Times Edu Press
IB Mathematics
AA & AI
Practice Papers & Solutions
00
Overview
Giới thiệu IB Mathematics
IB Mathematics có hai lộ trình chính: AA (Analysis & Approaches) và AI (Applications & Interpretation). Mỗi lộ trình phù hợp với định hướng học thuật và nghề nghiệp khác nhau.
AA · Analysis & Approaches
Focus: Pure maths, proof, algebra
SL Papers: P1 (no calc, 1h30m) + P2 (calc, 1h30m)
HL Papers: P1 (no calc, 2h) + P2 (calc, 2h) + P3 (1h)
Best for: Engineering, Physics, Pure Maths
AI · Applications & Interpretation
Focus: Real-world, statistics, modelling
SL Papers: P1 (no calc, 1h30m) + P2 (calc, 1h30m)
HL Papers: P1 (no calc, 1h30m) + P2 (calc, 1h30m) + P3 (1h)
Best for: Business, Social Sciences, Biology
Mẹo đạt điểm 7
- Formula booklet: Học cách TRA NHANH — không cần thuộc hết (booklet được cung cấp trong phòng thi)
- Paper 1 (no calc): Luyện tính tay, exact values, fractions
- Show working: Mỗi bước = marks. Không nhảy bước!
- GDC skills: Graph, solve, regression, normal distribution
- Read mark scheme: Hiểu cách chấm để biết cần viết gì
- Mark types: [M] marks = method · [A] marks = answer · [R] marks = reasoning
| Tiêu chí | AA (Analysis & Approaches) | AI (Applications & Interpretation) |
|---|---|---|
| Focus | Pure maths, proof, algebra | Real-world, statistics, modelling |
| Calculator | GDC cho P2 & P3 | GDC cho P2 & P3 |
| IA | Mathematical exploration (12–20 pages) | Mathematical exploration (same) |
| Best for | Engineering, Physics, Pure Maths | Business, Social Sciences, Biology |
01
Topic 1
Number & Algebra
1.1 · Sequences & Series
Arithmetic Progression (AP)
uₙ = u₁ + (n−1)d
Sₙ = n/2 · (2u₁ + (n−1)d)
Sₙ = n/2 · (u₁ + uₙ)
Geometric Progression (GP)
uₙ = u₁ · rⁿ⁻¹
Sₙ = u₁(1−rⁿ)/(1−r)
S∞ = u₁/(1−r) [|r| < 1]
Sigma Notation (HL)
∑[r=1→n] r = n(n+1)/2
∑ r² = n(n+1)(2n+1)/6
Binomial Theorem
(a+b)ⁿ = ∑ C(n,r) · aⁿ⁻ʳ · bʳ
C(n,r) = n! / (r!(n−r)!)
Tᵣ₊₁ = C(n,r) · aⁿ⁻ʳ · bʳ
Dùng để tìm hệ số của xᵏ trong khai triển đa thức
Exponents & Logarithms
aˣ = b ⟺ x = logₐ b
log(ab) = log a + log b
log(a/b) = log a − log b
log(aⁿ) = n log a
logₐ b = ln b / ln a
ln eˣ = x · e^(ln x) = x
Complex Numbers — AA HL only
z = a + bi
|z| = √(a²+b²)
arg(z) = arctan(b/a)
Polar: z = r cisθ = re^(iθ)
De Moivre: (r cisθ)ⁿ = rⁿ cis(nθ)
Practice Questions
✓ Solution
u₃ = u₁ + 2d = 11
u₇ = u₁ + 6d = 23
Trừ: 4d = 12 → d = 3
u₁ = 11 − 6 = 5
S₂₀ = 20/2 · (2×5 + 19×3)
= 10 · (10 + 57)
= 10 × 67
= 670
✓ Solution
(a) u₅ = 200(0.85)⁴ = 200 × 0.522 = 104.4
(b) 200(0.85)ⁿ⁻¹ < 50
0.85ⁿ⁻¹ < 0.25
(n−1)ln 0.85 < ln 0.25
n−1 > ln 0.25 / ln 0.85 = 8.53
∴ n ≥ 10
(c) S∞ = 200 / (1 − 0.85) = 200 / 0.15 = 1333.33
✓ Solution
Tᵣ₊₁ = C(7,r)(2x)⁷⁻ʳ(−1)ʳ Cần power của x = 3 → 7 − r = 3 → r = 4 T₅ = C(7,4)(2x)³(−1)⁴ = 35 × 8x³ × 1 = 280x³ ∴ Hệ số: 280
02
Topic 2
Functions
2.1 · Key Concepts
Domain: Tập xác định (input) · Range: Tập giá trị (output)
