IB Mathematics 4 — Questions and Answers
Question 1: What is the sum to infinity of the geometric series 12 + 4 + 4/3 + … ?
- 18 (Correct answer)
- 16
- 20
- 24
Correct answer: 18
r = 1/3 and a = 12; S∞ = a/(1−r) = 12/(1 − 1/3) = 12/(2/3) = 18.
Question 2: Which of the following is the derivative of y = e^(2x) · cos(x)?
- 2e^(2x)cos(x) − e^(2x)sin(x) (Correct answer)
- 2e^(2x)cos(x) + e^(2x)sin(x)
- e^(2x)(cos(x) − sin(x))
- 2e^(2x)sin(x)
Correct answer: 2e^(2x)cos(x) − e^(2x)sin(x)
By the product rule: dy/dx = 2e^(2x)cos(x) + e^(2x)(−sin(x)) = e^(2x)(2cos(x) − sin(x)).
Question 3: What is the area enclosed between y = x² and y = x over the interval where they intersect?
- 1/6 (Correct answer)
- 1/3
- 1/2
- 1/4
Correct answer: 1/6
Intersect at x = 0 and x = 1; area = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.
Question 4: The complex number z = 3 + 4i. What is |z|?
- 5 (Correct answer)
- 7
- 25
- √7
Correct answer: 5
|z| = √(3² + 4²) = √(9 + 16) = √25 = 5.
Question 5: A Poisson random variable X has mean λ = 3. What is P(X = 0)?
- e⁻³ (Correct answer)
- 3e⁻³
- 1 − e⁻³
- e³
Correct answer: e⁻³
P(X = 0) = e^(−λ)·λ⁰/0! = e^(−3).
Question 6: The angle between vectors a = (1, 0, 1) and b = (1, 1, 0) is:
- 60° (Correct answer)
- 45°
- 90°
- 30°
Correct answer: 60°
cos θ = (a·b)/(|a||b|) = 1/(√2·√2) = 1/2, so θ = 60°.
Question 7: What is the range of f(x) = 2sin(x) + 1?
- [−1, 3] (Correct answer)
- [−2, 2]
- [0, 2]
- [1, 3]
Correct answer: [−1, 3]
sin(x) ∈ [−1, 1], so 2sin(x) ∈ [−2, 2], thus f(x) = 2sin(x) + 1 ∈ [−1, 3].
IB Mathematics 4
What is the sum to infinity of the geometric series 12 + 4 + 4/3 + … ?