Baccalauréat Mauritanien (Mauritania)

What is the derivative of the real-valued function f(x) = x * ln(x) - x for all x > 0?

• Af'(x) = ln(x)
• Bf'(x) = ln(x) - 1
• Cf'(x) = 1/x
• Df'(x) = ln(x) + 1

What is the limit as x approaches 0 of the expression (e^x - 1) / x?

• A0
• B1
• Ce
• DThe limit does not exist (diverges to infinity)

What is the unique solution to the initial value problem y' - 3y = 0 with the initial condition y(0) = 5?

• Ay(x) = 5 * e^(-3x)
• By(x) = 3 * e^(5x)
• Cy(x) = 5 * e^(3x)
• Dy(x) = 5 + 3x

What is the value of the definite integral ∫_{1}^{e} (ln(x) / x) dx?

• A1
• Be - 1
• C2
• D1/2

What are the modulus and principal argument of the complex number z = 1 + i*sqrt(3)?

• AModulus = 2, Argument = pi/3
• BModulus = 2, Argument = pi/6
• CModulus = 4, Argument = pi/3
• DModulus = sqrt(2), Argument = 2*pi/3

What are the complex roots of the quadratic equation z^2 - 2z + 4 = 0 in the set of complex numbers C?

• Az = 1 ± 3i
• Bz = 1 ± i*sqrt(3)
• Cz = 2 ± 2i*sqrt(3)
• Dz = -1 ± i*sqrt(3)

In the complex plane, a transformation f associates to each point M of affix z the point M' of affix z' = i*z + 2 - i. What is the nature and geometric characteristic of f?

• AA translation of vector u(2, -1)
• BA homothety of ratio 1 and center (1, 1)
• CA rotation of angle pi/2 and center Omega with affix (3 + i)/2
• DA reflection across the real axis

An urn contains 4 red balls, 3 green balls, and 5 blue balls. If 2 balls are drawn simultaneously at random, what is the probability that both drawn balls are red?

• A1/22
• B1/6
• C2/11
• D1/11

A discrete random variable X follows a binomial distribution B(n = 5, p = 0.4). What is the mathematical expectation E(X)?

• A2.0
• B1.2
• C2.5
• D0.8

Let A and B be two events such that P(B) = 0.50 and P(A ∩ B) = 0.15. What is the conditional probability P(A | B)?

• A0.075
• B0.30
• C0.65
• D0.35

What is the abscissa (x-coordinate) of the inflection point of the curve representing f(x) = x^3 - 3x^2 + 4?

• Ax = 0
• Bx = 2
• Cx = 1
• Dx = -1

The function f is defined for x ≠ 2 by f(x) = (3x^2 - 2x + 4) / (x - 2). What is the equation of the oblique asymptote to its curve as x approaches +infinity?

• Ay = 3x - 2
• By = 3x
• Cy = 3x + 12
• Dy = 3x + 4

What is the greatest common divisor gcd(132, 84)?

• A12
• B6
• C24
• D4

What is the remainder of 3^2026 when divided by 7 (i.e., 3^2026 modulo 7)?

• A2
• B4
• C1
• D6

In an orthonormal frame of space, what is the distance from the point A(1, 2, 3) to the plane P with Cartesian equation 2x - y + 2z - 1 = 0?

• A5/9
• B3
• C5/3
• D7/3

What is the cross product u x v of vectors u = (1, 0, 2) and v = (0, 3, -1)?

• A(6, -1, -3)
• B(-6, -1, 3)
• C(6, 1, -3)
• D(-6, 1, 3)

A real sequence (u_n) is defined by u_0 = 1 and the recurrence relation u_{n+1} = (1/2) * u_n + 3 for all n >= 0. What is the limit L of this sequence as n approaches infinity?

• A6
• B3
• C2
• DThe sequence diverges to +infinity

Under what condition does the Intermediate Value Theorem (Théorème des valeurs intermédiaires / TVI corollary) guarantee that the equation f(x) = 0 has a UNIQUE solution on an interval [a, b]?

• Af is continuous and periodic on [a, b]
• Bf is continuous, strictly monotonic on [a, b], and f(a) * f(b) < 0
• Cf is differentiable with f'(x) = 0 for some x in [a, b]
• Df is bounded and positive on [a, b]

What is the value of the definite integral ∫_{0}^{1} x * e^x dx computed using integration by parts?

• Ae
• Be - 1
• C1
• D2

What is the value of the complex number (1 + i)^8?