Inverse f⁻¹: Hoán đổi x và y, giải tìm y. Đồ thị: phản chiếu qua y = x. Tính chất: f(f⁻¹(x)) = x
Composite fg: fg(x) = f(g(x)) — Thực hiện g trước, sau đó f
2.2 · Transformations
Graph Transformations
f(x) + a → dịch lên a
f(x − a) → dịch phải a
af(x) → giãn dọc ×a
f(ax) → co ngang ×1/a
−f(x) → phản chiếu Ox
f(−x) → phản chiếu Oy
2.3 · Types of Functions
Common Function Types
Quadratic: f(x) = ax²+bx+c · Vertex: (−b/2a, f(−b/2a)) · Δ = b²−4ac
Exponential: f(x) = kaˣ + c · Asymptote y=c · Growth a>1 · Decay 0<a<1
Logarithmic: f(x) = logₐ(x) · Inverse của exponential · Domain: x>0
Rational (HL): f(x) = (ax+b)/(cx+d) · VA: x=−d/c · HA: y=a/c
Practice Questions
✓ Solution
(a) Vertex: (2, 5) (b) Axis of symmetry: x = 2 (c) f(0) = 3(4) + 5 = 17 → y-intercept: (0, 17) (d) Range: y ≥ 5
✓ Solution
(a) fg(x) = f(x²−3) = 2(x²−3)+1 = 2x² − 5
(b) gf(x) = g(2x+1) = (2x+1)²−3
= 4x²+4x+1−3
= 4x²+4x−2
(c) y = 2x+1 → x = (y−1)/2
∴ f⁻¹(x) = (x−1)/2
✓ Solution
(a) P(0) = 500
(b) P(10) = 500e^0.3 = 500 × 1.3499 ≈ 675
(c) 1000 = 500e^(0.03t)
2 = e^(0.03t)
ln 2 = 0.03t
t = ln 2 / 0.03 ≈ 23.1 năm
03
Topic 3
Geometry & Trigonometry
3.1 · Trigonometry
Core Identities & Rules
sin²θ + cos²θ = 1
tanθ = sinθ/cosθ
Sine rule: a/sinA = b/sinB = c/sinC
Cosine rule: c² = a²+b²−2ab cosC
Area = ½ab sinC
Arc: s = rθ · Sector: A = ½r²θ
Compound Angle Identities — AA
sin(A±B) = sinA cosB ± cosA sinB
cos(A±B) = cosA cosB ∓ sinA sinB
sin 2A = 2 sinA cosA
cos 2A = cos²A − sin²A
cos 2A = 2cos²A − 1
cos 2A = 1 − 2sin²A
Vectors — AA & AI
|v| = √(x²+y²+z²)
v̂ = v/|v|
a·b = |a||b|cosθ
a·b = a₁b₁+a₂b₂+a₃b₃
Line: r = a + t·d
Angle: cosθ = |d₁·d₂| / (|d₁||d₂|)
Practice Questions
✓ Solution
c² = a² + b² − 2ab cosC
= 64 + 25 − 80 cos 60°
= 89 − 40 = 49
c = 7
Area = ½(8)(5) sin 60°
= 20 × (√3/2)
= 10√3 ≈ 17.3
✓ Solution
Đặt t = sinx: 2t² − t − 1 = 0 (2t + 1)(t − 1) = 0 t = −1/2 hoặc t = 1 sinx = 1: x = π/2 sinx = −1/2: x = 7π/6, 11π/6 ∴ Nghiệm: x = π/2, 7π/6, 11π/6
✓ Solution
a·b = 2(1) + 1(−2) + (−3)(4) = 2−2−12 = −12 |a| = √(4+1+9) = √14 |b| = √(1+4+16) = √21 cosθ = −12 / (√14 × √21) = −12 / √294 = −0.6993 θ = arccos(−0.6993) ≈ 134.4°
04
Topic 4
Statistics & Probability
4.1 · Descriptive Statistics
Measures of Centre & Spread
Mean μ = ∑x/n
Var σ² = ∑(x−μ)²/n
SD = √Variance
IQR = Q₃ − Q₁
Correlation r (Pearson's): −1 ≤ r ≤ 1. Gần ±1: strong. Gần 0: weak. Regression y = ax+b: dùng GDC; chỉ dự đoán trong range dữ liệu
4.2 · Probability
Probability Rules
P(A∪B) = P(A) + P(B) − P(A∩B)
P(A|B) = P(A∩B) / P(B)
Independent: P(A∩B) = P(A)×P(B)
Mut. exclusive: P(A∩B) = 0
4.3 · Distributions
Binomial X ~ B(n, p)
P(X=k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ
E(X) = np
Var(X) = np(1−p)
Normal X ~ N(μ, σ²)
Z = (X − μ)/σ
P(X < a) = P(Z < (a−μ)/σ)
GDC: normalcdf(lower, upper, μ, σ)
GDC: invNorm(area, μ, σ)
Poisson X ~ Po(λ) — AI HL
P(X=k) = e⁻λ · λᵏ / k!
