advanced probability, moments and conditional models

A temperature sensor reading T during concrete setting is uniformly distributed over the interval [15 deg C, 25 deg C]. What is the probability that T falls between 18 deg C and 23 deg C?

  1. 0.50 Correct answer.
  2. 0.40
  3. 0.60
  4. 0.25

Explanation

For a continuous uniform random variable X ~ U(a, b), the probability density function is f(t) = 1 / (b - a) on [a, b]. Here a = 15 and b = 25, so b - a = 10. The probability of falling in [18, 23] is P(18 <= T <= 23) = (23 - 18) / (25 - 15) = 5 / 10 = 0.50.

Question 36Analytical mechanics, oscillation and model reduction

In point mechanics (MPSI syllabus), a projectile is launched from flat ground with initial speed v_0 = 20 m/s at an angle alpha = 45 deg above the horizontal in a uniform gravitational field g = 10 m/s^2 (ignoring air drag). What is the horizontal range (portee) upon landing?

  1. 40 m Correct answer.
  2. 20 m
  3. 80 m
  4. 28.3 m

Explanation

The equations of motion are x(t) = v_0*cos(alpha)*t and y(t) = v_0*sin(alpha)*t - (1/2)*g*t^2. Landing occurs at y(t) = 0 (t > 0), giving flight time t_f = 2*v_0*sin(alpha) / g. The range is R = x(t_f) = (v_0^2 * sin(2*alpha)) / g. For alpha = 45 deg, sin(2*alpha) = sin(90 deg) = 1. Thus R = (20^2 * 1) / 10 = 400 / 10 = 40 m.

Question 37Analytical mechanics, oscillation and model reduction

An engineering inspection trolley of mass m = 100 kg starts from rest at the top of a slope of vertical drop h = 10 m and slope distance d = 25 m. It is subjected to a constant opposing friction force f = 200 N. Taking g = 10 m/s^2, what is the speed of the trolley at the bottom of the slope?

  1. 10 m/s Correct answer.
  2. 14.1 m/s
  3. 7.1 m/s
  4. 5.0 m/s

Explanation

Apply the work-energy theorem (theoreme de l'energie cinetique) between the top (rest, E_c1 = 0) and the bottom: Delta E_c = (1/2)*m*v^2 = W(gravity) + W(friction). Here W(gravity) = m*g*h = 100*10*10 = 10000 J, and W(friction) = -f*d = -200*25 = -5000 J. Hence (1/2)*m*v^2 = 10000 - 5000 = 5000 J. Solving for v: 50*v^2 = 5000 => v^2 = 100 => v = 10 m/s.

Question 38Analytical mechanics, oscillation and model reduction

A structural damping mechanism is modeled as a mass-spring-damper oscillator: m*x'' + lambda*x' + k*x = 0, with mass m = 1.0 kg and stiffness k = 400 N/m. What damping coefficient lambda_c achieves critical damping (regime critique)?

  1. 40 N*s/m Correct answer.
  2. 20 N*s/m
  3. 80 N*s/m
  4. 400 N*s/m

Explanation

The characteristic equation is m*r^2 + lambda*r + k = 0. The discriminant is Delta = lambda^2 - 4*m*k. Critical damping occurs at the boundary between oscillatory motion (Delta < 0) and non-oscillatory exponential decay (Delta > 0), when Delta = 0. Thus lambda_c = 2*sqrt(m*k) = 2*sqrt(1.0 * 400) = 2*20 = 40 N*s/m.

Question 39Analytical mechanics, oscillation and model reduction

A vertical rectangular retaining dam of width b = 10 m holds back water of density rho = 1000 kg/m^3 to a water depth H = 6 m. Assuming atmospheric pressure acts equally on both sides and taking g = 10 m/s^2, what is the total horizontal hydrostatic resultant force exerted by the water on the dam face?

