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Sample BAC 2 Série C4 Togo Practice Questions
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1A point in the complex plane has affix z = −5 + 12i. What is its distance from the origin?
A.17
B.169
C.13
D.7
Explanation: Distance from the origin is the modulus of the affix. Thus |z| = √((−5)² + 12²) = √169 = 13.
2The Euclidean algorithm starts with 126 = 2 × 54 + 18 and 54 = 3 × 18. What is PGCD(126, 54)?
A.18
B.54
C.6
D.3
Explanation: The greatest common divisor is the last nonzero remainder in the Euclidean algorithm. The next division has remainder zero, so the PGCD is 18.
3For the direct similitude z' = 2iz + 3 − i, which point is the invariant centre?
A.(3, −1)
B.(1/5, 7/5)
C.(−1, −1)
D.(1, 1)
Explanation: At the centre, z' = z, so (1 − 2i)z = 3 − i. Multiplying numerator and denominator by 1 + 2i gives z = (5 + 5i)/5 = 1 + i.
4The pair (2, −3) solves 11x + 7y = 1. Which formula gives all integer solutions?
A.x = 2 + 11k, y = −3 − 7k
B.x = 2 + 7k, y = −3 − 11k
C.x = 2 + 7k, y = −3 + 11k
D.x = 2k, y = −3k
Explanation: Subtracting the particular solution gives 11(x − 2) = −7(y + 3). Since 11 and 7 are coprime, x − 2 = 7k and y + 3 = −11k, for any integer k.
5For f(x) = ln(x² + 4), what is the slope of its tangent at x = 2?
A.1/8
B.4
C.1/2
D.ln(8)
Explanation: The chain rule gives f'(x) = 2x/(x² + 4). Evaluating at x = 2 gives 4/8 = 1/2.
6A sequence satisfies u₀ = 3 and uₙ₊₁ = uₙ/2 + 2. Which expression gives uₙ for every nonnegative integer n?
A.4 − 2⁻ⁿ
B.4 + 2⁻ⁿ
C.3 × 2⁻ⁿ
D.4 − 2ⁿ
Explanation: Subtract the fixed point 4: uₙ₊₁ − 4 = (uₙ − 4)/2. Therefore uₙ − 4 = (3 − 4)(1/2)ⁿ, giving uₙ = 4 − 2⁻ⁿ.
7Evaluate I = ∫₀¹ x e^(3x) dx by integration by parts.
A.e³/3
B.(e³ − 1)/9
C.(2e³ − 1)/9
D.(2e³ + 1)/9
Explanation: Taking u = x and dv = e^(3x)dx gives I = [xe^(3x)/3]₀¹ − ∫₀¹ e^(3x)/3 dx. Hence I = e³/3 − (e³ − 1)/9 = (2e³ + 1)/9.
8Six independent trials each have success probability 1/3. What is the probability of exactly two successes?
A.(1/3)²(2/3)⁴
B.15(1/3)²(2/3)⁴
C.6(1/3)(2/3)⁵
D.(1/3)⁶
Explanation: The count of successes follows a binomial distribution with n = 6 and p = 1/3. There are C(6, 2) = 15 positions for two successes, giving 15(1/3)²(2/3)⁴.
9A continuous function f is strictly decreasing on [1, 4], with f(1) = 6 and f(4) = −2. What can be concluded about f(x) = 0?
A.There is no solution because f decreases.
B.There are at least two solutions.
C.There is exactly one solution in (1, 4).
D.The solution must be x = 2.5.
Explanation: Continuity and opposite endpoint signs give at least one zero by the intermediate value theorem. Strict decrease gives at most one zero, so the solution exists and is unique.
10A cart of constant mass 2.5 kg has a resultant horizontal force of 7.5 N in an inertial frame. What horizontal acceleration follows?
A.3.0 m/s²
B.18.75 m/s²
C.0.333 m/s²
D.5.0 m/s²
Explanation: Newton's second law relates resultant force to mass times acceleration. Thus a = 7.5/2.5 = 3.0 m/s² in the force's direction.
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