E(X) = Var(X) = λ
Practice Questions
✓ Solution
(a) Z = (180−170)/8 = 1.25
P(Z > 1.25) = 1 − 0.8944 = 0.1056
(b) invNorm(0.3, 170, 8)
= 170 + 8 × (−0.5244)
= 170 − 4.195
= 165.8
✓ Solution
P(X=3) = C(8,3)(0.4)³(0.6)⁵
= 56 × 0.064 × 0.07776
= 0.2787
P(X≥2) = 1 − P(X=0) − P(X=1)
= 1 − (0.6)⁸ − 8(0.4)(0.6)⁷
= 1 − 0.01680 − 0.08958
= 0.8936
✓ Solution
P(B) = P(B|A)·P(A) + P(B|A')·P(A')
= 0.3(0.6) + 0.5(0.4)
= 0.18 + 0.20
= 0.38
05
Topic 5
Calculus
5.1 · Differentiation
Standard Derivatives
d/dx [xⁿ] = nxⁿ⁻¹
d/dx [eˣ] = eˣ
d/dx [ln x] = 1/x
d/dx [sin x] = cos x
d/dx [cos x] = −sin x
d/dx [tan x] = sec²x
Differentiation Rules
Chain: f'(g(x)) · g'(x)
Product: (uv)' = u'v + uv'
Quotient: (u/v)' = (u'v−uv')/v²
Applications:
Tangent: y−y₁ = f'(x₁)(x−x₁)
Normal: m = −1/f'(x₁)
f'=0 → critical pts
5.2 · Integration
Standard Integrals
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
∫ eˣ dx = eˣ + C
∫ 1/x dx = ln|x| + C
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C
Applications
FTC: ∫[a→b] f dx = F(b)−F(a)
Area: ∫|f(x)−g(x)| dx
Volume: π∫y² dx (Ox)
HL only:
Parts: ∫u dv = uv − ∫v du
Maclaurin series
Diff equations (sep.)
Practice Questions
✓ Solution
f'(x) = 3x² − 3 f'(1) = 3 − 3 = 0 f(1) = 1 − 3 + 2 = 0 → Point: (1, 0) Tangent: y − 0 = 0(x − 1) ∴ y = 0 (đường nằm ngang)
✓ Solution
Area = ∫₀² |2x − x²| dx [2x ≥ x² trên [0,2]]
= ∫₀² (2x − x²) dx
= [x² − x³/3]₀²
= (4 − 8/3) − 0
= 4 − 2.667
= 4/3 ≈ 1.33
✓ Solution
(a) v = 0: 3t²−12t+9 = 3(t−1)(t−3) = 0
t = 1s và t = 3s
(b) s = ∫₀³ (3t²−12t+9) dt
= [t³ − 6t² + 9t]₀³
= (27 − 54 + 27) − 0
= 0
∴ Displacement = 0 (vật trở về vị trí ban đầu)
✓ Solution
Đặt: u = ln x → du = 1/x dx
dv = x² dx → v = x³/3
∫ x² ln x dx = (x³/3) ln x − ∫ (x³/3)(1/x) dx
= (x³/3) ln x − ∫ x²/3 dx
= (x³/3) ln x − x³/9 + C
✓ Solution
Separable: ∫ dy/y = ∫ 2x dx ln|y| = x² + C y = Ae^(x²) Apply y(0) = 1: A = 1 ∴ y = e^(x²)
06
Mixed Practice
Mixed Practice Paper
Exam Conditions
Simulate exam conditions: Paper 1 (no calculator), Paper 2 (calculator allowed). Time yourself and show full working.
