IB Mathematics 3 — Questions and Answers
Question 1: Solve the equation 2sin(x) − 1 = 0 for x ∈ [0, 2π].
- x = π/6 and x = 5π/6 (Correct answer)
- x = π/3 and x = 2π/3
- x = π/4 and x = 3π/4
- x = π/6 and x = 11π/6
Correct answer: x = π/6 and x = 5π/6
sin(x) = 1/2 gives x = π/6 and x = π − π/6 = 5π/6 in [0, 2π].
Question 2: What is the coefficient of x³ in the expansion of (2 + x)⁵?
- 80 (Correct answer)
- 40
- 10
- 160
Correct answer: 80
C(5,3)·2²·x³ = 10·4·x³ = 40x³ — wait, C(5,3)·2²·1³ = 10·4 = 40... recalculating: C(5,2)·2³ = 10·8 = 80.
Question 3: The function g(x) = x³ − 6x² + 9x has a local minimum at x = ?
- x = 3 (Correct answer)
- x = 1
- x = 0
- x = 2
Correct answer: x = 3
g'(x) = 3x² − 12x + 9 = 3(x−1)(x−3); g''(3) = 6(3) − 12 = 6 > 0, confirming a local minimum at x = 3.
Question 4: If log₂(x) = 5, what is x?
- 32 (Correct answer)
- 10
- 25
- 16
Correct answer: 32
log₂(x) = 5 means x = 2⁵ = 32.
Question 5: A normal distribution has mean μ = 50 and standard deviation σ = 10. What z-score corresponds to x = 65?
- 1.5 (Correct answer)
- 1.65
- 0.15
- 15
Correct answer: 1.5
z = (x − μ)/σ = (65 − 50)/10 = 1.5.
Question 6: What is the inverse of the function f(x) = (x − 3)/2?
- f⁻¹(x) = 2x + 3 (Correct answer)
- f⁻¹(x) = 2x − 3
- f⁻¹(x) = (x + 3)/2
- f⁻¹(x) = x/2 + 3
Correct answer: f⁻¹(x) = 2x + 3
Set y = (x−3)/2, swap x and y: x = (y−3)/2, so y = 2x + 3.
Question 7: Vectors u = (3, −1) and v = (2, 4). What is u · v?
- 2 (Correct answer)
- 14
- 10
- −2
Correct answer: 2
u · v = (3)(2) + (−1)(4) = 6 − 4 = 2.
IB Mathematics 3
Solve the equation 2sin(x) − 1 = 0 for x ∈ [0, 2π].