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Bac Techniques Mathématiques (Algeria) - independent subject practice study materials
Bac Techniques Mathématiques (Algeria) - independent subject practice study resources

Bac Techniques Mathématiques (Algeria) - independent subject practice practice questions

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Question 1Calculus

Evaluate the limit lim(x→+∞) (5x³ − 2x + 1) / (2x³ + 7x² − 4).

  1. 0
  2. 5/2 Correct answer.
  3. 5/7
  4. +∞

Explanation

For a rational function as x → +∞, the limit equals the limit of the ratio of the highest-degree terms: lim(x→+∞) (5x³)/(2x³) = 5/2. Factor out x³ from both numerator and denominator to see that lower-degree terms (2/x², 1/x³, 7/x, 4/x³) approach zero.

Question 2Calculus

What is the limit of the function f(x) = (eˣ − 2x) / (eˣ + 1) as x → +∞?

  1. 0
  2. 2
  3. 1 Correct answer.
  4. +∞

Explanation

Factoring out eˣ from the numerator and denominator yields f(x) = [1 − 2(x/eˣ)] / [1 + e⁻ˣ]. By standard growth hierarchies (croissances comparées), lim(x→+∞) (x/eˣ) = 0 and lim(x→+∞) e⁻ˣ = 0, giving (1 − 0)/(1 + 0) = 1.

Question 3Calculus

Let f(x) = x ln(x) − x be defined on ]0, +∞[. What is the equation of the tangent line to the curve of f at the point with abscissa x = 1?

  1. y = -1 Correct answer.
  2. y = x - 1
  3. y = x - 2
  4. y = 0

Explanation

Differentiating gives f'(x) = 1·ln(x) + x·(1/x) − 1 = ln(x) + 1 − 1 = ln(x). At x = 1, the slope is f'(1) = ln(1) = 0 and the y-coordinate is f(1) = 1·0 − 1 = -1. The tangent line equation is y = f'(1)(x − 1) + f(1) = 0·(x − 1) − 1, which simplifies to the horizontal line y = -1.

Question 4Calculus

Consider the function f(x) = (ln x)² − ln x defined on ]0, +∞[. At which abscissa does the curve of f possess an inflection point?

  1. x = e
  2. x = √e
  3. x = e²
  4. x = e^(3/2) Correct answer.

Explanation

The first derivative is f'(x) = [2 ln(x) − 1] / x. Differentiating again using the quotient rule gives f''(x) = [(2/x)·x − (2 ln(x) − 1)·1] / x² = [3 − 2 ln(x)] / x². Setting f''(x) = 0 yields 3 − 2 ln(x) = 0, so ln(x) = 3/2, which gives x = e^(3/2) where f''(x) changes sign from positive to negative.

Question 5Calculus

What is the exact value of the definite integral I = ∫₁ᵉ x ln(x) dx?

  1. (e² − 1) / 2
  2. (e² − 1) / 4
  3. (2e² + 1) / 4
  4. (e² + 1) / 4 Correct answer.

Explanation

Integrate by parts with u(x) = ln(x) ⇒ u'(x) = 1/x, and v'(x) = x ⇒ v(x) = x²/2. Then I = [(x²/2) ln(x)]₁ᵉ − ∫₁ᵉ (x²/2)(1/x) dx = (e²/2 · 1 − 0) − (1/2) [x²/2]₁ᵉ = e²/2 − (e² − 1)/4 = (2e² − e² + 1)/4 = (e² + 1)/4.

Question 6Calculus

Find the unique solution to the differential equation y' + 2y = 4 satisfying the initial condition y(0) = 5.

  1. y(x) = 5e⁻²ˣ
  2. y(x) = 3e⁻²ˣ + 2 Correct answer.
  3. y(x) = 2e⁻²ˣ + 3
  4. y(x) = 3e²ˣ + 2

Explanation

The homogeneous solution is y_h(x) = C e⁻²ˣ. A particular constant solution to y' + 2y = 4 is y_p = 4/2 = 2. The general solution is y(x) = C e⁻²ˣ + 2. Applying y(0) = C + 2 = 5 gives C = 3, yielding y(x) = 3e⁻²ˣ + 2.

Question 7Calculus

What is the general solution of the second-order linear differential equation y'' + 9y = 0 on ℝ, where A and B are real constants?

  1. y(x) = A e³ˣ + B e⁻³ˣ
  2. y(x) = (Ax + B) e³ˣ
  3. y(x) = A cos(3x) + B sin(3x) Correct answer.
  4. y(x) = A cos(9x) + B sin(9x)

Explanation

The characteristic equation is r² + 9 = 0, which has pure imaginary roots r = ±3i. For complex roots α ± iβ with α = 0 and β = 3, the general real solution is y(x) = e^(0x) [A cos(3x) + B sin(3x)] = A cos(3x) + B sin(3x).

Question 8Calculus

Let (uₙ) be an arithmetic sequence defined for n ≥ 0 by uₙ = 3n + 2. What is the sum of the first 21 terms, S₂₀ = u₀ + u₁ + ... + u₂₀?

  1. 672 Correct answer.
  2. 640
  3. 714
  4. 602

Explanation

The sequence has initial term u₀ = 3(0) + 2 = 2, common difference d = 3, and last term u₂₀ = 3(20) + 2 = 62. The number of terms from 0 to 20 is 21. Using the arithmetic sum formula: S₂₀ = (number of terms / 2) · (first term + last term) = 21 · (2 + 62) / 2 = 21 · 32 = 672.