  1. 1.8 MN Correct answer.
  2. 3.6 MN
  3. 0.9 MN
  4. 600 kN

Explanation

Hydrostatic relative gauge pressure at depth z below the free surface is p(z) = rho*g*z. The elementary force on a horizontal strip of height dz is dF = p(z)*b*dz = rho*g*b*z dz. Integrating from z = 0 to z = H: F = rho*g*b * integral_0^H z dz = (1/2)*rho*g*b*H^2. Substituting values: F = 0.5 * 1000 * 10 * 10 * 6^2 = 50000 * 36 = 1800000 N = 1.8 MN.

Question 40Analytical mechanics, oscillation and model reduction

An open water reservoir has a small drainage orifice located h = 5.0 m below the free water surface. Assuming steady, incompressible, inviscid flow and taking g = 10 m/s^2, what is the efflux velocity v of the exiting water jet (Torricelli's law)?

  1. 10 m/s Correct answer.
  2. 5.0 m/s
  3. 14.1 m/s
  4. 7.1 m/s

Explanation

Applying Bernoulli's theorem along a streamline from the free surface (1) to the orifice exit (2): P_1 + (1/2)*rho*v_1^2 + rho*g*z_1 = P_2 + (1/2)*rho*v_2^2 + rho*g*z_2. Since both points are open to the atmosphere, P_1 = P_2 = P_atm. The reservoir surface area is large compared to the orifice, so v_1 approx 0. Setting z_1 - z_2 = h gives (1/2)*v_2^2 = g*h, hence v_2 = sqrt(2*g*h). For h = 5.0 m and g = 10 m/s^2: v_2 = sqrt(2 * 10 * 5.0) = sqrt(100) = 10 m/s.

Question 41Analytical mechanics, oscillation and model reduction

A civil observation satellite revolves in a circular orbit around Earth at radius r with orbital period T. If the satellite is transferred to a higher circular orbit with radius r' = 4*r, what is its new orbital period T'?

  1. 8*T Correct answer.
  2. 4*T
  3. 16*T
  4. 2*T

Explanation

By Kepler's third law for circular gravitational orbits around a central body of mass M, T^2 / r^3 = 4*pi^2 / (G*M) = constant. Therefore, (T' / T)^2 = (r' / r)^3. Substituting r' / r = 4 gives (T' / T)^2 = 4^3 = 64. Taking the square root yields T' / T = sqrt(64) = 8, so T' = 8*T.

Question 42Analytical mechanics, oscillation and model reduction

A small sensor unit of mass m = 2.0 kg moves on a frictionless horizontal plane attached to a cord passing through a hole in the surface. It revolves in a circle of radius r_1 = 0.50 m with speed v_1 = 2.0 m/s. The cord is pulled slowly from below, reducing the radius to r_2 = 0.25 m. What is the new speed v_2 of the sensor?

  1. 4.0 m/s Correct answer.
  2. 2.0 m/s
  3. 8.0 m/s
  4. 1.0 m/s

Explanation

The only horizontal force acting on the mass is the tension in the cord, which points directly towards the central hole (central force). Therefore, the net torque about the origin is zero (Gamma_O = 0), which implies that angular momentum L_O = m * r * v is conserved throughout the motion. Thus m * r_1 * v_1 = m * r_2 * v_2, which simplifies to v_2 = v_1 * (r_1 / r_2) = 2.0 * (0.50 / 0.25) = 4.0 m/s.

Question 43Analytical mechanics, oscillation and model reduction

A simple gravity pendulum consists of a point mass suspended by a light inextensible cord of length L = 0.993 m in a uniform gravitational field g = 9.81 m/s^2. Under the small-angle approximation (sin(theta) approx theta), what is the natural period of oscillation T_0?

  1. approx 2.00 s Correct answer.
  2. approx 1.00 s
  3. approx 3.14 s
  4. approx 0.50 s

Explanation

For small angular amplitudes, the equation of motion is theta'' + (g/L)*theta = 0, representing a simple harmonic oscillator with natural angular frequency omega_0 = sqrt(g/L). The natural period is T_0 = 2*pi / omega_0 = 2*pi*sqrt(L/g). Substituting the values: T_0 = 2*pi * sqrt(0.993 / 9.81) = 2*pi * sqrt(0.10122) approx 2*3.14159 * 0.31815 approx 2.00 s.

