Concours ENA Meknès — première année post-Baccalauréat (Morocco)

What is the value of the limit lim_{x -> +infinity} [sqrt(x^2 + 4x + 1) - x]?

• A0
• B2
• C4
• D+infinity

Determine the value of the limit lim_{x -> 0} [(e^(3x) - 1) / sin(2x)].

• A1
• B2/3
• C3/2
• D3

Let f be the real-valued function defined on ]-1/2, +infinity[ by f(x) = ln((2x + 1) / (x + 2)). What is the value of the first derivative f'(1)?

• A1/3
• B2/3
• C-1/3
• D1

What is the Cartesian equation of the tangent line to the curve representing f(x) = (3x - 2) * e^(2x) at the point of abscissa x = 0?

• Ay = -x + 2
• By = 3x - 2
• Cy = -2x - 2
• Dy = -x - 2

Consider the function f(x) = x^3 + 3x - 5 defined on R. According to the Intermediate Value Theorem (Théorème des Valeurs Intermédiaires), how many real roots does f(x) = 0 have on the open interval ]1, 2[?

• AZero roots
• BExactly one root
• CExactly two roots
• DInfinitely many roots

Using integration by parts, calculate the definite integral I = integral_{1}^{e} [x * ln(x)] dx.

• A(e^2 + 1) / 4
• B(e^2 - 1) / 4
• C(3e^2 + 1) / 4
• De^2 / 2

Which of the following functions represents the general primitive (antiderivative) of f(x) = (2x - 3) / (x^2 - 3x + 5) on R?

• AF(x) = (x^2 - 3x) / (x^2 - 3x + 5) + C
• BF(x) = arctan(x^2 - 3x + 5) + C
• CF(x) = ln(x^2 - 3x + 5) + C
• DF(x) = 1 / (x^2 - 3x + 5)^2 + C

Compute the definite integral J = integral_{0}^{1} [x / (x^2 + 1)^2] dx.

• A1/2
• B1/8
• Cln(2) / 2
• D1/4

What is the area (in area units) of the planar region bounded by the curve y = e^(2x), the horizontal x-axis, and the vertical lines x = 0 and x = 1?

• Ae^2 - 1
• B(e^2 - 1) / 2
• C(e^2 + 1) / 2
• De^2 / 2

What is the limit of the numerical sequence (u_n)_{n >= 1} defined by u_n = (5n^3 - 2n + 4) / (2n^3 + 7n^2 - 1) as n approaches +infinity?

• A5/2
• B0
• C+infinity
• D-4

A numerical sequence is defined by u_0 = 1 and u_{n+1} = (1/3) * u_n + 4 for all n in N. If the sequence converges, what is its limit L?

• A4
• B12
• C6
• D2

What is the sum of the infinite geometric series S = sum_{k=0}^{+infinity} (2/5)^k?

• A2/3
• B5/2
• C7/5
• D5/3

Consider the sequence u_n = sum_{k=1}^{n} [ 1 / sqrt(n^2 + k) ] for n >= 1. By applying the squeeze theorem (théorème des gendarmes), what is lim_{n -> +infinity} u_n?

• A0
• B1
• C1/2
• D+infinity

What is the exponential form of the complex number z = -sqrt(3) + i?

• A2 * e^(i * 5*pi / 6)
• B2 * e^(i * 2*pi / 3)
• Csqrt(2) * e^(i * 5*pi / 6)
• D2 * e^(-i * pi / 6)

Using De Moivre's formula or exponential representations, calculate the exact value of the complex power (1 - i)^12.

• A64
• B64i
• C-64
• D-64i

In the complex plane, points A and B have affixes z_A = 1 + 2i and z_B = 3 + 4i. What is the affix of the midpoint M of line segment [AB]?

• A4 + 6i
• B2 + 2i
• C1 + i
• D2 + 3i

What is the geometric locus of points M with affix z in the complex plane satisfying the equation |z - 3 + 2i| = 5?

• AThe line passing through (3, -2) perpendicular to the real axis
• BThe circle with center Omega(3, -2) and radius R = 5
• CThe circle with center Omega(-3, 2) and radius R = 5
• DThe circle with center Omega(3, -2) and radius R = 25

In a seed trial, 70% of seeds germinate. Among the germinated seeds, 90% remain free of the observed fungal disease. What is the probability that a randomly selected seed both germinates and remains disease-free?