Question 9Calculus

Consider the sequence defined by u₀ = 5 and u_{n+1} = √(uₙ + 2) for n ≥ 0. In proving by induction that uₙ > 2 for all n ≥ 0, which statement correctly justifies the inductive step?

  1. Showing that u_{k+1} − u_k = √(u_k + 2) − u_k > 0 by computing its derivative
  2. Assuming u_k < 2 and showing by contradiction that u₀ cannot equal 5
  3. Showing that since u₀ = 5 and u₁ = √7 ≈ 2.65, the sequence is strictly arithmetic
  4. Assuming u_k > 2, adding 2 gives u_k + 2 > 4, and taking the strictly increasing square root gives u_{k+1} = √(u_k + 2) > √4 = 2 Correct answer.

Explanation

In mathematical induction, the inductive step assumes the property holds for index k: u_k > 2. Adding 2 to both sides gives u_k + 2 > 4. Because the square root function is strictly increasing on [0, +∞[, applying it preserves the strict inequality: √(u_k + 2) > √4 = 2. Since u_{k+1} = √(u_k + 2), this establishes u_{k+1} > 2.

Question 10Calculus

Let f(x) = (2x − 3) e⁻ˣ be defined on ℝ. What is the derivative f'(x)?

  1. f'(x) = (2x - 1) e⁻ˣ
  2. f'(x) = (5 - 2x) e⁻ˣ Correct answer.
  3. f'(x) = (-2x + 1) e⁻ˣ
  4. f'(x) = 2 e⁻ˣ

Explanation

Apply the product rule (uv)' = u'v + uv' with u(x) = 2x − 3 and v(x) = e⁻ˣ: f'(x) = 2 · e⁻ˣ + (2x − 3) · (-e⁻ˣ) = [2 − (2x − 3)] e⁻ˣ = (2 − 2x + 3) e⁻ˣ = (5 − 2x) e⁻ˣ.

Question 11Calculus

According to the Monotone Convergence Theorem (théorème de la convergence monotone), which condition guarantees that a real numerical sequence (uₙ) converges to a finite real limit L?

  1. (uₙ) is strictly positive for all n ≥ 0
  2. (uₙ) is alternating in sign and bounded
  3. (uₙ) is increasing and bounded from above, or decreasing and bounded from below Correct answer.
  4. (uₙ) is an arithmetic sequence with positive common difference

Explanation

The fundamental Monotone Convergence Theorem states that every sequence that is monotonically increasing and bounded from above converges to a finite real limit (equal to its supremum), and every sequence that is monotonically decreasing and bounded from below converges to a finite real limit (equal to its infimum).

Question 12Algebra and Functions

Consider the polynomial function f(x) = x³ + 3x − 5 on the interval [1, 2]. Which condition guarantees that the equation f(x) = 0 has a unique solution on [1, 2]?

  1. f is continuous and strictly increasing on [1, 2], with f(1) = -1 < 0 and f(2) = 9 > 0 Correct answer.
  2. f is continuous on [1, 2] and f'(x) changes sign between x = 1 and x = 2
  3. f(1) and f(2) have the same sign, which guarantees an even number of solutions
  4. f is a polynomial of degree 3, which must always have three real roots on [1, 2]

Explanation

f is continuous on [1, 2] and its derivative f'(x) = 3x² + 3 > 0 for all x ∈ ℝ, making f strictly increasing. Since f(1) = -1 < 0 and f(2) = 9 > 0, the Intermediate Value Theorem and strict monotonicity together prove that f(x) = 0 has exactly one unique real root in ]1, 2[.

Question 13Algebra and Functions

Let f(x) = (2x² + 5x + 1) / (x + 1) defined for x ≠ -1. What is the equation of the oblique asymptote to the curve of f at ±∞?

  1. y = 2x + 3 Correct answer.
  2. y = 2x + 5
  3. y = 2x - 1
  4. y = x + 2

Explanation

Perform polynomial long division: 2x² + 5x + 1 = (x + 1)(2x + 3) − 2. Thus f(x) = 2x + 3 − 2/(x + 1). Since lim(x→±∞) [-2/(x + 1)] = 0, the line y = 2x + 3 is an oblique asymptote to the curve of f at both +∞ and -∞.

Question 14Algebra and Functions

In the complex number field ℂ, what is the algebraic form a + bi of the quotient z = (1 + 2i) / (3 − 4i)?

  1. 1/5 + (2/5)i
  2. -1/25 + (10/25)i
  3. -1/5 - (2/5)i
  4. -1/5 + (2/5)i Correct answer.

Explanation

Multiply numerator and denominator by the complex conjugate of the denominator, 3 + 4i: z = [(1 + 2i)(3 + 4i)] / [(3 − 4i)(3 + 4i)] = [3 + 4i + 6i + 8i²] / [9 + 16]. Since i² = -1, the numerator is 3 + 10i − 8 = -5 + 10i. Dividing by 25 gives -5/25 + (10/25)i = -1/5 + (2/5)i.

Question 15Algebra and Functions

What are the modulus r and principal argument θ of the complex number z = -√3 + i?

