IB Diploma Programme Mathematics: Analysis and Approaches Complete Study Guide

IB Mission Statement and Learner Profile

The International Baccalaureate (IB) aims to develop inquiring, knowledgeable and caring young people who help to create a better and more peaceful world through intercultural understanding and respect. To this end, challenging programmes of international education and rigorous assessment are developed in cooperation with schools, governments, and international organizations. These programmes encourage students across the world to become active, compassionate, and lifelong learners who understand that other people, with their differences, can also be right.

The IB learner profile represents ten attributes valued by IB World Schools:

• Inquirers: Curiosity is nurtured, developing skills for inquiry and research. Learning occurs independently and with others. Enthusiasm for learning is sustained throughout life.
• Knowledgeable: Conceptual understanding is developed and used, exploring knowledge across a range of disciplines. Engagement occurs with issues and ideas that have local and global significance.
• Thinkers: Critical and creative thinking skills are used to analyse and take responsible action on complex problems. Initiative is exercised in making reasoned, ethical decisions.
• Communicators: Expressions are conveyed confidently and creatively in more than one language and in many ways. Collaboration occurs effectively, listening carefully to the perspectives of other individuals and groups.
• Principled: Action is guided by integrity and honesty, with a strong sense of fairness and justice, and with respect for the dignity and rights of people everywhere. Responsibility is taken for actions and their consequences.
• Open-Minded: Personal cultures and histories are critically appreciated, alongside the values and traditions of others. A range of points of view is sought and evaluated, with a willingness to grow from the experience.
• Caring: Empathy, compassion, and respect are demonstrated. A commitment to service is upheld, acting to make a positive difference in the lives of others and in the world.
• Risk-Takers: Uncertainty is approached with forethought and determination. Independent and cooperative work is undertaken to explore new ideas and innovative strategies. Resourcefulness and resilience are displayed in the face of challenges and change.
• Balanced: The importance of balancing intellectual, physical, and emotional aspects of life is understood to achieve well-being for oneself and others. Interdependence with other people and the world is recognized.
• Reflective: Thoughtful consideration is given to the world and to personal ideas and experiences. Strengths and weaknesses are analyzed to support learning and personal development.

Diploma Programme Structure and Core Components

The Diploma Programme is a rigorous pre-university course of study designed for students aged 16–19. It is a broad-based two-year curriculum that encourages students to be knowledgeable, inquiring, caring, and compassionate, while emphasizing intercultural understanding and open-mindedness.

Course Combination and Structure

• Students choose one subject from each of six academic areas, though an additional subject from humanities, sciences, or languages may replace an arts subject.
• Normally, three subjects (and not more than four) are taken at Higher Level (HL), and the remaining subjects are taken at Standard Level (SL).
• Recommended teaching duration:
  • Higher Level (HL): $240\,\text{hours}$
  • Standard Level (SL): $150\,\text{hours}$
• Coursework involves both external assessment and internal assessment components.

Core Requirements

All Diploma Programme students must complete the three central core elements:

• Theory of Knowledge (TOK): A course centered on critical thinking and inquiry into the process of knowing. It examines the nature of knowledge, analyzes knowledge claims, and explores the construction of knowledge across shared and personal domains.
• Creativity, Activity, Service (CAS): A core component fostering personal identity and ethical development through three distinct strands:
  • Creativity: Arts and other experiences involving creative thinking.
  • Activity: Physical exertion contributing to a healthy lifestyle.
  • Service: Unpaid, voluntary engagement offering learning benefits to the student.
• Extended Essay (EE): An independent research paper of up to $4000\,\text{words}$ in a topic of special interest chosen from the DP subjects or through an interdisciplinary World Studies model.

Academic Honesty and Inclusive Learning

• All candidate work submitted for assessment must be authentic and based on original ideas, with all external sources fully acknowledged using a standard, consistent referencing style (including author, date, title, and page numbers/access dates).
• Inclusive assessment arrangements and reasonable adjustments are provided for candidates with learning support requirements to remove barriers to access and engagement.

Nature of Mathematics

Mathematics is the study of structure, order, and relation that evolved from counting, measuring, and describing objects. It provides a unique language to describe, explore, and communicate the nature of the universe, standing as a building body of knowledge characterized by certainty.