• A0.07
• B0.30
• C0.63
• D0.90

Find the unique solution of the second-order differential equation y'' + 9y = 0 satisfying y(0) = 2 and y'(0) = 3.

• Ay(x) = 2*cos(3x) + 3*sin(3x)
• By(x) = 2*cos(9x) + sin(9x)
• Cy(x) = 2*cos(3x) + sin(3x)
• Dy(x) = 2*e^(3x) + e^(-3x)

What is the exact real solution to the logarithmic equation ln(x - 2) + ln(x + 1) = ln(4)?

• Ax = 2
• Bx = -2
• Cx = 3 and x = -2
• Dx = 3

What are the coordinates of the inflection point of the cubic curve defined by f(x) = 2x^3 - 9x^2 + 12x - 4?

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

An urn contains 6 black balls and 4 white balls. A student draws 3 balls simultaneously at random. How many distinct drawing combinations contain exactly 2 black balls and 1 white ball?

• A60
• B24
• C120
• D15

A fair standard six-sided die is rolled twice independently. What is the probability that the sum of the two numbers obtained is equal to 8?

• A1/6
• B1/9
• C5/36
• D7/36

In an agronomic trial, an olive tree seed has an independent germination probability of p = 0.8. If 4 seeds are sown under identical conditions, what is the probability that exactly 3 seeds germinate?

• A0.5120
• B0.1024
• C0.2048
• D0.4096

An agricultural cooperative packages olive oil using two production lines, M1 and M2. Line M1 produces 60% of the bottles with a defect rate of 2%, while line M2 produces 40% of the bottles with a defect rate of 5%. A defective bottle is selected at random. According to Bayes' formula, what is the probability that it came from line M1?

• A0.500
• B0.375
• C0.625
• D0.240

Evaluate the convergent series sum from n = 1 to infinity of 1/[n(n+2)].

• A3/4
• B1/2
• C1
• D3/2

Let f(x) = x^3 + 2x - 1 be a strictly increasing bijection from R to R, and let f^(-1) be its reciprocal function. Given that f(1) = 2, what is the value of the derivative (f^(-1))'(2)?

• A5
• B1/2
• C1/5
• D1/3

A sequence is defined by u_0 = 1 and u_{n+1} = sqrt(6 + u_n) for all n in N. Knowing that (u_n) is bounded and convergent, what is its limit L?

• Asqrt(6)
• B6
• C-2
• D3

What are the two complex roots in C of the quadratic equation z^2 - 6z + 25 = 0?

• Az = 6 + 8i and z = 6 - 8i
• Bz = 3 + 4i and z = 3 - 4i
• Cz = -3 + 4i and z = -3 - 4i
• Dz = 3 + 8i and z = 3 - 8i

A discrete random variable X takes values 0, 1, and 2 with probabilities P(X = 0) = 0.2, P(X = 1) = 0.5, and P(X = 2) = 0.3. What is the variance Var(X)?

• A0.49
• B1.10
• C1.70
• D0.70

A transverse mechanical wave propagates along a taut horizontal string with a constant speed v = 20 m/s. What is the time delay tau for the wave disturbance created at the source S to reach a point M situated at distance d = 5.0 m from S?

• A0.10 s
• B0.25 s
• C4.0 s
• D100 s

A sound wave of frequency N = 1000 Hz propagates through ambient air where the speed of sound is v = 340 m/s. What is the spatial period (wavelength lambda) of this wave in air?

• A34 cm
• B3.4 cm
• C2.94 m
• D340 m

A progressive sinusoidal wave has a wavelength of lambda = 8.0 cm. Two points, M1 and M2, located along the propagation line are separated by distance d = 20.0 cm. What is the vibrational state of M1 and M2 relative to each other?

• AThey vibrate strictly in phase
• BThey vibrate in quadrature (phase difference of pi / 2)
• CThey vibrate in phase opposition (opposition de phase)
• DTheir vibrations are completely uncorrelated

A monochromatic laser beam of wavelength lambda = 600 nm illuminates a narrow vertical slit of width a = 0.12 mm. A flat screen is placed parallel to the slit at a distance D = 2.0 m. What is the linear width L of the central diffraction fringe observed on the screen?

• A1.0 cm
• B5.0 mm
• C1.0 mm
• D2.0 cm

A radioactive iodine isotope used in agronomic tracer studies has a radioactive half-life t_{1/2} = 8.0 days. Taking ln(2) approx 0.693, what is the value of its radioactive decay constant lambda?