  1. r = 2 and θ = π/6
  2. r = 2 and θ = 5π/6 Correct answer.
  3. r = 4 and θ = 5π/6
  4. r = 2 and θ = -π/6

Explanation

The modulus is r = √((-√3)² + 1²) = √(3 + 1) = √4 = 2. Factoring gives z = 2(-√3/2 + i/2). Thus cos(θ) = -√3/2 and sin(θ) = 1/2. Because cos(θ) < 0 and sin(θ) > 0, θ lies in the second quadrant: θ = π − π/6 = 5π/6.

Question 16Algebra and Functions

What are the solutions in ℂ to the quadratic equation z² − 4z + 13 = 0?

  1. z₁ = 4 + 6i and z₂ = 4 - 6i
  2. z₁ = -2 + 3i and z₂ = -2 - 3i
  3. z₁ = 2 + 6i and z₂ = 2 - 6i
  4. z₁ = 2 + 3i and z₂ = 2 - 3i Correct answer.

Explanation

Compute the discriminant: Δ = (-4)² − 4(1)(13) = 16 − 52 = -36 = (6i)². The two complex conjugate roots are z = [-(-4) ± 6i] / 2 = (4 ± 6i)/2 = 2 ± 3i.

Question 17Algebra and Functions

What is the exact algebraic value of the complex power (1 + i√3)⁶?

  1. 64 Correct answer.
  2. -64
  3. 64i
  4. 32 + 32i√3

Explanation

Convert 1 + i√3 to exponential form: modulus r = √(1 + 3) = 2 and argument θ = π/3, so 1 + i√3 = 2e^(i π/3). Applying de Moivre's formula: (1 + i√3)⁶ = [2e^(i π/3)]⁶ = 2⁶ e^(i 6π/3) = 64 e^(i 2π). Since e^(i 2π) = 1, the result is 64 · 1 = 64.

Question 18Algebra and Functions

What is the mean (average) value μ of the function f(x) = 3x² − 2x on the interval [0, 2]?

  1. 1
  2. 4
  3. 3
  4. 2 Correct answer.

Explanation

The average value of a continuous function on [a, b] is μ = [1 / (b − a)] ∫ₐᵇ f(x) dx. Here b − a = 2 − 0 = 2. The definite integral is ∫₀² (3x² − 2x) dx = [x³ − x²]₀² = (2³ − 2²) − 0 = 8 − 4 = 4. Dividing by the interval length 2 gives μ = 4 / 2 = 2.

Question 19Mechanics, fields, thermal behavior, waves and modern models

A capacitor with capacitance C = 100 μF is charged under a constant potential difference U = 20 V. What is the electrostatic energy E_e stored in the electric field of the capacitor?

  1. 2.0 mJ
  2. 40 mJ
  3. 200 mJ
  4. 20 mJ Correct answer.

Explanation

The electrostatic energy stored in a capacitor is E_e = (1/2) C U². Substituting C = 100 × 10⁻⁶ F and U = 20 V gives E_e = (1/2) · (100 × 10⁻⁶) · (20)² = 50 × 10⁻⁶ · 400 = 20,000 × 10⁻⁶ J = 0.020 J = 20 mJ.

Question 20Mechanics, fields, thermal behavior, waves and modern models

A coil of inductance L = 0.2 H and internal resistance r = 5 Ω is connected in series with a resistor R = 45 Ω and a DC generator of EMF E = 10 V. What are the circuit time constant τ and the steady-state current I_max?

  1. τ = 20 ms and I_max = 0.22 A
  2. τ = 4 ms and I_max = 0.2 A Correct answer.
  3. τ = 4 ms and I_max = 2.0 A
  4. τ = 10 ms and I_max = 0.5 A

Explanation

The total series resistance is R_tot = R + r = 45 + 5 = 50 Ω. The inductive time constant is τ = L / R_tot = 0.2 H / 50 Ω = 0.004 s = 4 ms. In the steady state (t → ∞), the inductor behaves as a simple resistor with di/dt = 0, so I_max = E / R_tot = 10 V / 50 Ω = 0.2 A.

Question 21Mechanics, fields, thermal behavior, waves and modern models

The radioactive decay constant of Iodine-131 is λ = 9.98 × 10⁻⁷ s⁻¹. What is its radioactive half-life t_{1/2} in days (given ln(2) ≈ 0.693 and 1 day = 86,400 s)?

  1. 4.0 days
  2. 8.0 days Correct answer.
  3. 14.2 days
  4. 30.0 days

Explanation

The half-life and radioactive decay constant are related by t_{1/2} = ln(2) / λ. Substituting gives t_{1/2} = 0.693 / (9.98 × 10⁻⁷ s⁻¹) ≈ 6.944 × 10⁵ s. Converting to days: 694,400 s / 86,400 s/day ≈ 8.04 ≈ 8.0 days.

Question 22Mechanics, fields, thermal behavior, waves and modern models

For a nucleus ^A_Z X containing Z protons (mass m_p) and N = A − Z neutrons (mass m_n) with measured nuclear mass m_nucleus, what are the mass defect Δm and the binding energy E_b?

