Concours parallèle EMI — DEUG/DEUST/Licence (Morocco)

What is the limit of the real sequence defined by u_n = ( (n + 3) / n )^(2n) as n tends to infinity?

• Ae^6
• Be^3
• Ce^2
• De^9

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

• A1/2
• B1/4
• C1/6
• D1/12

For which real values of the parameter p does the alternating series sum_{n=1}^infty ( (-1)^(n-1) ) / n^p converge conditionally (converges, but does not converge absolutely)?

• Ap > 1
• Bp <= 0
• C0 < p <= 1
• Dp >= 2

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

• AR = 4
• BR = 1/2
• CR = 1
• DR = 1/4

Consider the real function f(x) = ln(1 - x + x^2) defined on (-1, 1). What is the coefficient of x^6 in its Maclaurin power series expansion?

• A-1/2
• B-1/3
• C1/6
• D2/3

What is the exact value of the improper Dirichlet integral I = int_0^infty (sin(x) / x) dx?

• Api / 2
• Bpi
• C1
• Dsqrt(pi) / 2

Determine the limit of the multivariable function f(x, y) = (x^3 * y) / (x^4 + y^2) as (x, y) tends to (0, 0).

• AThe limit does not exist because directional limits depend on the slope.
• B1/2
• C0
• D1

For the scalar field f(x, y) = x^2 * y + exp(x * y), what is the directional derivative D_v f(1, 0) in the direction of the unit vector v = (3/5, 4/5)?

• A3/5
• B1
• C2
• D8/5

Calculate the double integral iint_D (x^2 + y^2) dx dy, where D is the upper half-disk defined by x^2 + y^2 <= R^2 and y >= 0.

• A(pi * R^4) / 4
• B(pi * R^4) / 2
• C(pi * R^3) / 3
• D(pi * R^4) / 8

Using Green's theorem (théorème de Green-Riemann), evaluate the line integral oint_C (y^3 dx - x^3 dy) counterclockwise along the circle C: x^2 + y^2 = a^2.

• A0
• B-(3/2) * pi * a^4
• C(3/2) * pi * a^4
• D-3 * pi * a^2

What is the general solution of the first-order linear differential equation y' + (2/x) * y = x^3 for x > 0?

• Ay(x) = x^4 / 4 + C / x^2
• By(x) = x^4 / 6 + C * x^2
• Cy(x) = x^4 / 6 + C / x^2
• Dy(x) = x^3 / 5 + C / x

Find the general solution of the second-order linear differential equation y'' - 4y' + 4y = e^(2x).

• Ay(x) = (c_1 + c_2 * x) * e^(2x) + e^(2x)
• By(x) = c_1 * e^(2x) + c_2 * e^(-2x) + (1/2) * x^2 * e^(2x)
• Cy(x) = (c_1 + c_2 * x + x^2) * e^(2x)
• Dy(x) = (c_1 + c_2 * x + (1/2) * x^2) * e^(2x)

Consider the sequence of functions f_n(x) = (n * x) / (1 + n^2 * x^2) defined on the closed interval [0, 1]. Which of the following statements is true regarding its convergence?

• Af_n converges pointwise to 0 on [0, 1], but does not converge uniformly on [0, 1].
• Bf_n converges uniformly to 0 on [0, 1].
• Cf_n does not converge pointwise at x = 0.
• Df_n converges uniformly to 1/2 on [0, 1].

What is the general coefficient a_n of x^n in the Maclaurin expansion of (1 - x)^(-1/2) for |x| < 1?

• A(2n)! / (n!)^2
• B(2n)! / (4^n * (n!)^2)
• C1 / (2n + 1)
• D(-1)^n * (2n)! / (4^n * (n!)^2)

Consider the differential 1-form omega = (2x * y + cos(y)) dx + (x^2 - x * sin(y) + 3y^2) dy on R^2. Which of the following is a potential function f(x, y) satisfying df = omega?

• Af(x, y) = x^2 * y^2 - x * cos(y) + y^3 + C
• Bf(x, y) = 2x^2 * y + x * sin(y) + y^3 + C
• Cf(x, y) = x^2 * y + x * cos(y) + y^3 + C
• Df(x, y) = x^2 * y - x * cos(y) + 3y^3 + C

What is the volume of the solid region bounded below by the paraboloid z = x^2 + y^2 and above by the plane z = 4?

• A16 * pi
• B(16/3) * pi
• C4 * pi
• D8 * pi

Using the method of Lagrange multipliers, what is the maximum value of f(x, y) = x * y on the ellipse (x^2 / a^2) + (y^2 / b^2) = 1, where a > 0 and b > 0?

• A(a * b) / 2
• Ba * b
• C(a^2 + b^2) / 4
• Dsqrt(a * b)

Given the Euler Gamma function Gamma(z) = int_0^infty t^(z-1) e^(-t) dt, what is the value of the integral int_0^infty x^(-1/2) * e^(-x) dx?