Question 44Analytical mechanics, oscillation and model reduction

Water (density rho = 1000 kg/m^3) flows steadily through a horizontal pipe. At section 1 (cross-sectional area A_1 = 0.04 m^2), the fluid velocity is v_1 = 2.0 m/s. The pipe narrows to section 2 with area A_2 = 0.01 m^2. Assuming inviscid flow, what is the static pressure drop Delta P = P_1 - P_2 across the constriction (Venturi effect)?

  1. 30 kPa Correct answer.
  2. 15 kPa
  3. 60 kPa
  4. 3.75 kPa

Explanation

By mass conservation for an incompressible fluid, the volume flow rate is constant: Q = A_1*v_1 = A_2*v_2, giving v_2 = v_1*(A_1 / A_2) = 2.0 * (0.04 / 0.01) = 8.0 m/s. Applying Bernoulli's equation along a horizontal streamline (z_1 = z_2): P_1 + (1/2)*rho*v_1^2 = P_2 + (1/2)*rho*v_2^2. Thus Delta P = P_1 - P_2 = (1/2)*rho*(v_2^2 - v_1^2) = 0.5 * 1000 * (8.0^2 - 2.0^2) = 500 * (64 - 4) = 500 * 60 = 30000 Pa = 30 kPa.

Question 45Analytical mechanics, oscillation and model reduction

A precast concrete foundation block of mass m = 500 kg rests on a horizontal concrete platform. The static coefficient of friction between the two surfaces is mu_s = 0.60. Taking g = 10 m/s^2, what is the minimum horizontal force F_min required to set the block into motion (Coulomb's friction law)?

  1. 3.0 kN Correct answer.
  2. 5.0 kN
  3. 1.5 kN
  4. 8.3 kN

Explanation

In Coulomb's law of dry static friction, motion initiates when the tangential pulling force F exceeds the maximum static friction force: F_min = f_{s,max} = mu_s * N. For a mass on a horizontal plane without vertical applied forces, the normal reaction is N = m * g = 500 kg * 10 m/s^2 = 5000 N = 5.0 kN. Thus F_min = 0.60 * 5000 N = 3000 N = 3.0 kN.

Question 46Analytical mechanics, oscillation and model reduction

A particle moves along the x-axis subject to a 1D conservative force associated with the potential energy E_p(x) = 2*x^3 - 9*x^2 + 12*x (in Joules). What are the equilibrium positions, and which one is stable?

  1. x = 1 (unstable) and x = 2 (stable) Correct answer.
  2. x = 1 (stable) and x = 2 (unstable)
  3. x = 0 (stable) and x = 3 (unstable)
  4. Both x = 1 and x = 2 are stable

Explanation

The conservative force is F(x) = -dE_p / dx = -(6*x^2 - 18*x + 12) = -6*(x^2 - 3*x + 2) = -6*(x - 1)*(x - 2). Equilibrium points satisfy F(x) = 0, giving x = 1 and x = 2. Stability depends on the second derivative: d^2E_p / dx^2 = 12*x - 18. At x = 1: d^2E_p / dx^2 = 12 - 18 = -6 < 0, meaning E_p has a local maximum, so x = 1 is an unstable equilibrium. At x = 2: d^2E_p / dx^2 = 24 - 18 = +6 > 0, meaning E_p has a local minimum (potential well), so x = 2 is a stable equilibrium.

Question 47Analytical mechanics, oscillation and model reduction

A rectangular hollow concrete maritime caisson of total mass M = 120 metric tonnes (1.20 * 10^5 kg) and rectangular base dimensions 10.0 m x 4.0 m (base area S_b = 40.0 m^2) floats in calm seawater of density rho_sea = 1025 kg/m^3. What is the submerged draft (tirant d'eau T)?