  1. Δm = m_nucleus - (Z m_p + N m_n) and E_b = Δm / c²
  2. Δm = Z m_p - N m_n and E_b = Δm c
  3. Δm = [Z m_p + (A - Z) m_n] - m_nucleus and E_b = Δm c² Correct answer.
  4. Δm = A m_n - m_nucleus and E_b = (1/2) Δm c²

Explanation

The mass of an assembled stable nucleus is always less than the sum of the masses of its constituent free nucleons. The mass defect is Δm = [Z·m_p + (A − Z)·m_n] − m_nucleus > 0. According to Einstein's mass-energy equivalence, the total nuclear binding energy required to dissociate the nucleus into free nucleons is E_b = Δm · c².

Question 23Mechanics, fields, thermal behavior, waves and modern models

On the Aston curve representing binding energy per nucleon (E_b / A) versus mass number A, which group of nuclei exhibits the highest binding energy per nucleon (~8.8 MeV/nucleon) and maximum nuclear stability?

  1. Very light nuclei with A < 4 such as Deuterium and Tritium
  2. Superheavy transuranic nuclei with A > 230 such as Uranium and Plutonium
  3. Light nuclei with equal proton and neutron numbers in the range A = 10 to 20
  4. Intermediate-mass nuclei with mass numbers around A ≈ 56 to 62, such as Iron-56 (⁵⁶₂₆Fe) Correct answer.

Explanation

The Aston curve reaches its broad maximum around mass number A ≈ 56 to 62 (most notably Iron-56 and Nickel-62) with a binding energy of approximately 8.8 MeV per nucleon. These nuclei are the most thermodynamically stable. Lighter nuclei can release energy via nuclear fusion, while heavier nuclei release energy via nuclear fission.

Question 24Physical chemistry, amounts, reactions and equilibria

A projectile is launched from ground level with initial velocity v₀ at an elevation angle α above the horizontal in a uniform gravitational field g (ignoring air drag). What is the expression for the maximum height H attained?

  1. H = (v₀² sin α) / g
  2. H = (v₀² cos² α) / (2g)
  3. H = (v₀² sin² α) / (2g) Correct answer.
  4. H = (v₀² sin(2α)) / g

Explanation

At the trajectory's vertex, the vertical velocity component is zero: v_y(t) = v₀ sin(α) − gt = 0 ⇒ t_apex = (v₀ sin α)/g. Substituting t_apex into the vertical position equation y(t) = -(1/2)gt² + (v₀ sin α)t yields H = -(1/2)g [(v₀ sin α)/g]² + (v₀ sin α)[(v₀ sin α)/g] = (v₀² sin² α) / (2g).

Question 25Physical chemistry, amounts, reactions and equilibria

A block of mass m = 1 kg is released from rest at point A at height h = 5 m. As it slides down to ground level B, friction performs work W_nc = -10 J. Taking g = 10 m/s², what is the kinetic energy E_k of the block at point B?

  1. 50 J
  2. 60 J
  3. 40 J Correct answer.
  4. 10 J

Explanation

By the mechanical energy theorem, ΔE_m = E_m(B) − E_m(A) = W_nc. At point A, E_m(A) = E_p(A) + E_k(A) = mgh + 0 = 1 · 10 · 5 = 50 J. At ground level B, E_p(B) = 0, so E_m(B) = E_k(B). Thus E_k(B) − 50 = -10 J ⇒ E_k(B) = 50 − 10 = 40 J.

Question 26Physical chemistry, amounts, reactions and equilibria

An undamped horizontal spring-mass system has a mass m = 0.25 kg attached to a spring of stiffness k = 100 N/m. What is the natural period of oscillation T₀?

  1. T₀ = π / 10 s (≈ 0.314 s) Correct answer.
  2. T₀ = π / 5 s (≈ 0.628 s)
  3. T₀ = 2π s (≈ 6.28 s)
  4. T₀ = 0.05 s

Explanation

The natural angular frequency of a spring-mass oscillator is ω₀ = √(k/m) = √(100 / 0.25) = √400 = 20 rad/s. The natural period is T₀ = 2π / ω₀ = 2π / 20 = π / 10 s ≈ 0.314 s.

Question 27Physical chemistry, amounts, reactions and equilibria

In a series RLC circuit, how does the total circuit resistance R influence free electrical oscillations, and under what condition does the underdamped (pseudo-periodic) regime occur?

  1. Resistance increases the oscillation frequency without causing any electromagnetic energy dissipation
  2. Free oscillations continue indefinitely with strictly constant amplitude regardless of the value of R
  3. Resistance dissipates electromagnetic energy as heat via the Joule effect; the underdamped regime occurs when R < 2√(L/C), producing decaying oscillations Correct answer.
  4. Resistance acts only on the capacitor, causing the magnetic energy of the inductor to grow exponentially without bound

Explanation

Total resistance R dissipates the electromagnetic energy stored in the LC tank through Joule heating (P_J = R·i²). The characteristic equation r² + (R/L)r + 1/(LC) = 0 has complex roots when the discriminant Δ = (R/L)² − 4/(LC) < 0, which corresponds to R < 2√(L/C). This condition defines the underdamped (pseudo-periodic) regime where the capacitor voltage oscillates with exponentially decaying amplitude.

Question 28Physical chemistry, amounts, reactions and equilibria

According to Soddy's conservation laws for nuclear reactions, which daughter nucleus is produced by the alpha (α) decay of Uranium-238 (²³⁸₉₂U)?

