Concours National Commun (CNC) — CPGE MP/PSI/TSI (Morocco)

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

• AR = 3
• BR = 1/3
• CR = 9
• DR = inf

What is the sum of the alternating harmonic series sum_{n=1}^inf ((-1)^(n-1) / n) = 1 - 1/2 + 1/3 - 1/4 + ...?

• Aln(2)
• B1
• Ce
• D1/2

For what values of the real parameter alpha does the Bertrand series sum_{n=2}^inf (1 / (n * (ln n)^alpha)) converge?

• AConverges for alpha >= 1 and diverges for alpha < 1
• BConverges if and only if alpha > 1
• CConverges for all alpha > 0
• DDiverges for all real values of alpha

What is the asymptotic expansion of the harmonic series partial sum H_n = sum_{k=1}^n (1 / k) through the 1/n^2 term?

• AH_n = ln n + gamma + O(1/n)
• BH_n = n * ln n + gamma + O(1/n^2)
• CH_n = ln n + gamma + 1/(2n) - 1/(12n^2) + O(1/n^4)
• DH_n = ln n + 1/n + O(1/n^2)

What is the value of the improper integral int_0^inf e^(-3t) dt?

• A3
• B1
• C0
• D1/3

What is the exact value of the improper integral int_0^inf t^3 * e^(-2t) dt?

• A3/8
• B3/4
• C3/16
• D6

What is the value of the Gaussian integral I = int_{-inf}^inf e^(-x^2 / 2) dx?

• Api
• Bsqrt(2 * pi)
• Csqrt(pi) / 2
• D2 * pi

What is the value of the Dirichlet integral int_0^inf (sin x / x) dx?

• Api
• B1
• Cpi / 2
• Dpi / 4

What is the general solution of the second-order homogeneous linear differential equation y'' - 5y' + 6y = 0?

• Ay(t) = C1 * e^(-2t) + C2 * e^(-3t)
• By(t) = (C1 + C2 * t) * e^(2.5t)
• Cy(t) = C1 * cos(2t) + C2 * sin(3t)
• Dy(t) = C1 * e^(2t) + C2 * e^(3t)

What is a particular solution y_p(t) to the differential equation y'' + 4y = 8 * cos(2t)?

• Ay_p(t) = 2t * sin(2t)
• By_p(t) = 2 * cos(2t)
• Cy_p(t) = -2t * cos(2t)
• Dy_p(t) = 4t * sin(2t)

Solve the initial value problem x * y' + 2y = 4x^2 for x > 0, with y(1) = 2. What is y(x)?

• Ay(x) = 2x^2
• By(x) = x^2 + 1/x^2
• Cy(x) = x^2 + 1
• Dy(x) = (4/3)*x^2 + (2/3)*x

For the linear differential system X' = AX with A = [[0, 1], [-4, 0]], what is the nature and stability of the equilibrium point at the origin (0, 0)?

• AAsymptotically stable node with real negative eigenvalues
• BUnstable saddle point with eigenvalues of opposite sign
• CStable center (neutral Lyapunov stability) with trajectories forming closed concentric el…
• DUnstable focus with spiral trajectories diverging outwards

Let T: R^5 -> R^3 be a linear map whose kernel has dimension dim(ker(T)) = 2. What can be concluded about rank(T) and the surjectivity of T?

• Arank(T) = 2 and T is injective
• Brank(T) = 1 and T is neither injective nor surjective
• Crank(T) = 5 and T cannot be defined
• Drank(T) = 3 and T is surjective

What is the determinant of the 3x3 Vandermonde matrix V(a, b, c) whose rows are [1, a, a^2], [1, b, b^2], and [1, c, c^2]?

• A(b - a) * (c - a) * (c - b)
• B(a + b) * (b + c) * (c + a)
• C(a - b) * (b - c) * (c - a)
• Da^2 * b^2 * c^2 - (a + b + c)

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

• Alambda_1 = 2, lambda_2 = 2
• Blambda_1 = 1, lambda_2 = 3
• Clambda_1 = -1, lambda_2 = -3
• Dlambda_1 = 0, lambda_2 = 4

What are the eigenvalues of the upper triangular matrix A = [[3, 2], [0, 1]]?

