Mali Baccalauréat Série TSE (Sciences Exactes)

Given the complex number z = 1 + i*sqrt(3), what is the trigonometric (polar) form of z?

• A2 * (cos(pi/3) + i * sin(pi/3))
• B2 * (cos(pi/6) + i * sin(pi/6))
• Csqrt(2) * (cos(pi/3) + i * sin(pi/3))
• D4 * (cos(pi/3) + i * sin(pi/3))

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

• A16
• B16i
• C-16
• D8 + 8i

In the complex plane, what is the geometric locus of points M representing complex numbers z such that |z - 2 + 3i| = 5?

• AA circle centered at Omega(2, -3) with radius R = 5
• BA circle centered at Omega(-2, 3) with radius R = 5
• CA straight line perpendicular to the segment from (2, 0) to (0, -3)
• DA circle centered at Omega(2, -3) with radius R = 25

For a real constant a, lim_{x->0} (e^(ax) - 1) / x = 6. What is a?

• A6
• B1/6
• Ce^6
• D0

What is the derivative of the function f(x) = ln(x^2 + 1) for all real x?

• A2x / (x^2 + 1)
• B1 / (x^2 + 1)
• C2 / (x^2 + 1)
• Dx / (x^2 + 1)

What is the general solution of the differential equation y' - 3y = 0 on R?

• Ay(x) = C * e^(3x), where C is a real constant
• By(x) = C * e^(-3x), where C is a real constant
• Cy(x) = 3x + C, where C is a real constant
• Dy(x) = C * e^(x/3), where C is a real constant

What is the particular solution to the differential equation y'' + 4y = 0 satisfying y(0) = 1 and y'(0) = 2?

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

Evaluate the definite integral I = int_0^1 (2x + 1) * e^(x^2 + x) dx.

• Ae^2 - 1
• Be^2
• Ce^2 - e
• D(e^2 - 1) / 2

Using integration by parts, evaluate J = int_1^e x * ln(x) dx.

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

What is the remainder of 3^100 when divided by 7?

• A4
• B2
• C1
• D3

Using the Euclidean algorithm, what is the greatest common divisor (GCD) of 348 and 124?

• A4
• B2
• C8
• D12

According to Bezout's theorem, two integers a and b are coprime if and only if there exist integers u and v such that:

• Aa*u + b*v = 1
• Ba*u - b*v = 0
• Ca*u + b*v = gcd(a, b)^2
• Da / u + b / v = 1

In 3-dimensional space, what is the Cartesian equation of the plane P passing through point A(1, -2, 3) with normal vector n(2, 3, -1)?

• A2x + 3y - z + 7 = 0
• B2x + 3y - z - 7 = 0
• C2x - 2y + 3z - 7 = 0
• Dx - 2y + 3z - 14 = 0

What is the distance from point M(1, 2, -1) to the plane P with equation 2x - y + 2z - 5 = 0?

• A7 / 3
• B2
• C5 / 3
• D3

What is the dot product (scalar product) of vectors u(1, -3, 2) and v(4, 2, 1) in an orthonormal coordinate system?

• A0
• B1
• C-1
• D12

A random variable X follows a binomial distribution B(n = 10, p = 0.3). What is the expected value (mean) E(X) and variance V(X)?

• AE(X) = 3 and V(X) = 2.1
• BE(X) = 3 and V(X) = 0.9
• CE(X) = 3 and V(X) = 3
• DE(X) = 0.3 and V(X) = 0.21

An urn contains 5 red balls and 3 green balls. Two balls are drawn simultaneously at random. What is the probability of drawing two balls of different colors?

• A15 / 28
• B13 / 28
• C15 / 56
• D5 / 14

Given two events A and B such that P(A) = 0.4, P(B) = 0.5, and P(A cap B) = 0.2, what is the conditional probability P(A | B)?

• A0.40
• B0.50
• C0.20
• D0.80

What is the horizontal asymptote of the function f(x) = (3x^2 - 5x + 2) / (x^2 + 4) as x tends to +infinity?

• Ay = 3
• By = 0
• Cy = -5
• Dx = 3

What are the inflection points of the curve representing f(x) = x^3 - 3x^2 + 2?

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

What is the sum of the infinite geometric series S = 1 + 1/3 + 1/9 + 1/27 + ...?