  1. ²³⁴₉₀Th Correct answer.
  2. ²³⁸₉₃Np
  3. ²³⁴₉₁Pa
  4. ²³⁶₉₀Th

Explanation

An alpha particle is a Helium-4 nucleus (⁴₂He). By Soddy's laws, total nucleon number A and proton number Z are conserved: ²³⁸₉₂U → ⁴₂He + ^A_Z X. Thus A = 238 − 4 = 234 and Z = 92 − 2 = 90. The chemical element with atomic number 90 is Thorium (Th), producing ²³⁴₉₀Th.

Question 29Physical chemistry, amounts, reactions and equilibria

In beta-minus (β⁻) radioactive decay, what fundamental nuclear conversion occurs inside the parent nucleus, and what particles are emitted?

  1. A proton transforms into a neutron, emitting a positron and an electron neutrino
  2. An alpha particle is expelled from the nucleus along with high-frequency gamma radiation
  3. Two neutrons fuse into a deuteron, emitting low-energy characteristic X-rays
  4. A neutron transforms into a proton, emitting an electron (β⁻ particle) and an electron antineutrino Correct answer.

Explanation

Beta-minus decay occurs in neutron-rich nuclei. A neutron converts into a proton mediated by the weak interaction: ¹₀n → ¹₁p + ⁰₋₁e + ν̄_e. The emitted particles are a high-speed electron (the β⁻ particle) and an electron antineutrino (ν̄_e), increasing the atomic number Z by 1 while keeping mass number A constant.

Question 30Physical chemistry, amounts, reactions and equilibria

For a chemical reaction taking place in a constant volume V, how is the volumetric rate of reaction v(t) defined, and how is it determined graphically from the progress curve x(t)?

  1. v(t) = (1/V) dx/dt, determined by the slope of the tangent to the curve x(t) divided by volume V Correct answer.
  2. v(t) = V (dx/dt), determined by multiplying the area under the curve x(t) by V
  3. v(t) = -d[A]/dt, determined exclusively by the y-intercept of the absorbance curve
  4. v(t) = x(t) / t, determined by drawing a secant line from the origin to (t, x(t))

Explanation

The volumetric reaction rate is defined as v(t) = (1/V)·(dx/dt), where x is the reaction progress in moles and V is the total solution volume. Graphically, the derivative dx/dt corresponds to the slope of the tangent line drawn to the progress curve x = f(t) at time t; dividing this slope by V yields the volumetric rate.

Question 31Physical chemistry, amounts, reactions and equilibria

In chemical kinetics, what is the definition of the half-reaction time t_{1/2}, and how does increasing the reaction temperature generally affect it?

  1. The time required for all reactants to be completely consumed; increasing temperature increases t_{1/2}
  2. The time required for progress to reach half of its final value (x(t_{1/2}) = x_f / 2); increasing temperature decreases t_{1/2} Correct answer.
  3. The time at which the volumetric rate reaches its maximum value; increasing temperature has no effect on t_{1/2}
  4. The time required for the catalyst to regenerate; increasing temperature doubles t_{1/2}

Explanation

By definition, the half-reaction time t_{1/2} is the time needed for the reaction progress x to reach half of its final value: x(t_{1/2}) = x_f / 2. Temperature is a kinetic factor: elevating the temperature increases reactant thermal energy and effective collision frequency, speeding up the reaction and therefore shortening the time required to reach half-progress (t_{1/2} decreases).

Question 32Physical chemistry, amounts, reactions and equilibria

At 25°C, the ionic product of pure water is K_e = 1.0 × 10⁻¹⁴. What is the pH of a 0.01 mol/L aqueous solution of sodium hydroxide (NaOH), a strong base?

  1. 12.0 Correct answer.
  2. 2.0
  3. 7.0
  4. 10.0

Explanation

Sodium hydroxide is a strong base that dissociates completely in water: NaOH → Na⁺ + OH⁻. Hence [OH⁻] = C_b = 0.01 mol/L = 10⁻² mol/L. The hydronium ion concentration is [H₃O⁺] = K_e / [OH⁻] = 10⁻¹⁴ / 10⁻² = 10⁻¹² mol/L. Therefore, pH = -log[H₃O⁺] = -log(10⁻¹²) = 12.0 (or pH = 14 + log(C_b) = 14 − 2 = 12.0).

Question 33Physical chemistry, amounts, reactions and equilibria

An aqueous solution of ethanoic acid (acetic acid, pK_a = 4.8) has a measured pH of 5.8. According to the Henderson-Hasselbalch relationship pH = pK_a + log([CH₃COO⁻]/[CH₃COOH]), what is the ratio of conjugate base to weak acid, and which chemical species predominates?

  1. [A⁻]/[HA] = 1, neither species predominates
  2. [A⁻]/[HA] = 0.1, the acidic form CH₃COOH predominates
  3. [A⁻]/[HA] = 2, the acidic form CH₃COOH predominates
  4. [A⁻]/[HA] = 10, the basic conjugate form CH₃COO⁻ predominates Correct answer.

Explanation

Rearranging the Henderson-Hasselbalch equation gives log([CH₃COO⁻]/[CH₃COOH]) = pH − pK_a = 5.8 − 4.8 = 1.0. Taking the inverse base-10 logarithm yields [CH₃COO⁻]/[CH₃COOH] = 10¹ = 10. Because pH > pK_a, the conjugate base (ethanoate ion CH₃COO⁻) predominates over the undissociated acid by a factor of 10 to 1.

