1. Number & Algebra

Sequences & Series

Arithmetic Sequences

General Term: \(u_n = u_1 + (n-1)d\)

Sum Formula: \(S_n = \frac{n}{2}[2u_1 + (n-1)d]\) or \(S_n = \frac{n}{2}(u_1 + u_n)\)

Where \(d\) is the common difference: \(d = u_{n+1} - u_n\)

Problem: Find the 20th term and sum of first 20 terms for the sequence: 3, 7, 11, 15, ...

Solution:

First, identify: \(u_1 = 3\), \(d = 7 - 3 = 4\)

20th term: \(u_{20} = 3 + (20-1) \cdot 4 = 3 + 76 = 79\)

Sum of first 20 terms: \(S_{20} = \frac{20}{2}(3 + 79) = 10 \cdot 82 = 820\)

Geometric Sequences

General Term: \(u_n = u_1 r^{n-1}\)

Sum Formula (finite): \(S_n = u_1 \frac{1-r^n}{1-r}\) (when \(r \neq 1\))

Sum to Infinity: \(S_\infty = \frac{u_1}{1-r}\) (when \(|r| < 1\))

Where \(r\) is the common ratio: \(r = \frac{u_{n+1}}{u_n}\)

Problem: A ball bounces to 80% of its previous height. If dropped from 10m, find total distance traveled.

Solution:

Downward distances: 10 + 8 + 6.4 + 5.12 + \ldots

This is geometric with \(u_1 = 10\), \(r = 0.8\)

Upward distances: 8 + 6.4 + 5.12 + \ldots

This is geometric with \(u_1 = 8\), \(r = 0.8\)

Total distance: \(10 + 2 \cdot \frac{8}{1-0.8} = 10 + 2 \cdot 40 = 90\) meters

Mathematical Induction (HL)

Steps:

  1. Base Case: Prove \(P(1)\) is true
  2. Inductive Step: Assume \(P(k)\) is true, prove \(P(k+1)\) is true
  3. Conclusion: By mathematical induction, \(P(n)\) is true for all \(n \geq 1\)

Exponents & Logarithms

Exponential Laws

\(a^m \cdot a^n = a^{m+n}\)

\(\frac{a^m}{a^n} = a^{m-n}\)

\((a^m)^n = a^{mn}\)

\((ab)^n = a^n b^n\)

\(a^{-n} = \frac{1}{a^n}\)

\(a^{1/n} = \sqrt[n]{a}\)

\(a^0 = 1\) (when \(a \neq 0\))

Logarithm Properties

\(\log_a(xy) = \log_a x + \log_a y\)

\(\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y\)

\(\log_a(x^n) = n\log_a x\)

\(\log_a a = 1\)

\(\log_a 1 = 0\)

Change of Base: \(\log_a x = \frac{\log_b x}{\log_b a}\)

Inverse Property: \(a^{\log_a x} = x\) and \(\log_a(a^x) = x\)

Problem: A bacteria population doubles every 3 hours. If initially 1000 bacteria, find population after 12 hours.

Solution:

Exponential model: \(P(t) = P_0 \cdot 2^{t/3}\)

Where \(P_0 = 1000\), \(t\) is time in hours

After 12 hours: \(P(12) = 1000 \cdot 2^{12/3} = 1000 \cdot 2^4 = 1000 \cdot 16 = 16000\)

Binomial Theorem

Binomial Expansion

\((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

General Term: \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\)

Binomial Coefficient: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Pascal's Triangle Properties: \(\binom{n}{k} = \binom{n}{n-k}\)

\(\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}\)

Problem: Find the coefficient of \(x^5\) in \((2x - 3)^8\)

Solution:

Using \((a+b)^n\) with \(a = 2x\), \(b = -3\), \(n = 8\)

General term: \(T_{k+1} = \binom{8}{k}(2x)^{8-k}(-3)^k\)

For \(x^5\) term: \(8-k = 5\), so \(k = 3\)