• A3 / 2
• B4 / 3
• C2
• D3

Let f(x) = ln(x) / x on ]0, +infinity[. What is the coordinates of the absolute maximum of f?

• A(e, 1/e)
• B(1, 0)
• C(e^2, 2/e^2)
• D(sqrt(e), 1/(2*sqrt(e)))

In the set of complex numbers C, what are the solutions to the quadratic equation z^2 - 2z + 5 = 0?

• Az = 1 + 2i and z = 1 - 2i
• Bz = 2 + 4i and z = 2 - 4i
• Cz = -1 + 2i and z = -1 - 2i
• Dz = 1 + 4i and z = 1 - 4i

If a sequence (u_n) is defined by u_0 = 2 and u_{n+1} = 3*u_n - 4 for all n >= 0, what is the explicit expression of u_n in terms of n?

• Au_n = 2
• Bu_n = 2 * 3^n
• Cu_n = 3^n + 1
• Du_n = 2 * 3^n - 4n

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

• A2
• B0
• C+infinity
• D4

A projectile is launched from ground level with an initial velocity v_0 = 20 m/s at an angle alpha = 30 degrees above the horizontal. Taking g = 10 m/s^2 and neglecting air resistance, what is the maximum height H reached by the projectile?

• A5.0 m
• B10.0 m
• C20.0 m
• D2.5 m

A satellite of mass m describes a uniform circular orbit of radius r around the Earth (mass M_T). Taking G as Newton's universal gravitational constant, what is the orbital speed v of the satellite?

• Asqrt(G * M_T / r)
• Bsqrt(G * M_T / r^2)
• CG * M_T / r
• Dsqrt(2 * G * M_T / r)

Kepler's third law states that for any planet orbiting the Sun, T^2 / a^3 is constant. What is this constant in terms of the solar mass M_S and gravitational constant G?

• A4 * pi^2 / (G * M_S)
• B2 * pi / (G * M_S)
• CG * M_S / (4 * pi^2)
• D4 * pi^2 * G / M_S

A proton with elementary charge q = 1.6 * 10^-19 C enters perpendicularly into a uniform magnetic field B = 0.50 T. If the magnetic force acting on the proton is F = 1.6 * 10^-13 N, what is the speed v of the proton?

• A2.0 * 10^6 m/s
• B1.0 * 10^6 m/s
• C4.0 * 10^6 m/s
• D5.0 * 10^5 m/s

An ideal solenoid of length L = 0.50 m has N = 1000 turns and carries a direct current I = 2.0 A. Given mu_0 = 4*pi * 10^-7 T*m/A, what is the magnitude of the magnetic field B inside the solenoid?

• A5.0 * 10^-3 T (5.0 mT)
• B2.5 * 10^-3 T (2.5 mT)
• C1.0 * 10^-2 T (10 mT)
• D1.26 * 10^-4 T (0.126 mT)

A flat circular coil with N = 100 turns and cross-sectional area S = 0.020 m^2 is placed perpendicular to a uniform magnetic field. The magnetic field decreases uniformly from 0.80 T to 0.20 T in Delta t = 0.30 s. What is the induced electromotive force (EMF) magnitude |e| in the coil?

• A4.0 V
• B2.0 V
• C0.40 V
• D40 V

An inductor with self-inductance L = 0.20 H carries an electric current I = 3.0 A. What is the magnetic energy E_m stored in this inductor?

• A0.90 J
• B1.80 J
• C0.60 J
• D0.30 J

In a series RL circuit subjected to a voltage step, the coil has inductance L = 50 mH and the circuit has total resistance R = 25 ohms. What is the time constant tau of the circuit?

• A2.0 ms
• B0.50 ms
• C1.25 s
• D50 ms

A series RLC circuit comprises a resistor R = 50 ohms, an inductor L = 0.10 H, and a capacitor C = 10 microfarads (10 * 10^-6 F). What is the angular resonance frequency omega_0 of the circuit?

• A1000 rad/s
• B100 rad/s
• C318 rad/s
• D10,000 rad/s

At the resonance frequency omega_0 in a series RLC circuit, what is the value of the total impedance Z of the circuit?