• Api
• Bsqrt(pi)
• Csqrt(pi) / 2
• D1 / sqrt(pi)

Consider the series of functions sum_{n=1}^infty ( (-1)^n ) / (n + x) defined on [0, +infty). Which statement correctly describes its convergence?

• AThe series converges normally on [0, +infty).
• BThe series diverges for all x > 0.
• CThe series converges uniformly on [0, +infty), but does not converge normally.
• DThe series converges pointwise on [0, +infty), but not uniformly.

Using the Fourier series expansion of the 2pi-periodic even function f(x) = |x| on [-pi, pi], what is the sum of the series sum_{k=0}^infty 1 / (2k + 1)^2?

• Api^2 / 6
• Bpi^2 / 12
• Cpi^2 / 4
• Dpi^2 / 8

Applying the Cauchy-Schwarz inequality in the Hilbert space L^2([0, 1]), what upper bound can be established for the integral I = int_0^1 x * exp(x) dx?

• Asqrt((e^2 - 1) / 6)
• Bsqrt((e^2 - 1) / 3)
• C(e - 1) / sqrt(3)
• De / 3

Using the Gauss divergence theorem (théorème de flux-divergence), calculate the outward flux of the vector field F(x, y, z) = (x^3, y^3, z^3) through the sphere S: x^2 + y^2 + z^2 = R^2.

• A(4/5) * pi * R^5
• B(12/5) * pi * R^5
• C4 * pi * R^4
• D3 * pi * R^5

What is the general solution of the Euler-Cauchy differential equation x^2 * y'' - 2x * y' + 2y = 0 for x > 0?

• Ay(x) = c_1 * x + c_2 * x * ln(x)
• By(x) = c_1 * cos(ln x) + c_2 * sin(ln x)
• Cy(x) = c_1 * x + c_2 * x^2
• Dy(x) = c_1 * x^2 + c_2 * x^3

Consider the coordinate transformation from R^2 to R^2 given by x = u^2 - v^2 and y = 2u * v. What is the absolute value of the Jacobian determinant |det J(u, v)|?

• A2 * (u^2 + v^2)
• B4 * (u^2 - v^2)
• Cu^2 + v^2
• D4 * (u^2 + v^2)

What is the exact sum of the infinite series sum_{n=1}^infty n / 2^n?

• A2
• B1
• C4
• D3/2

Using the Laplace transform definition L{t^n} = int_0^infty e^(-s*t) * t^n dt = n! / s^(n+1), calculate the value of int_0^infty t^3 * e^(-2t) dt.

• A3/4
• B3/8
• C1/4
• D3/16

For the differential equation (3x^2 * y + 2x * y + y^3) dx + (x^2 + y^2) dy = 0, which integrating factor mu(x) depending solely on x makes the equation exact?

• Amu(x) = e^x
• Bmu(x) = x^3
• Cmu(x) = e^(3x)
• Dmu(x) = e^(-2x)

According to Taylor's theorem with integral remainder (formule de Taylor avec reste intégral de Laplace), what is the exact expression for the remainder R_n(x) of a function f in C^(n+1)([a, b])?

• Aint_a^x (x - t)^n * f^(n+1)(t) dt
• B(1 / (n+1)!) * int_a^x (x - t)^(n+1) * f^(n+1)(t) dt
• C(1 / n!) * int_a^x (t - a)^n * f^(n+1)(t) dt
• D(1 / n!) * int_a^x (x - t)^n * f^(n+1)(t) dt

For the function of two variables f(x, y) = x^3 + y^3 - 3x * y, identify the nature of the critical point (1, 1) and find the local extremum value.

• ALocal minimum with f(1, 1) = -1
• BSaddle point with f(1, 1) = 0
• CLocal maximum with f(1, 1) = 1
• DLocal minimum with f(1, 1) = 0

Using Stokes' theorem (théorème de Stokes-Ampère), evaluate the circulation oint_C F . dr of the vector field F(x, y, z) = (y, z, x) along the triangular path C with vertices (1, 0, 0), (0, 1, 0), (0, 0, 1) oriented counterclockwise when viewed from high on the positive z-axis.

• A0
• B-3/2
• C3/2
• D-sqrt(3)

Let E be a real vector space of dimension 5. Let F and G be two subspaces of E with dim(F) = 3 and dim(G) = 4. What is the minimum possible dimension of the intersection subspace F cap G?

• A0
• B1
• C2
• D3

Let u: R^6 -> R^4 be a linear map whose image Im(u) has dimension 3. By the rank-nullity theorem (théorème du rang), what is the dimension of the kernel Ker(u)?

• A1
• B2
• C4
• D3

What are the eigenvalues of the real symmetric 2x2 matrix A = [[1, 2], [2, 1]]?

• A3 and -1
• B1 and -1
• C2 and -2
• D5 and 1

Let A be an invertible 3x3 matrix whose characteristic polynomial is p_A(X) = X^3 - 4X^2 + 5X - 2. By the Cayley-Hamilton theorem, which expression represents the inverse matrix A^(-1)?