• A16i
• B-16
• C-16i
• D16

Which of the following pairs of integers (x, y) is a particular solution to the linear Diophantine equation 5x + 3y = 1?

• A(2, -3)
• B(-2, 3)
• C(1, -1)
• D(3, -4)

Let f be a differentiable, strictly monotonic function on an interval I with f'(x) != 0 for all x in I. If f(2) = 5 and f'(2) = 4, what is the value of the derivative of the inverse function (f^(-1))'(5)?

• A-4
• B1/4
• C1/5
• D4

What is the sum of the infinite convergent geometric series S = ∑_{k=0}^{infinity} (1/3)^k = 1 + 1/3 + 1/9 + 1/27 + ...?

• A1/2
• B3
• C3/2
• D4/3

In 3D Cartesian space, what are the center Omega and radius R of the sphere defined by the equation x^2 + y^2 + z^2 - 4x + 2y - 6z + 5 = 0?

• AOmega(4, -2, 6) and R = 5
• BOmega(2, -1, 3) and R = 9
• COmega(-2, 1, -3) and R = 3
• DOmega(2, -1, 3) and R = 3

What is the theoretical variance Var(X) of a continuous random variable X uniformly distributed over the interval [a, b]?

• A(b - a)^2 / 12
• B(b - a) / 2
• C(b + a)^2 / 12
• D(b - a)^2 / 6

A projectile is launched from ground level with an initial velocity v0 at an angle alpha relative to the horizontal across flat ground in a uniform gravitational field g (neglecting air resistance). At what launch angle alpha is the maximum horizontal range achieved?

• Aalpha = 30°
• Balpha = 45°
• Calpha = 60°
• Dalpha = 90°

A simple pendulum of length L has an oscillation period T0 for small angular amplitudes. If the length of the string is quadrupled to 4L while keeping g constant, what will be the new period T'?

• AT' = 4 * T0
• BT' = T0 / 2
• CT' = 2 * T0
• DT' = 16 * T0

A vehicle of mass m = 1,000 kg travelling along a horizontal road at v1 = 20 m/s brakes and comes to a complete stop (v2 = 0 m/s). What is the total work done by the braking resistive forces?

• A+200 kJ
• B-400 kJ
• C-100 kJ
• D-200 kJ

A particle with positive charge q and mass m enters a uniform magnetic field B with a velocity v perpendicular to the magnetic field lines. What is the radius R of its circular trajectory?

• AR = (m * v) / (q * B)
• BR = (q * B) / (m * v)
• CR = (m * B) / (q * v)
• DR = (q * v) / (m * B)

A capacitor of capacitance C initially uncharged is connected in series with a resistor R across an ideal DC voltage source of electromotive force E. What is the voltage across the capacitor at time t = tau = R * C?

• Au_C = 0.50 * E
• Bu_C ≈ 0.63 * E
• Cu_C ≈ 0.37 * E
• Du_C = 0.99 * E

An ideal inductor of inductance L = 0.5 H carries a steady direct current I = 4 A. What is the magnetic energy stored in the inductor?

• A1.0 J
• B2.0 J
• C4.0 J
• D8.0 J

In a series RLC circuit driven by a sinusoidal AC voltage source, what happens at electrical resonance (angular frequency omega0 = 1 / sqrt(L * C))?

• AThe circuit current reaches a minimum value
• BThe total impedance of the circuit becomes infinite
• CThe phase difference between voltage and current is pi/2
• DThe total impedance of the circuit is minimal and equal to R

In a Young's double-slit interference experiment with monochromatic light of wavelength lambda, slit separation a, and screen distance D, the fringe width is i = (lambda * D) / a. What happens to the fringe width if the distance D between the slits and the screen is doubled?

• AThe fringe width is doubled
• BThe fringe width is halved
• CThe fringe width is quadrupled
• DThe fringe width remains unchanged

A photon of yellow light has a wavelength in vacuum of lambda = 589 nm (5.89 x 10^(-7) m). Given Planck's constant h = 6.63 x 10^(-34) J*s and c = 3.00 x 10^8 m/s, what is the energy of this photon in Joules?

• A1.16 x 10^(-19) J
• B3.38 x 10^(-19) J
• C5.89 x 10^(-19) J
• D2.11 x 10^(-18) J

A radioactive sample contains N0 nuclei at t = 0 and decays according to N(t) = N0 * e^(-lambda * t), where lambda is the radioactive decay constant. What is the relationship between the radioactive half-life T_{1/2} and the decay constant lambda?