• AZ = R (impedance is purely resistive and minimal)
• BZ = 0 (zero impedance)
• CZ = infinity (maximum impedance)
• DZ = sqrt(R^2 + (L * omega_0)^2)

A frictionless horizontal spring-mass system consists of a mass m = 0.25 kg attached to a spring of spring constant k = 100 N/m. What is the natural period of oscillation T_0? (Take pi ≈ 3.14)

• A0.314 s (0.10*pi s)
• B0.628 s (0.20*pi s)
• C3.14 s (1.0*pi s)
• D0.157 s (0.05*pi s)

A simple pendulum has an oscillation period of T = 2.0 s at a location where g = 9.80 m/s^2. Taking pi^2 ≈ 9.87, what is the length L of the pendulum string?

• A0.99 m
• B0.50 m
• C1.98 m
• D0.25 m

In a Young's double-slit experiment, monochromatic light of wavelength lambda = 600 nm (6.0 * 10^-7 m) illuminates two slits separated by distance a = 0.30 mm (3.0 * 10^-4 m). The screen is at distance D = 1.5 m. What is the fringe width (interfrange i)?

• A3.0 mm
• B1.5 mm
• C6.0 mm
• D0.30 mm

A light ray in air (n_1 = 1.00) enters a glass prism (n_2 = 1.50) with an angle of incidence i_1 = 30 degrees. According to Snell-Descartes law of refraction, what is sin(i_2)?

• A1 / 3 (approximately 0.333)
• B1 / 2 (0.500)
• C3 / 4 (0.750)
• D2 / 3 (approximately 0.667)

According to Einstein's photoelectric equation, what is the maximum kinetic energy E_c,max of photoelectrons emitted from a metal with extraction work function W_0 when illuminated by light of frequency nu?

• AE_c,max = h * nu - W_0
• BE_c,max = h * nu + W_0
• CE_c,max = W_0 - h * nu
• DE_c,max = (h * nu) / W_0

A radioactive isotope has a half-life of T = 8.0 days. If the initial activity of a sample is A_0 = 1600 Bq, what is its activity after 24 days?

• A200 Bq
• B400 Bq
• C100 Bq
• D533 Bq

The mass defect Delta m of a helium-4 nucleus (^4_2He) is 0.0304 u. Taking 1 atomic mass unit (u) equivalent to 931.5 MeV/c^2, what is the total binding energy E_l of this nucleus?

• A28.3 MeV
• B7.08 MeV
• C931.5 MeV
• D113.2 MeV

During beta-minus (beta-) radioactive decay, an unstable nucleus emits an electron and an antineutrino. Which nucleon conversion occurs inside the nucleus?

• AA neutron transforms into a proton: n -> p + e^- + anti-nu_e
• BA proton transforms into a neutron: p -> n + e^+ + nu_e
• CTwo protons fuse into an alpha particle
• DA proton absorbs an electron without emitting any neutrino

Radium-226 (^226_88Ra) undergoes spontaneous alpha decay (^4_2He). According to Soddy's conservation laws for nucleons (A) and charge (Z), what is the resulting daughter nucleus?

• A^222_86Rn (Radon-222)
• B^222_87Fr (Francium-222)
• C^230_90Th (Thorium-230)
• D^226_89Ac (Actinium-226)

For the stoichiometric reaction A + 2B -> C in aqueous solution, how is the volumetric rate of consumption of reactant B related to the volumetric rate of formation of product C?

• A-d[B]/dt = 2 * (d[C]/dt)
• B-d[B]/dt = d[C]/dt
• C-d[B]/dt = (1/2) * (d[C]/dt)
• D-d[B]/dt = 4 * (d[C]/dt)

Which combination of experimental modifications will systematically accelerate the rate of a homogeneous liquid-phase chemical reaction?

• AIncreasing temperature and increasing reactant initial concentrations
• BDecreasing temperature and diluting the reaction mixture
• CDecreasing reactant concentrations while cooling the solution
• DAdding an inert solvent while keeping temperature constant

What is the pH of an aqueous solution of hydrochloric acid (HCl, a strong monoprotic acid) with concentration C_a = 2.5 * 10^-3 mol/L at 25 degrees C? (Take log(2.5) ≈ 0.40)

• A2.60
• B3.40
• C1.60
• D3.00

What is the pH of an aqueous sodium hydroxide solution (NaOH, strong base) with concentration C_b = 1.0 * 10^-2 mol/L at 25 degrees C (where K_e = 1.0 * 10^-14)?

• A12.0
• B2.0
• C7.0
• D14.0