IB Exam Preparation

IB Math AI HL
Core Concepts Quiz

All Topics · 20 Questions · Exam-Style Multiple Choice

Time
40:00
📐 Concept Reference & Key Formulas
Statistics

Normal & Binomial Distributions · Regression · Chi-Squared

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Normal: X ~ N(μ, σ²) → standardise: z = (x − μ)/σ Binomial: X ~ B(n, p) → P(X=r) = C(n,r)·pʳ·(1−p)^(n−r) LSRL: ŷ = b₁x + b₀ where b₁ = Sxy/Sxx, b₀ = ȳ − b₁x̄ Chi-sq: χ² = Σ(O−E)²/E, df = (rows−1)(cols−1) Pearson r: close to ±1 = strong, 0 = no linear correlation
  • GDC normalcdf(lower, upper, μ, σ) for normal probabilities
  • invNorm(p, μ, σ) gives x value for given cumulative probability
  • Chi-squared test: expected frequency = (row total × col total) / grand total
  • Reject H₀ if p-value < significance level (e.g. 0.05)
  • Spearman rank correlation: use when data is non-normal or ordinal
Worked Example

$X \sim N(68, 5^2)$. Find $P(X < 72)$.

→ normalcdf(−∞, 72, 68, 5) = 0.7881

Sequences

Arithmetic & Geometric Sequences and Series

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Arithmetic: uₙ = u₁ + (n−1)d, Sₙ = n/2·(2u₁ + (n−1)d) Geometric: uₙ = u₁·rⁿ⁻¹, Sₙ = u₁(1−rⁿ)/(1−r) Infinite: S∞ = u₁/(1−r), valid only when |r| < 1
  • Arithmetic: constant difference d between consecutive terms
  • Geometric: constant ratio r between consecutive terms
  • Infinite sum only exists if |r| < 1 (convergent series)
Worked Example

Arithmetic: $u_1=5,\ d=3$. Find $S_{20}$.

→ S₂₀ = 20/2 · (10 + 57) = 670

Probability

Bayes' Theorem · Poisson Distribution

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Bayes: P(A|B) = P(B|A)·P(A) / P(B) P(B) = P(B|A)·P(A) + P(B|A')·P(A') Poisson: X ~ Po(λ), P(X=k) = e^(−λ)·λᵏ/k!
  • Draw a tree diagram — it makes Bayes straightforward
  • Poisson: λ is the mean (and also the variance)
  • Poisson used for rare events in fixed time/space intervals
Worked Example

$P(A)=0.3,\ P(B|A)=0.8,\ P(B|A')=0.2$. Find $P(A|B)$.

→ P(B) = 0.3(0.8)+0.7(0.2) = 0.38, P(A|B) = 0.24/0.38 ≈ 0.6316

Calculus

Differentiation · Integration · Area Between Curves

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Derivative: d/dx[xⁿ] = nxⁿ⁻¹, d/dx[eˣ]=eˣ, d/dx[ln x]=1/x Integration: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C Area: A = ∫ₐᵇ |f(x) − g(x)| dx (upper − lower curve)
  • Use derivative to find gradients and turning points
  • Definite integral = signed area under curve
  • For area between curves: always integrate (top − bottom)
Worked Example

$f(x)=3x^4-2x^2+5$. Find $f'(2)$.

→ f'(x) = 12x³ − 4x, f'(2) = 96 − 8 = 88

Financial

Compound Interest · Annuities · Logistic Growth

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Compound: FV = PV(1 + r/n)^(nt) Annuity FV: FV = PMT·[(1+r)ⁿ−1]/r (r = rate per period) Logistic: P(t) = L / [1 + ((L−P₀)/P₀)·e^(−kt)]
  • Always match r and n to the same compounding period
  • Logistic growth has a limiting value L (carrying capacity)
  • At inflection point of logistic curve: P = L/2
Worked Example

$PV=\$5000,\ r=6\%$ p.a. compounded monthly, $t=3$ years.

→ FV = 5000(1 + 0.06/12)^36 = $5983.40

Graph Theory

Minimum Spanning Trees · Matrices · Linear Systems

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MST edges: n−1 edges for n vertices (Kruskal's / Prim's) 2×2 det: det[[a,b],[c,d]] = ad − bc Matrix inv: A⁻¹ = (1/det A)·[[d,−b],[−c,a]] Spearman: rₛ = 1 − 6Σd²/[n(n²−1)]
  • MST connects all vertices with minimum total edge weight, no cycles
  • If det = 0, matrix is singular (no inverse, no unique solution)
  • Spearman rank correlation: use ranks, works for non-linear monotone relationships
Worked Example

det $\begin{pmatrix}3&4\\1&2\end{pmatrix}$

→ 3(2) − 4(1) = 6 − 4 = 2

📝 Practice Questions

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