• AT_{1/2} = lambda / ln(2)
• BT_{1/2} = 1 / lambda
• CT_{1/2} = ln(2) / lambda
• DT_{1/2} = 2 * ln(lambda)

A satellite of mass m moves in a uniform circular orbit of radius r around a planet of mass M. Applying Newton's second law and universal gravitation F = G * M * m / r^2, which of the following is Kepler's third law for this circular orbit?

• AT^2 / r^2 = 4*pi^2 / (G * M)
• BT^3 / r^2 = 2*pi / (G * M)
• CT / r^3 = 4*pi^2 * G * M
• DT^2 / r^3 = 4*pi^2 / (G * M)

A tight string of length L is fixed at both ends. For the fundamental standing wave mode (first harmonic), what is the relationship between the wavelength lambda1 and the length of the string L?

• Alambda1 = 2 * L
• Blambda1 = L
• Clambda1 = L / 2
• Dlambda1 = 4 * L

For a chemical reaction a A + b B -> products, how is the volumetric rate of disappearance of reactant A defined mathematically at any instant t in a constant volume V?

• Av_disp(A) = +(1/a) * d[A]/dt
• Bv_disp(A) = -d[A]/dt
• Cv_disp(A) = -[A] / t
• Dv_disp(A) = (d[A]/dt)^2

What is the effect of adding a catalyst to a reversible chemical reaction at constant temperature?

• AIt shifts the chemical equilibrium toward greater product yield
• BIt increases the enthalpy change Delta H of the reaction
• CIt lowers the activation energy of both forward and reverse reactions without altering th…
• DIt permanently increases the standard Gibbs free energy change Delta G°

Consider the Haber-Bosch synthesis of ammonia: N2(g) + 3 H2(g) ⇌ 2 NH3(g) (Delta H < 0, exothermic). According to Le Chatelier's principle, which change will shift the equilibrium position to produce MORE ammonia?

• AIncreasing the temperature
• BDecreasing the total pressure
• CRemoving nitrogen gas N2
• DIncreasing the total pressure

What is the pH of an aqueous solution of hydrochloric acid (HCl, a strong monoprotic acid) with a concentration of C = 1.0 x 10^(-3) mol/L at 25 °C?

• A3.0
• B1.0
• C11.0
• D7.0

According to the Henderson-Hasselbalch equation pH = pKa + log10([A-] / [HA]), what is the pH of an equimolar buffer solution containing equal concentrations of acetic acid (HA) and sodium acetate (A-)?

• ApH = 7.0
• BpH = pKa
• CpH = pKa + 1
• DpH = 14 - pKa

During the titration of 20 mL of 0.10 mol/L ethanoic acid (CH3COOH, weak acid, pKa = 4.8) by 0.10 mol/L sodium hydroxide (NaOH, strong base), what is the pH at the equivalence point?

• ApH < 7 (acidic)
• BpH = 7.0 (strictly neutral)
• CpH > 7 (basic)
• DpH = 4.8

What are the primary kinetic and thermodynamic characteristics of the direct Fischer esterification reaction between a carboxylic acid and an alcohol (R-COOH + R'-OH ⇌ R-COOR' + H2O)?

• AFast, complete (total), and strongly exothermic
• BFast, limited, and endothermic
• CSlow, complete, and athermic
• DSlow, limited (reversible reaching equilibrium), and athermic

What reaction occurs during the saponification of a fatty ester (triglyceride) with sodium hydroxide (NaOH)?

• ATotal and irreversible alkaline hydrolysis producing soap (fatty acid carboxylate salt) a…
• BReversible esterification producing wax and water
• CAddition of hydrogen across double bonds producing margarine
• DCombustion producing carbon dioxide and soot

What is the product obtained when a primary alcohol (R-CH2OH, such as ethanol) undergoes mild controlled oxidation with a limited amount of acidified potassium dichromate (the oxidant in default)?

• AA ketone
• BAn aldehyde
• CAn alkene
• DAn ether

Two point masses m1 = 2 kg and m2 = 6 kg are located on the x-axis at x1 = 0 m and x2 = 8 m, respectively. What is the coordinate xG of their center of mass G?

• AxG = 4 m
• BxG = 8 m
• CxG = 6 m
• DxG = 2 m

According to Faraday's law and Lenz's law, how is the induced electromotive force e across a conducting circuit related to the magnetic flux Phi through the circuit?

• Ae = +dPhi/dt; induced currents reinforce the flux change
• Be = Phi * t; induced currents have constant magnitude
• Ce = -Phi / dt; induced currents depend on circuit resistance only
• De = -dPhi/dt; induced currents oppose the change in magnetic flux that produced them