• Alambda_1 = 3, lambda_2 = 2
• Blambda_1 = 2, lambda_2 = 1
• Clambda_1 = 3, lambda_2 = 1
• Dlambda_1 = 4, lambda_2 = 0

Under what condition is an n x n complex matrix A in M_n(C) diagonalizable?

• AIts minimal polynomial has degree equal to n
• BIts characteristic polynomial has only non-zero roots
• CAll of its eigenvalues are real numbers
• DIts minimal polynomial mu_A(X) is split over C with all roots having multiplicity 1 (simp…

Let J = [[lambda, 1], [0, lambda]] be a 2x2 Jordan block. What is the matrix power J^k for any positive integer k >= 1?

• A[[lambda^k, k * lambda^(k-1)], [0, lambda^k]]
• B[[lambda^k, 1], [0, lambda^k]]
• C[[lambda^k, lambda^k], [0, lambda^k]]
• D[[lambda^k, 0], [0, lambda^k]]

According to the Cayley-Hamilton theorem, if A is a 2x2 matrix with characteristic polynomial chi_A(lambda) = lambda^2 - 4*lambda + 3, what is the matrix A^2 - 4A + 3I?

• AI (the identity matrix)
• BThe zero matrix 0
• C4A
• D-3I

In a real inner product space (E, <., .>), under what necessary and sufficient condition does equality hold in the Cauchy-Schwarz inequality |<x, y>| = ||x|| * ||y||?

• AIf and only if x and y are orthogonal (<x, y> = 0)
• BIf and only if ||x|| = ||y||
• CIf and only if the vectors x and y are linearly dependent (collinear)
• DFor all unit vectors x and y

Apply the Gram-Schmidt orthonormalization process to the vectors v1 = (1, 0, 0) and v2 = (1, 2, 0) in R^3 with the standard dot product. What is the resulting orthonormal family (e1, e2)?

• Ae1 = (1, 0, 0), e2 = (1/2, 1, 0)
• Be1 = (1, 0, 0), e2 = (1/sqrt(5), 2/sqrt(5), 0)
• Ce1 = (1, 0, 0), e2 = (0, 0, 1)
• De1 = (1, 0, 0), e2 = (0, 1, 0)

What is the orthogonal projection of the vector v = (3, 4, 1) onto the line directed by u = (1, 2, 2) in R^3?

• Ap_u(v) = (13/9, 26/9, 26/9)
• Bp_u(v) = (13/3, 26/3, 26/3)
• Cp_u(v) = (1, 2, 2)
• Dp_u(v) = (3, 4, 1)

What does the spectral theorem (theoreme spectral) state regarding any real symmetric matrix S in M_n(R)?

• AAll eigenvalues of S are purely imaginary numbers
• BAll eigenvalues of S are real, and S can be diagonalized in an orthonormal basis of eigen…
• CS is diagonalizable if and only if all of its eigenvalues are strictly positive
• DS is invertible and its determinant is always positive

Consider the quadratic form q(x, y) = 2x^2 + 2xy + 3y^2 on R^2. Applying Sylvester's criterion, what is the signature and definiteness of q?

• AIndefinite with signature (1, 1)
• BNegative definite with signature (0, 2)
• CPositive definite with signature (2, 0)
• DPositive semidefinite with rank 1

What is the gradient vector grad f(1, 1) of the scalar field f(x, y) = x^2 * y + 3 * x * y^3 at the point (1, 1)?

• A(2, 9)
• B(5, 9)
• C(4, 10)
• D(5, 10)

Classify the critical point of f(x, y) = x^2 - y^2 + 4xy at the origin (0, 0) using the Hessian matrix.

• ASaddle point (point-selle / col)
• BStrict local minimum
• CStrict local maximum
• DNon-isolated degenerate line of extrema

Consider the series of functions sum_{n=1}^inf (cos(nx) / n^2) for x in R. What type of convergence does it exhibit on R?

• APointwise convergence only, but not uniform convergence on R
• BNormal convergence (convergence normale), and therefore uniform and absolute convergence…
• CDivergence for all x = 2k*pi
• DUniform convergence only on compact intervals [-pi/2, pi/2]

What is the closed-form sum function S(x) of the power series sum_{n=0}^inf (x^(2n+1) / (2n+1)) for |x| < 1?