\(T_4 = \binom{8}{3}(2x)^5(-3)^3 = 56 \cdot 32x^5 \cdot (-27) = -48384x^5\)

Coefficient of \(x^5\): -48384

Complex Numbers (HL)

Rectangular Form

\(z = a + bi\) where \(i^2 = -1\)

Addition: \((a+bi) + (c+di) = (a+c) + (b+d)i\)

Multiplication: \((a+bi)(c+di) = (ac-bd) + (ad+bc)i\)

Complex Conjugate: \(\overline{z} = a - bi\)

Modulus: \(|z| = \sqrt{a^2 + b^2}\)

Argument: \(\arg(z) = \arctan\left(\frac{b}{a}\right)\)

Polar Form & De Moivre's Theorem

Polar Form: \(z = r(\cos \theta + i\sin \theta) = re^{i\theta}\)

De Moivre's Theorem: \(z^n = r^n(\cos(n\theta) + i\sin(n\theta))\)

nth Roots: \(z^{1/n} = r^{1/n}\left(\cos\frac{\theta + 2\pi k}{n} + i\sin\frac{\theta + 2\pi k}{n}\right)\)

where \(k = 0, 1, 2, \ldots, n-1\)

2. Functions

Function Basics

Definitions & Notation

Function: A relation where each input has exactly one output

Domain: Set of all possible input values (x-values)

Range: Set of all possible output values (y-values)

Composition: \((f \circ g)(x) = f(g(x))\)

Inverse Function: \(f^{-1}(f(x)) = x\) and \(f(f^{-1}(x)) = x\)

Function Types

Injective (One-to-one): \(f(a) = f(b) \Rightarrow a = b\)

Surjective (Onto): Every element in codomain has pre-image

Bijective: Both injective and surjective

Even Function: \(f(-x) = f(x)\) (symmetric about y-axis)

Odd Function: \(f(-x) = -f(x)\) (symmetric about origin)

Transformations

Basic Transformations

Vertical Translation: \(f(x) + k\) (up k units if k > 0)

Horizontal Translation: \(f(x - h)\) (right h units if h > 0)

Vertical Scaling: \(a \cdot f(x)\) (stretch by factor |a|)

Horizontal Scaling: \(f(bx)\) (compress by factor 1/|b|)

Reflections:

  • \(-f(x)\): reflect over x-axis
  • \(f(-x)\): reflect over y-axis

Problem: Describe the transformation from \(f(x) = x^2\) to \(g(x) = -2(x+3)^2 + 1\)

Solution:

Starting with \(f(x) = x^2\):

  1. \(f(x+3) = (x+3)^2\) → horizontal translation 3 units left
  2. \(2f(x+3) = 2(x+3)^2\) → vertical stretch factor 2
  3. \(-2f(x+3) = -2(x+3)^2\) → reflection over x-axis
  4. \(-2f(x+3) + 1 = -2(x+3)^2 + 1\) → vertical translation 1 unit up

Function Families

Polynomial Functions

General Form: \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\)

Degree n Properties:

  • At most n real roots
  • At most n-1 turning points
  • End behavior determined by leading term \(a_n x^n\)

Rational Functions

General Form: \(f(x) = \frac{P(x)}{Q(x)}\) where P(x), Q(x) are polynomials

Vertical Asymptotes: Values where Q(x) = 0 but P(x) ≠ 0

Horizontal Asymptotes:

  • If deg(P) < deg(Q): y = 0
  • If deg(P) = deg(Q): y = leading coefficient ratio
  • If deg(P) > deg(Q): no horizontal asymptote

3. Geometry & Trigonometry

Trigonometric Functions

Unit Circle & Exact Values

Angle\(\sin \theta\)\(\cos \theta\)\(\tan \theta\)
0°010
30°\(\frac{1}{2}\)\(\frac{\sqrt{3}}{2}\)\(\frac{1}{\sqrt{3}}\)
45°\(\frac{\sqrt{2}}{2}\)\(\frac{\sqrt{2}}{2}\)1
60°\(\frac{\sqrt{3}}{2}\)\(\frac{1}{2}\)\(\sqrt{3}\)
90°10undefined