• A(1/2) * (A^2 - 4A - 5I)
• B(1/2) * (A^2 - 4A + 5I)
• C-(1/2) * (A^2 - 4A + 5I)
• DA^2 - 4A + 5I

Which of the following conditions is both necessary and sufficient for an n x n matrix A with entries in a field K to be diagonalizable over K?

• AThe characteristic polynomial of A has n distinct roots in K.
• BA is an invertible matrix with non-zero determinant.
• CThe minimal polynomial of A is split over K and has only simple roots.
• DThe trace of A is equal to the sum of the diagonal entries.

Apply the Gram-Schmidt orthogonalization process to the vectors v_1 = (1, 1, 0) and v_2 = (1, 0, 1) in R^3 equipped with the canonical Euclidean inner product. What is the second orthogonal vector u_2 (prior to normalization)?

• A(0, -1, 1)
• B(1/2, 1/2, 1)
• C(1, -1, 2)
• D(1/2, -1/2, 1)

Let N in M_n(R) be a nilpotent matrix (there exists an integer k >= 1 such that N^k = 0). What are the values of Tr(N) and det(I_n + N)?

• ATr(N) = 0 and det(I_n + N) = 1
• BTr(N) = 0 and det(I_n + N) = 0
• CTr(N) = 1 and det(I_n + N) = 1
• DTr(N) = n and det(I_n + N) = 1

Using Gauss's method of reduction into squares (réduction de Gauss), determine the signature of the quadratic form q(x, y, z) = x^2 + 2y^2 + 5z^2 + 2xy - 4yz on R^3.

• A(2, 1)
• B(3, 0)
• C(1, 2)
• D(2, 0)

What is the 3x3 standard matrix of the orthogonal projection P onto the 1-dimensional subspace W = span{v} in R^3, where v = (1, 1, 1)^T?

• A[[1, 0, 0], [0, 1, 0], [0, 0, 1]]
• B(1/2) * [[1, 1, 0], [1, 1, 0], [0, 0, 0]]
• C(1/3) * [[1, 1, 1], [1, 1, 1], [1, 1, 1]]
• D(1/sqrt(3)) * [[1, 1, 1], [1, 1, 1], [1, 1, 1]]

Let M be a block upper-triangular matrix of the form M = [[A, B], [0, D]], where A is an invertible 2x2 matrix with det(A) = 3, B is a 2x3 matrix, and D is a 3x3 matrix with det(D) = -4. What is the determinant of M?

• A-1
• B7
• C12
• D-12

According to the Spectral Theorem for real symmetric matrices, which statement is universally true for any matrix A in S_n(R)?

• AAll eigenvalues of A are real, and there exists an orthogonal matrix P such that P^T * A…
• BAll eigenvalues of A are strictly positive, and A is always invertible.
• CThe minimal polynomial of A always equals its characteristic polynomial.
• DA is diagonalizable over C, but may fail to have a full set of real eigenvectors.

Calculate the determinant of the 3x3 Vandermonde matrix V = [[1, 1, 1], [1, 2, 4], [1, 4, 16]], with distinct nodes x_1 = 1, x_2 = 2, and x_3 = 4.

• A12
• B6
• C8
• D2

Let E be a finite-dimensional vector space and p: E -> E be a linear projector (satisfying p^2 = p). What is the trace of p, Tr(p)?

• A0
• Bdim(Ker(p))
• Cdim(Im(p))
• Ddim(E)

What is the matrix exponential exp(t * N) for the 2x2 nilpotent matrix N = [[0, 2], [0, 0]] and t in R?

• A[[1, 0], [0, 1]]
• B[[e^t, 2t], [0, e^t]]
• C[[1, 2 * e^t], [0, 1]]
• D[[1, 2t], [0, 1]]

Let E be a finite-dimensional Euclidean space and u: E -> E be an endomorphism with adjoint u^*. Which fundamental orthogonality relation holds between the range and kernel?

• A(Im(u))^perp = Ker(u^*)
• B(Ker(u))^perp = Ker(u^*)
• CIm(u) = (Ker(u))^perp
• D(Im(u))^perp = Im(u^*)

Consider the cyclic permutation matrix J = [[0, 1, 0], [0, 0, 1], [1, 0, 0]] in M_3(C). What are the eigenvalues of J in the complex field C?

• A1, -1, and 0
• B1, e^(i * 2pi / 3), and e^(i * 4pi / 3)
• C0, 1, and 2
• De^(i * pi / 3), -1, and e^(-i * pi / 3)

A matrix A in M_4(C) has characteristic polynomial p_A(X) = (X - 2)^4 and minimal polynomial mu_A(X) = (X - 2)^2. What is the maximum size of a Jordan block in the Jordan normal form of A?

• A4
• B1
• C2
• D3

In a real Euclidean vector space (E, <., .>), under what exact geometric condition does the Cauchy-Schwarz equality |<u, v>| = ||u|| * ||v|| hold for non-zero vectors u and v?

• Au and v are orthogonal (<u, v> = 0)
• B||u|| = ||v||
• Cu + v is orthogonal to u - v
• Du and v are linearly dependent (collinear)