• Aln(1 - x^2)
• B1 / (1 - x^2)
• CS(x) = (1/2) * ln((1 + x) / (1 - x)) = artanh(x)
• Darctan(x)

What is the directional derivative of f(x, y) = x^2 + 2y^2 at point P(1, 2) in the unit direction u = (3/5, 4/5)?

• A10
• B34/5
• C42/5
• D38/5

Let u and v be commuting endomorphisms of an n-dimensional vector space E (i.e. u o v = v o u). If u is diagonalizable with n pairwise distinct eigenvalues, what can be deduced about v?

• Au and v are simultaneously diagonalizable in a common basis of eigenvectors, and v is a p…
• Bv must be the zero endomorphism or an identity multiple
• Cv is nilpotent with nilpotency index n
• Dv cannot be diagonalized unless u = v

Neglecting buoyancy, a small spherical particle of radius r and mass m falls in a fluid of dynamic viscosity eta under gravity g. Assuming linear Stokes drag F_d = -6*pi*eta*r*v, what is the terminal velocity v_t?

• Av_t = (m * g) / (6 * pi * eta * r)
• Bv_t = (6 * pi * eta * r) / (m * g)
• Cv_t = sqrt((2 * m * g) / (pi * eta * r))
• Dv_t = (m * g * r) / (6 * pi * eta)

In the Northern Hemisphere, a projectile is fired horizontally due North with relative velocity v_r. Due to the Coriolis acceleration a_c = 2 * Omega x v_r, in which direction is the projectile deflected?

• ADeflected to the left (westward)
• BDeflected to the right (eastward)
• CDeflected purely downward with zero lateral displacement
• DDeflected purely upward with zero lateral displacement

In central gravitational motion, the effective potential is V_eff(r) = -(G*M*m / r) + (L^2 / (2*m*r^2)), where L is the conserved angular momentum. What is the radius r_0 of a stable circular orbit?

• Ar_0 = L / (G * M * m)
• Br_0 = (G * M * m^2) / L^2
• Cr_0 = L^2 / (G * M * m^2)
• Dr_0 = sqrt(L / (G * M * m))

A uniform solid cylinder of mass M and radius R (moment of inertia J = (1/2)*M*R^2) rolls without slipping down an incline of angle theta. What is the acceleration a of its center of mass?

• Aa = g * sin(theta)
• Ba = (1/2) * g * sin(theta)
• Ca = (5/7) * g * sin(theta)
• Da = (2/3) * g * sin(theta)

The moment of inertia of a thin uniform rod of mass M and length L about its center of mass is J_G = (1/12)*M*L^2. Applying Steiner's parallel axis theorem, what is its moment of inertia J_end about a perpendicular axis through one of its ends?

• AJ_end = (1/3) * M * L^2
• BJ_end = (1/4) * M * L^2
• CJ_end = (1/2) * M * L^2
• DJ_end = (7/12) * M * L^2

What are the natural angular frequency omega_0 and period T_0 of a simple pendulum of length L in the small-angle approximation?

• Aomega_0 = g/L and T_0 = 2*pi*(L/g)
• Bomega_0 = sqrt(g/L) and T_0 = 2*pi * sqrt(L/g)
• Comega_0 = sqrt(L/g) and T_0 = 2*pi * sqrt(g/L)
• Domega_0 = 2*pi * sqrt(g/L) and T_0 = sqrt(L/g)

A series RLC circuit has resistance R = 10 Ohm, inductance L = 100 mH, and capacitance C = 10 microF. What are the resonance angular frequency omega_0 and quality factor Q?

• Aomega_0 = 100 rad/s and Q = 1
• Bomega_0 = 1000 rad/s and Q = 100
• Comega_0 = 1000 rad/s and Q = 10
• Domega_0 = 10000 rad/s and Q = 10

According to Gauss's law, what is the radial electric field E(r) of a uniformly charged solid non-conducting sphere of radius R with total charge Q?