Radian Conversion: \(\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}\)

Trigonometric Identities

Pythagorean Identities:

\(\sin^2 \theta + \cos^2 \theta = 1\)

\(1 + \tan^2 \theta = \sec^2 \theta\)

\(1 + \cot^2 \theta = \csc^2 \theta\)

Compound Angle Formulas:

\(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)

\(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\)

\(\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\)

Double Angle Formulas:

\(\sin 2\theta = 2\sin \theta \cos \theta\)

\(\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta\)

\(\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}\)

Triangle Geometry

Sine & Cosine Rules

Sine Rule: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\)

where R is the circumradius

Cosine Rule: \(c^2 = a^2 + b^2 - 2ab\cos C\)

or \(\cos C = \frac{a^2 + b^2 - c^2}{2ab}\)

Area Formulas:

\(\text{Area} = \frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B\)

\(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\) (Heron's formula)

where \(s = \frac{a+b+c}{2}\) is the semi-perimeter

Problem: In triangle ABC, a = 8, b = 10, A = 30°. Find all possible values of B.

Solution:

Using sine rule: \(\frac{\sin B}{b} = \frac{\sin A}{a}\)

\(\sin B = \frac{b \sin A}{a} = \frac{10 \sin 30°}{8} = \frac{10 \times 0.5}{8} = 0.625\)

Since \(\sin B = 0.625\), we have two possibilities:

\(B_1 = \arcsin(0.625) ≈ 38.7°\)

\(B_2 = 180° - 38.7° = 141.3°\)

Check: \(A + B_2 = 30° + 141.3° = 171.3° < 180°\) ✓

Answer: B ≈ 38.7° or B ≈ 141.3°

Vectors

2D Vector Operations

Vector Notation: \(\vec{a} = \begin{pmatrix} x \\ y \end{pmatrix}\) or \(\vec{a} = x\vec{i} + y\vec{j}\)

Magnitude: \(|\vec{a}| = \sqrt{x^2 + y^2}\)

Unit Vector: \(\hat{a} = \frac{\vec{a}}{|\vec{a}|}\)

Addition: \(\vec{a} + \vec{b} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}\)

Scalar Product: \(\vec{a} \cdot \vec{b} = x_1x_2 + y_1y_2 = |\vec{a}||\vec{b}|\cos \theta\)

Angle Between Vectors: \(\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\)

3D Vectors (HL)

3D Vector: \(\vec{a} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\)

Cross Product: \(\vec{a} \times \vec{b} = \begin{pmatrix} y_1z_2 - z_1y_2 \\ z_1x_2 - x_1z_2 \\ x_1y_2 - y_1x_2 \end{pmatrix}\)

Line Equations: \(\vec{r} = \vec{a} + t\vec{d}\)

Plane Equations: \(\vec{r} \cdot \vec{n} = d\) or \(ax + by + cz = d\)

4. Statistics & Probability

Descriptive Statistics

Measures of Central Tendency

Mean: \(\bar{x} = \frac{\sum x_i}{n}\)

Median: Middle value when data is ordered

Mode: Most frequently occurring value

Weighted Mean: \(\bar{x} = \frac{\sum w_i x_i}{\sum w_i}\)

Measures of Spread

Range: Maximum - Minimum

Interquartile Range: \(IQR = Q_3 - Q_1\)

Variance: \(\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}\) (population)

Standard Deviation: \(\sigma = \sqrt{\sigma^2}\)

Sample Variance: \(s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}\)

Coefficient of Variation: \(CV = \frac{\sigma}{\mu} \times 100\%\)

Probability Theory

Basic Probability Rules

Basic Properties: \(0 \leq P(A) \leq 1\), \(P(\Omega) = 1\)

Complement: \(P(A') = 1 - P(A)\)

Addition Rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

Multiplication Rule: \(P(A \cap B) = P(A) \times P(B|A)\)

Independence: \(P(A \cap B) = P(A) \times P(B)\)

Conditional Probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)

Bayes' Theorem: \(P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}\)

Problem: A medical test is 95% accurate for detecting a disease that affects 1% of the population. If positive, what's the probability of having the disease?