Question 34Physical chemistry, amounts, reactions and equilibria

Which statement correctly defines the action of a catalyst in a chemical reaction system?

  1. A catalyst shifts the chemical equilibrium position toward products, increasing the final yield
  2. A catalyst is consumed as a primary reactant and appears in the balanced chemical equation
  3. A catalyst speeds up the reaction by providing an alternative pathway with lower activation energy without altering the final thermodynamic equilibrium Correct answer.
  4. A catalyst increases the total standard reaction enthalpy ΔH°

Explanation

A catalyst is a chemical substance that accelerates a chemical reaction by opening an alternative reaction mechanism with lower activation energy. It is regenerated at the end of the catalytic cycle and does not alter the thermodynamic state, meaning it changes neither the final composition nor the equilibrium constant K.

Question 35Physical chemistry, amounts, reactions and equilibria

During the acid-base titration of a weak acid (such as ethanoic acid) by a strong base (sodium hydroxide), what is the pH condition at the equivalence point E, and which colored indicator is most suitable?

  1. pH_E > 7 (basic); Phenolphthalein (color change interval 8.2–10.0) Correct answer.
  2. pH_E = 7.0 (neutral); Bromothymol blue (color change interval 6.0–7.6)
  3. pH_E < 7 (acidic); Methyl orange (color change interval 3.1–4.4)
  4. pH_E = pK_a; Methyl red (color change interval 4.2–6.2)

Explanation

At equivalence, the weak acid HA is quantitatively converted into its conjugate weak base A⁻ (ethanoate ion). Because A⁻ hydrolyzes with water to produce OH⁻ ions, the equivalence pH is strictly basic (pH_E > 7, typically around 8.5–9.0). An indicator is suitable if its turning interval encompasses pH_E; phenolphthalein (8.2–10.0) is the standard choice.

Question 36Physical chemistry, amounts, reactions and equilibria

An equimolar mixture of a carboxylic acid and a primary alcohol undergoes esterification at 100°C. What are the key thermodynamic and kinetic characteristics of this reaction, and what is its final equilibrium yield?

  1. The reaction is rapid, complete (100% yield), and highly exothermic
  2. The reaction is slow, reversible, athermic, and reaches a dynamic equilibrium with an ester yield of approximately 67% Correct answer.
  3. The reaction is explosive, endothermic, and produces a yield of exactly 33%
  4. The reaction is irreversible and catalyzed by sodium hydroxide

Explanation

Esterification of a carboxylic acid with a primary alcohol is slow, reversible (limited by the reverse hydrolysis reaction), and practically athermic (ΔH ≈ 0). Starting from an equimolar mixture of primary alcohol and acid, it reaches a dynamic chemical equilibrium with an ester yield of approximately 2/3 (~67%).

Question 37Physical chemistry, amounts, reactions and equilibria

In a commercial pressurized water nuclear reactor (PWR), what are the distinct functional roles of the control rods (made of boron or cadmium) and the moderator (pressurized light water)?

  1. Control rods generate thermal neutrons; the moderator absorbs gamma rays
  2. Control rods cool the fuel assemblies; the moderator accelerates neutrons to relativistic speeds
  3. Control rods store radioactive fission products; the moderator converts neutrons into protons
  4. The moderator slows down fast fission neutrons to thermal energy levels; control rods absorb excess neutrons to regulate or halt the fission chain reaction Correct answer.

Explanation

Nuclear fission of Uranium-235 is induced most effectively by slow thermal neutrons (~0.025 eV). The moderator collides with fast neutrons emitted by fission to slow them down without capturing them. Control rods contain high neutron capture cross-section materials (boron, cadmium) that absorb excess neutrons to maintain criticality (k = 1) or safely shut down the reactor.

Question 38Physical chemistry, amounts, reactions and equilibria

When a damped mechanical or electrical oscillator is driven by a periodic external harmonic source of frequency f, under what condition does velocity resonance occur, and how does the system behave?

  1. When the excitation frequency equals the oscillator's natural frequency (f = f₀), maximizing the velocity amplitude and placing velocity in phase with the driving force Correct answer.
  2. When the excitation frequency is double the natural frequency (f = 2f₀), resulting in total destructive cancellation
  3. When the damping coefficient becomes infinitely large, eliminating all phase differences
  4. When the excitation frequency approaches zero, maximizing the stored kinetic energy

Explanation

Velocity resonance (or current resonance in electrical RLC circuits) occurs when the driving frequency matches the undamped natural frequency: f = f₀ = (1 / 2π)√(k / m). At this exact frequency, mechanical impedance is minimized (reduced purely to the damping coefficient), the velocity response amplitude is maximal, and the velocity is in phase with the driving force.

Question 39Physical chemistry, amounts, reactions and equilibria

[Process Engineering] In chemical reaction engineering, how is the fractional conversion X_A of a limiting reactant A defined between the inlet molar flow rate F_{A0} (or initial moles n_{A0}) and the outlet flow rate F_A (or remaining moles n_A)?

  1. X_A = F_A / F_{A0}
  2. X_A = (F_A - F_{A0}) / F_A
  3. X_A = F_{A0} / (F_{A0} + F_A)
  4. X_A = (F_{A0} - F_A) / F_{A0} Correct answer.