• AE(r) = 0 for r < R, and E(r) = Q / (4*pi*epsilon_0 * r^2) for r > R
• BE(r) is constant everywhere inside the sphere
• CE(r) is inversely proportional to r inside the sphere
• DE(r) increases linearly with r inside (E ~ r), and decreases as 1/r^2 outside (E ~ 1/r^2)

What is the magnetic field B inside an ideal infinite solenoid with n turns per unit length carrying a steady current I?

• AB = mu_0 * n * I, directed axially along the solenoid interior
• BB = (mu_0 * I) / (2*pi*r), directed azimuthally
• CB = (1/2) * mu_0 * n * I, directed radially
• DB = 0 inside and non-zero outside

A particle of mass m and charge q enters a uniform magnetic field B with perpendicular and parallel velocity components v_perp and v_parallel. What are the helix radius r and pitch p?

• Ar = m*v_parallel/(|q|*B), p = 2*pi*m*v_perp/(|q|*B)
• Br = |q|*B/(m*v_perp), p = v_parallel*|q|*B/m
• Cr = m*v_perp/(|q|*B), p = 2*pi*m*v_parallel/(|q|*B)
• Dr = m*(v_perp+v_parallel)/(|q|*B), p = 0

Why did James Clerk Maxwell introduce the displacement current density j_D = epsilon_0 * (dE/dt) into the Maxwell-Ampere equation curl B = mu_0 * j + mu_0 * epsilon_0 * (dE/dt)?

• ATo satisfy Faraday's law of electromagnetic induction
• BTo guarantee the non-existence of magnetic monopoles
• CTo ensure conservation of electric charge div j + d(rho)/dt = 0, because div(curl B) is i…
• DTo account for the Hall effect in metallic conductors

For a monochromatic plane electromagnetic wave in vacuum with electric field amplitude E_0, what is the time-averaged magnitude of the Poynting vector <Pi> (wave intensity)?

• A<Pi> = E_0^2 / (mu_0 * c)
• B<Pi> = epsilon_0 * c * E_0
• C<Pi> = (E_0 * B_0) / (4 * mu_0)
• D<Pi> = (1/2) * epsilon_0 * c * E_0^2 = E_0^2 / (2 * mu_0 * c)

In a Young's double-slit experiment, the slits are separated by a = 0.50 mm and the observation screen is placed at D = 2.0 m. Using light of wavelength lambda = 500 nm, what is the fringe spacing i?

• Ai = 2.0 mm
• Bi = 0.50 mm
• Ci = 1.0 mm
• Di = 4.0 mm

In a Young's interference setup using wavelength lambda = 600 nm, a thin glass slide of thickness e = 6.0 micrometers and refractive index n = 1.50 is inserted in front of one slit. By how many fringes does the central interference pattern shift?

• A10 fringes
• B5 fringes
• C2.5 fringes
• D15 fringes

In Fraunhofer diffraction by a single slit of width a = 0.10 mm illuminated by light of wavelength lambda = 600 nm, a lens of focal length f' = 1.0 m focuses the pattern on a screen. What is the linear total width L of the central diffraction peak?

• AL = 6.0 mm
• BL = 24 mm
• CL = 12 mm
• DL = 1.2 mm

When a Michelson interferometer is adjusted in the air wedge (coin d'air) configuration, what is the geometric nature of the observed interference fringes?

• AConcentric circular rings localized at infinity
• BNon-localized hyperbolic curves filling all space
• CRandom speckle patterns with varying contrast
• DStraight, parallel, equidistant fringes localized on the mirrors

For a monoatomic ideal gas with heat capacity ratio gamma = C_p / C_v = 5/3, what are the molar heat capacities C_{v,m} and C_{p,m} according to Mayer's relation C_{p,m} - C_{v,m} = R?

• AC_{v,m} = (3/2)*R and C_{p,m} = (5/2)*R
• BC_{v,m} = (5/2)*R and C_{p,m} = (7/2)*R
• CC_{v,m} = 3R and C_{p,m} = 4R
• DC_{v,m} = (1/2)*R and C_{p,m} = (3/2)*R

A diatomic ideal gas (gamma = 1.4 = 7/5) initially at temperature T1 = 300 K undergoes a reversible adiabatic expansion from volume V1 = 1.0 L to V2 = 32 L. What is the final temperature T2?

• A150 K
• B75 K
• C60 K
• D120 K