• A5.54 day^(-1)
• B0.0866 day^(-1)
• C0.0433 day^(-1)
• D11.5 day^(-1)

A radioactive tracer has a half-life of 8 days and an initial activity of 1,600 Bq. What activity remains after 24 days?

• A200 Bq
• B400 Bq
• C50 Bq
• D800 Bq

A metal has a work function of 2.2 eV and is illuminated by photons of energy 3.0 eV. According to Einstein's photoelectric equation, what is the maximum kinetic energy of emitted electrons?

• A5.2 eV
• B2.2 eV
• C0.8 eV
• DNo electrons are emitted

Carbon-14 has a half-life of t_{1/2} = 5730 years. An archaeological wood relic discovered in the Volubilis agricultural plain exhibits a Carbon-14 activity equal to 25% of the activity of modern living wood. What is the estimated age of this relic?

• A2865 years
• B5730 years
• C8595 years
• D11,460 years

An RC series circuit consists of an ideal DC voltage generator of electromotive force E = 12 V, an initially uncharged capacitor of capacitance C = 100 muF, and a resistor R = 10 kOmega. What is the value of the circuit time constant tau?

• A1.0 s
• B0.10 s
• C10 s
• D1.0 ms

What is the electrostatic energy E_C stored in a capacitor of capacitance C = 50 muF when the voltage across its terminals reaches u_C = 12 V?

• A6.0 * 10^(-4) J
• B7.2 * 10^(-3) J
• C3.6 * 10^(-3) J
• D3.0 * 10^(-4) J

An RL series circuit is connected to an ideal DC generator of emf E = 10 V. The inductor has inductance L = 0.20 H and internal resistance r = 5.0 Omega, and the series resistor is R = 45 Omega. What is the steady-state current I_0 established in the circuit?

• A0.22 A
• B0.20 A
• C2.0 A
• D0.50 A

For the RL circuit described previously (L = 0.20 H, series resistor R = 45 Omega, and internal resistance r = 5.0 Omega), what is the circuit time constant tau?

• A40 ms
• B1.0 ms
• C25 ms
• D4.0 ms

An ideal LC oscillator comprises an inductor of inductance L = 10 mH and a capacitor of capacitance C = 2.5 muF. Taking pi^2 approx 10, what is the natural oscillation period T_0 of the circuit?

• A1.0 ms
• B3.14 ms
• C0.50 ms
• D10 ms

In an ideal undamped LC circuit, which of the following statements correctly describes the distribution of total electromagnetic energy E_T when the electric charge q(t) on the capacitor is momentarily zero?

• AThe total electromagnetic energy is zero
• BAll energy is stored as magnetic energy in the inductor: E_T = (1/2) * L * I_max^2
• CAll energy is stored as electrostatic energy in the capacitor: E_T = (1/2) * C * U_max^2
• DThe energy is equally divided between the capacitor and inductor

A crate of mass m = 2.0 kg slides down a straight plane inclined at angle alpha = 30 degrees to the horizontal. The coefficient of kinetic friction between the crate and the plane is mu = 0.20. Taking g = 9.8 m/s^2, sin(30°) = 0.500, and cos(30°) = 0.866, what is the acceleration a of the crate?

• A4.90 m/s^2
• B1.70 m/s^2
• C3.20 m/s^2
• D6.60 m/s^2

A projectile is launched from ground level with initial speed v_0 = 20 m/s at an elevation angle alpha = 45 degrees over horizontal ground in a uniform gravitational field with g = 10 m/s^2. Neglecting air resistance, what is the horizontal range R of the projectile?

• A40 m
• B20 m
• C80 m
• D10 m

An agricultural cart of mass m = 50 kg starts from rest and descends a frictionless curved ramp from an initial height h = 5.0 m above the ground. Taking g = 10 m/s^2, what is its speed v upon reaching ground level?

• A5.0 m/s
• B50 m/s
• C25 m/s
• D10 m/s

An agricultural remote-sensing satellite revolves in a circular orbit of radius r around the Earth (mass M). Which of the following relationships correctly describes its orbital speed v and orbital period T according to Newton's universal law of gravitation?

• Av = sqrt(G * M * r) and T = 2*pi*r / v
• Bv = sqrt(G * M / r) and T^2 / r^3 = 4*pi^2 / (G * M)
• Cv = G * M / r^2 and T = 2*pi*r * v
• Dv = sqrt(G * M / r^2) and T^3 / r^2 = 4*pi^2 / (G * M)