Explanation

Fractional conversion X_A measures the proportion of reactant A that has been transformed into products. It is defined as the moles of A reacted divided by the moles of A fed to the reactor: X_A = (F_{A0} − F_A) / F_{A0} = 1 − (F_A / F_{A0}). It is a dimensionless quantity ranging from 0 (no reaction) to 1 (complete consumption of reactant A).

Question 40Statics, strength and mechanical transmission

[Civil Engineering] For a planar pin-jointed truss with stable geometry and independent constraints, which necessary counting condition relates b bars, j joints, and r external reaction components for static determinacy?

  1. b + j = 2r
  2. b + r = 2j Correct answer.
  3. 2b + r = 3j
  4. b - r = j

Explanation

Each planar pin joint supplies two equilibrium equations, giving 2j equations. The unknowns are b member forces plus r reaction components, so the counting condition is b + r = 2j. This equality is necessary but not sufficient by itself: the truss must also have stable geometry and independent constraints, as stated in the question.

Question 41Statics, strength and mechanical transmission

[Civil Engineering] In a planar isostatic truss, joint A is connected to two bars: bar 1 is horizontal along the x-axis, and bar 2 is inclined at θ = 45° above the horizontal. A downward vertical load F = 10 kN is applied directly at joint A. Assuming tensile forces are positive, what is the axial force N₂ in bar 2?

  1. N₂ = -10 kN (compression)
  2. N₂ = +10 kN (tension)
  3. N₂ = +10√2 kN ≈ +14.14 kN (tension) Correct answer.
  4. N₂ = -10√2 kN ≈ -14.14 kN (compression)

Explanation

Writing equilibrium for joint A along the vertical y-axis: ∑ F_y = 0 ⇒ N₂ sin(45°) − F = 0 ⇒ N₂ sin(45°) = 10 kN. Since sin(45°) = √2 / 2 = 1 / √2, we have N₂ = 10 / (1 / √2) = +10√2 kN ≈ +14.14 kN. Because N₂ > 0, bar 2 is in tension.

Question 42Statics, strength and mechanical transmission

[Civil Engineering] When analyzing an isostatic planar truss using the method of sections (Ritter's method), what is the maximum number of unknown member forces that can be determined by cutting through a single section?

  1. 1 member force
  2. 2 member forces
  3. 4 member forces
  4. 3 member forces Correct answer.

Explanation

In planar statics, the equilibrium of an isolated portion of a structure is governed by three independent equations of equilibrium: ∑ F_x = 0, ∑ F_y = 0, and ∑ M_O = 0 (about any reference point O). Therefore, a section cut can solve for a maximum of 3 unknown, non-concurrent, and non-parallel internal member forces.

Question 43Statics, strength and mechanical transmission

[Mechanical Engineering] In the planar kinematics of an isolated rigid link undergoing general plane motion, what is the Instantaneous Center of Rotation (CIR, Centre Instantané de Rotation) I, and how does it determine point velocities?

  1. The CIR is the link's center of mass where linear acceleration is identically zero
  2. The CIR is the point where angular velocity ω vanishes while linear velocity becomes infinite
  3. The CIR is the unique point in the link's plane of motion with zero instantaneous velocity (v_I = 0); the velocity vector of any point M is perpendicular to segment [IM] with magnitude v_M = ω · IM Correct answer.
  4. The CIR is the fixed hinge attached to the machine frame that remains stationary at all times

Explanation

At any given instant, general planar motion of a rigid body link can be modeled as pure rotation about a single point called the Instantaneous Center of Rotation (CIR, denoted I), where instantaneous velocity is zero (v_I = 0). The velocity vector of any point M on the body is oriented perpendicularly to line IM in the direction of angular rotation ω, with magnitude given by v_M = ω · IM.

Question 44Statics, strength and mechanical transmission

[Mechanical Engineering] A simple gear train comprises driving gear 1 (Z₁ = 20 teeth), intermediate idler gear 2 (Z₂ = 40 teeth), and driven gear 3 (Z₃ = 60 teeth). If driving gear 1 rotates at ω₁ = 1,200 rpm clockwise, what are the rotational speed ω₃ and direction of driven gear 3?

  1. ω₃ = 600 rpm counterclockwise
  2. ω₃ = 400 rpm clockwise Correct answer.
  3. ω₃ = 400 rpm counterclockwise
  4. ω₃ = 200 rpm clockwise

Explanation

The global transmission ratio of a simple gear train with k external meshes is r = (-1)^k · (Z_driving / Z_driven) = ω_out / ω_in. Here there are two external meshes (1 with 2, and 2 with 3), so k = 2 and (-1)² = +1. The intermediate idler gear 2 cancels out of the ratio: r = +1 · (Z₁ / Z₃) = 20 / 60 = 1/3. Therefore, ω₃ = (1/3) · 1,200 rpm = 400 rpm, rotating in the same clockwise direction as gear 1.

Question 45Statics, strength and mechanical transmission

[Mechanical Engineering] In an epicyclic (planetary) gear train with sun gear 1 (Z₁ teeth), planet carrier (porte-satellite) ps, and stationary internal ring gear 3 (Z₃ teeth), which expression represents Willis's relation for the kinematic speed ratio relative to the planet carrier?

