IB Mathematics 3

Question 1

Solve the equation 2sin(x) − 1 = 0 for x ∈ [0, 2π].

Accepted Answer

x = π/6 and x = 5π/6

Answer

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?

Accepted Answer

80

Answer

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 = ?

Accepted Answer

x = 3

Answer

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?

Accepted Answer

32

Answer

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?

Accepted Answer

1.5

Answer

z = (x − μ)/σ = (65 − 50)/10 = 1.5.

IB - International Baccalaureate Practice Test