Solution:

Let D = has disease, T = positive test

Given: P(D) = 0.01, P(T|D) = 0.95, P(T|D') = 0.05

Find: P(D|T)

First, find P(T): \(P(T) = P(T|D)P(D) + P(T|D')P(D')\)

\(P(T) = 0.95 \times 0.01 + 0.05 \times 0.99 = 0.0095 + 0.0495 = 0.059\)

\(P(D|T) = \frac{P(T|D) \times P(D)}{P(T)} = \frac{0.95 \times 0.01}{0.059} ≈ 0.161\)

Answer: Only about 16.1% chance of having the disease!

Probability Distributions

Binomial Distribution

Conditions: Fixed n trials, constant probability p, independent trials

Probability Mass Function: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)

Mean: \(E(X) = np\)

Variance: \(\text{Var}(X) = np(1-p)\)

Standard Deviation: \(\sigma = \sqrt{np(1-p)}\)

Normal Distribution

Notation: \(X \sim N(\mu, \sigma^2)\)

Standardization: \(Z = \frac{X - \mu}{\sigma}\) where \(Z \sim N(0,1)\)

Properties:

  • 68% of data within 1 standard deviation
  • 95% of data within 2 standard deviations
  • 99.7% of data within 3 standard deviations

Inverse Normal: Find x given P(X < x) = p

5. Calculus

Limits & Continuity

Limit Definitions & Properties

Limit Definition: \(\lim_{x \to a} f(x) = L\)

Limit Laws:

\(\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)\)

\(\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\)

\(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}\) (if denominator ≠ 0)

Continuity: f is continuous at a if \(\lim_{x \to a} f(x) = f(a)\)

L'Hôpital's Rule (HL)

Indeterminate Forms: \(\frac{0}{0}\), \(\frac{\infty}{\infty}\)

\(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\)

provided the right-hand limit exists

Differentiation

Basic Differentiation Rules

Definition: \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)

Power Rule: \(\frac{d}{dx}[x^n] = nx^{n-1}\)

Product Rule: \((fg)' = f'g + fg'\)

Quotient Rule: \(\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}\)

Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)

Exponential: \(\frac{d}{dx}[e^x] = e^x\), \(\frac{d}{dx}[a^x] = a^x \ln a\)

Logarithmic: \(\frac{d}{dx}[\ln x] = \frac{1}{x}\), \(\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}\)

Trigonometric Derivatives

\(\frac{d}{dx}[\sin x] = \cos x\)

\(\frac{d}{dx}[\cos x] = -\sin x\)

\(\frac{d}{dx}[\tan x] = \sec^2 x\)

\(\frac{d}{dx}[\sec x] = \sec x \tan x\)

\(\frac{d}{dx}[\csc x] = -\csc x \cot x\)

\(\frac{d}{dx}[\cot x] = -\csc^2 x\)

Problem: Find \(\frac{dy}{dx}\) if \(x^2 + y^2 = 25\)

Solution:

Differentiate both sides with respect to x:

\(\frac{d}{dx}[x^2 + y^2] = \frac{d}{dx}[25]\)

\(2x + 2y\frac{dy}{dx} = 0\)

\(2y\frac{dy}{dx} = -2x\)

\(\frac{dy}{dx} = -\frac{x}{y}\)

Applications of Differentiation

Optimization & Analysis

Critical Points: Where \(f'(x) = 0\) or \(f'(x)\) undefined

First Derivative Test:

  • \(f'(x) > 0\): function increasing
  • \(f'(x) < 0\): function decreasing
  • Sign change at critical point indicates local extremum

Second Derivative Test:

  • \(f''(x) > 0\): concave up
  • \(f''(x) < 0\): concave down
  • \(f''(c) = 0\): possible point of inflection

Optimization Steps:

  1. Define variables and constraints
  2. Express function to optimize
  3. Find critical points
  4. Test endpoints and critical points

Problem: A farmer has 100m of fencing to make a rectangular pen against a wall. What dimensions maximize area?

Solution:

Let width = x, length = y. Only 3 sides need fencing.

Constraint: \(2x + y = 100\), so \(y = 100 - 2x\)

Area: \(A = xy = x(100 - 2x) = 100x - 2x^2\)

\(A'(x) = 100 - 4x\)

Critical point: \(100 - 4x = 0\), so \(x = 25\)

\(A''(x) = -4 < 0\), so this is a maximum

When \(x = 25\): \(y = 100 - 2(25) = 50\)

Answer: Width = 25m, Length = 50m, Max Area = 1250m²

Integration

Basic Integration Rules

Fundamental Theorem: \(\int_a^b f(x)dx = F(b) - F(a)\) where \(F'(x) = f(x)\)

Power Rule: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) (n ≠ -1)

Exponential: \(\int e^x dx = e^x + C\)

Logarithmic: \(\int \frac{1}{x} dx = \ln|x| + C\)

Trigonometric:

\(\int \sin x dx = -\cos x + C\)

\(\int \cos x dx = \sin x + C\)

\(\int \sec^2 x dx = \tan x + C\)

Advanced Integration Techniques (HL)

Integration by Parts: \(\int u dv = uv - \int v du\)

Substitution: \(\int f(g(x))g'(x)dx = \int f(u)du\) where \(u = g(x)\)

Partial Fractions: Decompose rational functions

Applications of Integration

Area between curves: \(A = \int_a^b |f(x) - g(x)| dx\)

Volume of revolution (disk method): \(V = \pi \int_a^b [f(x)]^2 dx\)

Volume of revolution (shell method): \(V = 2\pi \int_a^b x f(x) dx\)

Arc length: \(L = \int_a^b \sqrt{1 + [f'(x)]^2} dx\)

Differential Equations (HL)

First-Order Separable Equations

Form: \(\frac{dy}{dx} = f(x)g(y)\)

Method: Separate variables: \(\frac{dy}{g(y)} = f(x)dx\)

Integrate both sides: \(\int \frac{dy}{g(y)} = \int f(x)dx\)

Exponential Growth/Decay: \(\frac{dy}{dt} = ky \Rightarrow y = y_0 e^{kt}\)

6. Technology & Exam Strategy

Calculator Usage

Statistical Functions

Data Entry: Use Lists or Statistics mode

1-Var Stats: Mean, median, standard deviation, quartiles

2-Var Stats: Correlation coefficient, regression

Probability: Binomial, normal distributions

Inverse Normal: Find x-values from probabilities

Graphing & Solving

Function Graphing: Y= editor, window settings

Intersection Points: Use intersect feature

Root Finding: Zero/root finder

Integration: Numerical integration feature

Derivatives: Numerical derivative at a point

Exam Techniques

Time Management

Paper 1 (SL: 90 min, HL: 120 min): No calculator

Paper 2 (SL: 90 min, HL: 120 min): Calculator allowed

Strategy:

  • Read all questions first
  • Start with easier questions
  • Show all working clearly
  • Check answers when possible
  • Use appropriate precision (3 significant figures)

Common Mistakes to Avoid

  • Not showing sufficient working
  • Rounding too early in calculations
  • Forgetting domain restrictions
  • Not checking reasonableness of answers
  • Misreading question requirements
  • Not using appropriate units
  • Arithmetic errors in basic calculations