  1. (ω₃ + ω_ps) / (ω₁ + ω_ps) = Z₃ / Z₁
  2. (ω₁ - ω₃) / ω_ps = (Z₁ + Z₃) / Z₁
  3. (ω₃ - ω₁) / ω_ps = Z₁ / Z₃
  4. (ω₃ - ω_ps) / (ω₁ - ω_ps) = -Z₁ / Z₃ Correct answer.

Explanation

Willis's formula evaluates the gear train ratio in the frame of reference attached to the moving planet carrier (porte-satellite): (ω_outlet − ω_ps) / (ω_inlet − ω_ps) = (-1)^k · (Z_driving / Z_driven). In a standard planetary train with sun 1 and ring 3, there is one external mesh (sun 1 with planet 2) and one internal mesh (planet 2 with ring 3). Because only external meshes introduce a factor of -1, k = 1, giving (ω₃ − ω_ps) / (ω₁ − ω_ps) = -Z₁ / Z₃.

Question 46Statics, strength and mechanical transmission

[Mechanical Engineering] A mechanical link is subjected to only two external forces F_A and F_B applied at two distinct points A and B. For this link to remain in static equilibrium, what conditions must the two forces satisfy?

  1. They must be perpendicular to each other and possess identical numerical magnitudes
  2. They must share the exact same line of action (line AB), have equal magnitudes, and point in opposite directions (F_A = -F_B) Correct answer.
  3. They must intersect at the center of mass and both point inward
  4. They must generate a non-zero couple to prevent rotational drift

Explanation

According to the principle of static equilibrium for a body subjected to two forces (two-force member): for force equilibrium ∑ F = 0, the two forces must be equal in magnitude and opposite in direction (F_A = -F_B). For moment equilibrium ∑ M_A = 0, the moment of F_B about point A must be zero (BA × F_B = 0), which requires that the line of action of both forces coincide with the straight line AB connecting their points of application.

Question 47Statics, strength and mechanical transmission

[Mechanical Engineering] In kinematic analysis of planar and spatial mechanisms, which standard kinematic pair (liaison mécanique) removes 5 degrees of freedom and allows exactly one degree of freedom consisting of pure rotation around an axis?

  1. Pivot joint (liaison pivot / revolute joint) Correct answer.
  2. Prismatic joint (liaison glissière)
  3. Spherical joint (liaison rotule / ball joint)
  4. Planar joint (liaison appui-plan)

Explanation

A pivot (revolute) joint eliminates all three translational degrees of freedom (T_x = 0, T_y = 0, T_z = 0) and two rotational degrees of freedom (e.g., R_y = 0, R_z = 0), leaving strictly one degree of freedom: rotation around the joint axis (R_x = 1).

Question 48Electrical circuits, machines and digital logic

[Electrical Engineering] In a balanced three-phase electrical power system connected in star (Y), what are the exact relationships between the line-to-line RMS voltage U and the line-to-neutral phase voltage V, and between line current I and phase current J?

  1. U = V and I = √3 J
  2. U = V / √3 and I = J
  3. U = 3V and I = J / 3
  4. U = √3 V and I = J Correct answer.

Explanation

In a star (Y) connection, each phase winding is connected between one line and the neutral point. The vector difference between two phase voltages 120° apart gives line-to-line voltage U = √3 V. Because each line conductor connects directly in series with one phase winding, line current equals phase current: I = J.

Question 49Electrical circuits, machines and digital logic

[Electrical Engineering] A balanced three-phase factory load consumes active power P = 20 kW with lagging power factor cos(φ) = 0.80 under line voltage U = 400 V at 50 Hz. To elevate the power factor to cos(φ') = 0.95, what reactive power Q_C must be supplied by parallel capacitor banks?

  1. Q_C = 20.0 kVAR
  2. Q_C = 15.0 kVAR
  3. Q_C = P [tan(φ) - tan(φ')] ≈ 8.43 kVAR Correct answer.
  4. Q_C = 2.50 kVAR

Explanation

Capacitors generate leading reactive power that cancels inductive lagging reactive power without modifying active power P. For cos(φ) = 0.80, tan(φ) = sin(φ)/cos(φ) = 0.60/0.80 = 0.75. For cos(φ') = 0.95, φ' ≈ 18.19° and tan(φ') ≈ 0.3287. The required reactive compensation is Q_C = P [tan(φ) − tan(φ')] = 20 kW · [0.75 − 0.3287] = 20 · 0.4213 ≈ 8.43 kVAR.

Question 50Electrical circuits, machines and digital logic

[Electrical Engineering] A three-phase squirrel-cage asynchronous (induction) motor has p = 2 pole pairs and is supplied by a 50 Hz grid. At rated full load, the rotor speed is n = 1,440 rpm. What are the synchronous speed n_s and the fractional slip g?

  1. n_s = 1,500 rpm and g = 0.04 (4%) Correct answer.
  2. n_s = 3,000 rpm and g = 0.52 (52%)
  3. n_s = 1,500 rpm and g = 0.06 (6%)
  4. n_s = 1,000 rpm and g = -0.44

Explanation

Synchronous speed is n_s = 60 · f / p = 60 · 50 / 2 = 1,500 rpm. Motor slip is the relative speed difference between the stator rotating field and the rotor: g = (n_s − n) / n_s = (1,500 − 1,440) / 1,500 = 60 / 1,500 = 0.04, or 4%.