  1. approx 2.93 m Correct answer.
  2. approx 3.00 m
  3. approx 2.50 m
  4. approx 1.46 m

Explanation

According to Archimedes' principle, a floating body in static equilibrium displaces a weight of fluid equal to its own total weight: M * g = rho_sea * V_submerged * g. For a prismatic vertical caisson, V_submerged = S_b * T. Therefore, M = rho_sea * S_b * T, which yields T = M / (rho_sea * S_b). Substituting values: T = 1.20 * 10^5 kg / (1025 kg/m^3 * 40.0 m^2) = 120000 / 41000 approx 2.9268 m approx 2.93 m.

Question 48Analytical mechanics, oscillation and model reduction

On a frictionless horizontal guide rail, cart 1 of mass m_1 = 2.0 kg moving at speed v_1 = 6.0 m/s undergoes a head-on perfectly elastic 1D collision with cart 2 of mass m_2 = 1.0 kg initially at rest (v_2 = 0). What are the post-collision velocities v_1' and v_2'?

  1. v_1' = 2.0 m/s and v_2' = 8.0 m/s Correct answer.
  2. v_1' = 0 m/s and v_2' = 6.0 m/s
  3. v_1' = 4.0 m/s and v_2' = 4.0 m/s
  4. v_1' = -2.0 m/s and v_2' = 10.0 m/s

Explanation

In a 1D elastic collision, both total linear momentum and kinetic energy are conserved. The standard collision formulas with v_2 = 0 give: v_1' = ((m_1 - m_2) / (m_1 + m_2)) * v_1 = ((2.0 - 1.0) / (2.0 + 1.0)) * 6.0 = (1/3) * 6.0 = 2.0 m/s; and v_2' = ((2 * m_1) / (m_1 + m_2)) * v_1 = ((2 * 2.0) / 3.0) * 6.0 = (4/3) * 6.0 = 8.0 m/s. Check momentum: 2*6 = 12 kg*m/s initially; 2*2 + 1*8 = 12 kg*m/s finally. Check kinetic energy: 0.5*2*36 = 36 J; 0.5*2*4 + 0.5*1*64 = 4 + 32 = 36 J.

Question 49Analytical mechanics, oscillation and model reduction

In a damped harmonic oscillator test on a footbridge cable, two consecutive positive displacement peaks separated by one pseudo-period T are measured as x_1 = 10.0 mm and x_2 = 6.0 mm. What is the logarithmic decrement (decrement logarithmique delta)?

  1. approx 0.511 Correct answer.
  2. approx 0.400
  3. approx 0.693
  4. approx 1.667

Explanation

The logarithmic decrement delta measures the rate of decay of an underdamped oscillator, defined as delta = ln(x(t) / x(t + T)) = ln(x_1 / x_2). Substituting the measured peak amplitudes: delta = ln(10.0 / 6.0) = ln(5/3) approx ln(1.6667) approx 0.5108. For small damping, delta approx 2*pi*xi, where xi is the damping ratio.

Question 50Electromagnetic fields, thermal physics and wave models

One mole (n = 1.0 mol) of an ideal gas at temperature T = 300 K undergoes a reversible isothermal expansion from initial volume V_1 to final volume V_2 = 2*V_1. Taking R = 8.314 J/(mol*K) and ln(2) approx 0.693, what is the work W received by the gas?

  1. -1728 J Correct answer.
  2. +1728 J
  3. -2494 J
  4. 0 J

Explanation

In thermodynamics with standard thermodynamic conventions (work received is positive, work performed is negative), W = -integral_{V_1}^{V_2} P dV. For an ideal gas undergoing an isothermal process, P = n*R*T / V, so W = -n*R*T * ln(V_2 / V_1). Substituting the given values: W = -1.0 * 8.314 * 300 * ln(2) = -2494.2 * 0.69315 approx -1728 J. The gas delivers 1728 J of work to the external surroundings.