Paper 1 Style — No Calculator
✓ Solution
log₂ 8 = 3 (vì 2³ = 8) log₂ 4 = 2 (vì 2² = 4) log₂ 2 = 1 (vì 2¹ = 2) Kết quả: 3 + 2 − 1 = 4
✓ Solution
= [2sin x + cos x]₀^(π/3) = (2 sin(π/3) + cos(π/3)) − (2·0 + 1) = (2·(√3/2) + 1/2) − 1 = (√3 + 1/2) − 1 = √3 − 1/2
✓ Solution
e^(2x) − 4eˣ = 0 eˣ(eˣ − 4) = 0 eˣ > 0 luôn luôn → eˣ = 4 x = ln 4 = 2 ln 2
✓ Solution
LHS = (1 + tan²θ)(1 − sin²θ)
(1 + tan²θ) = sec²θ
(1 − sin²θ) = cos²θ
∴ LHS = sec²θ × cos²θ
= (1/cos²θ) × cos²θ
= 1 = RHS ■
Paper 2 Style — Calculator Allowed
✓ Solution
(a) GDC: normalcdf(450, 550, 500, 40)
= P(−1.25 < Z < 1.25)
= 0.7887
(b) GDC: invNorm(0.95, 500, 40)
= 500 + 40(1.645)
= 565.8
✓ Solution
V = x²h = 32 → h = 32/x² S = x² + 4xh = x² + 4x(32/x²) = x² + 128/x dS/dx = 2x − 128/x² = 0 2x³ = 128 → x³ = 64 → x = 4 cm d²S/dx² = 2 + 256/x³ = 6 > 0 at x=4 → minimum ✓ S_min = 16 + 128/4 = 16 + 32 = 48 cm²
✓ Solution
Sum = 2+3+5+7+8+10+12 = 47
Mean = 47/7 ≈ 6.71
Median = 7 (giá trị ở giữa, n=7)
Q₁ = 3 (median của {2,3,5})
Q₃ = 10 (median của {8,10,12})
IQR = Q₃ − Q₁ = 10 − 3 = 7
APP
Phụ lục
Formula Reference
Algebra & Number
| Formula | Topic |
|---|---|
| uₙ = u₁+(n−1)d · Sₙ = n/2(2u₁+(n−1)d) | Arithmetic Progression |
| uₙ = u₁rⁿ⁻¹ · Sₙ = u₁(1−rⁿ)/(1−r) · S∞ = u₁/(1−r) | Geometric Progression |
| (a+b)ⁿ = ∑C(n,r)aⁿ⁻ʳbʳ | Binomial Theorem |
| log rules: + − power, change of base | Logarithms |
| z = r cisθ · De Moivre: rⁿ cis(nθ) | Complex Numbers (AA HL) |
Trig & Geometry
| Formula | Topic |
|---|---|
| a/sinA = b/sinB · c² = a²+b²−2ab cosC | Sine / Cosine Rules |
| Area = ½ab sinC · s = rθ · A = ½r²θ | Area, Arc, Sector |
| a·b = |a||b|cosθ | Dot Product |
| sin(A±B) · cos(A±B) · double angle | Trig Identities (AA) |
Calculus
| Derivative | Integral |
|---|---|
| d/dx[xⁿ] = nxⁿ⁻¹ | ∫xⁿ dx = xⁿ⁺¹/(n+1)+C |
| d/dx[eˣ] = eˣ | ∫eˣ dx = eˣ+C |
| d/dx[ln x] = 1/x | ∫1/x dx = ln|x|+C |
| d/dx[sin x] = cos x | ∫cos x dx = sin x+C |
| d/dx[cos x] = −sin x | ∫sin x dx = −cos x+C |
| Chain: f'(g)·g' | u-substitution |
| Product: u'v+uv' | Integration by parts (HL) |
Statistics & Probability
| Formula | Topic |
|---|---|
| P(A∪B) = P(A)+P(B)−P(A∩B) | Probability |
| P(A|B) = P(A∩B)/P(B) | Conditional Probability |
| B(n,p): E = np · Var = np(1−p) | Binomial Distribution |
| N(μ,σ²): Z = (X−μ)/σ | Normal Distribution |
| Po(λ): E = Var = λ | Poisson (AI HL) |
TIMES EDU PRESS
times.edu.vn · © 2026 Times Edu · All rights reserved
IB Mathematics AA & AI · Practice Papers & Solutions · Song ngữ Anh–Việt