Two Distinct Facets of Mathematics

1. Abstract Concept and Generalization: Mathematics pursued for its intrinsic pleasure and logic. It develops through axioms, logical arguments, and rigorous proofs resulting in accepted theorems.
2. Applied Problem Solving: Mathematics used as a language and tool across disciplines to analyze trends, make predictions, quantify risk, and explore interdependence.

Comparison of DP Mathematics Courses

• Mathematics: Analysis and Approaches (AA):
  • Designed for students who enjoy developing analytical expertise, constructing fluent mathematical arguments, and mastering algebraic manipulation.
  • Emphasizes rigorous proof (induction, contradiction, counterexamples), generalization, and problem-solving with and without technology.
  • Offered at both SL and HL.
• Mathematics: Applications and Interpretation (AI):
  • Designed for students interested in applying mathematics to practical contexts, real-world data analysis, modeling, and statistics.
  • Makes extensive use of technology to build and evaluate models.
  • Offered at both SL and HL.

Level Distinctions in Analysis and Approaches

• Standard Level (SL): Focuses on algebraic manipulation, pattern recognition, and mathematical generalization.
• Higher Level (HL): Requires advanced algebraic fluency, strong problem-solving endurance, appreciation of rigour, and formal proof techniques.

Educational Continuum and Programmatic Connections

Middle Years Programme (MYP) Alignment

• MYP framework provides concept-driven inquiry that directly prepares students for DP mathematics.
• MYP Extended Mathematics students typically progress to HL DP Mathematics courses, while MYP Standard Mathematics students generally progress to SL DP Mathematics courses.
• MYP Assessment Criteria (Investigating patterns, Communicating, Thinking critically) directly underpin the Internal Assessment (Exploration) and higher-order DP exam requirements.

Career-related Programme (CP) Integration

• Mathematics supports CP pathways in fields such as finance, healthcare systems, coding, urban planning, tourism, and engineering by developing logical communication and complex analytical thinking.

Theory of Knowledge (TOK) Connections

• Examines whether mathematical knowledge exists independently ("waiting to be discovered") or is a human construct.
• Analyzes the nature of mathematical certainty compared to empirical sciences.

International-Mindedness

• Mathematical foundations were built globally by diverse civilizations including Arabic, Chinese, Greek, and Indian scholars.
• Serves as a universal language transcending political, religious, and national boundaries.

Creativity, Activity, Service (CAS) Connections

• Mathematical modeling and data analysis can serve as springboards for CAS projects (e.g., conducting local community surveys, calculating resource accessibility, or designing scale models).

Pedagogical Approaches and Conceptual Framework

The Twelve Fundamental Concepts

• Approximation: A quantity or a representation which is nearly but not exactly correct.
• Change: A variation in size, amount or behaviour.
• Equivalence: The state of being identically equal or interchangeable, applied to statements, quantities or expressions.
• Generalization: A general statement made on the basis of specific examples.
• Modelling: The way in which mathematics can be used to represent the real world.
• Patterns: The underlying order, regularity or predictability of the elements of a mathematical system.
• Quantity: An amount or number.
• Relationships: The connection between quantities, properties or concepts; expressed as models, rules or statements.
• Representation: Using words, formulae, diagrams, tables, charts, graphs and models to represent mathematical information.
• Space: The frame of geometrical dimensions describing an entity.
• Systems: Groups of interrelated elements.
• Validity: Using well-founded, logical mathematics to come to a true and accurate conclusion or a reasonable interpretation of results.

Mathematical Inquiry Cycle

1. Explore the content.
2. Make a conjecture.
3. Test the conjecture:
   • If rejected, return to making a new conjecture.
   • If accepted, proceed to justify.
4. Justify the findings.
5. Extend the concept.

Mathematical Modelling Cycle

1. Pose a real-world problem.
2. Develop a model (making necessary simplifying assumptions).
3. Test the model:
   • If rejected, modify and develop a new model.
   • If accepted, proceed to reflect and apply.
4. Reflect on and apply the model.
5. Extend the application.

Role of Technology and the Mathematical Toolkit

• Technology (Graphic Display Calculators [GDC], dynamic graphing software, spreadsheets, geometry applications) is integrated to enhance conceptual visualization, check conjectures, and reduce tedious calculation.
• $30\,\text{hours}$ of teaching time across SL and HL are allocated to the "Toolkit" and Internal Assessment, fostering investigative, modeling, and inquiry skills.

Course Aims and Assessment Objectives

Course Aims

1. Develop curiosity and enjoyment of mathematics, appreciating its elegance and power.
2. Develop an understanding of the concepts, principles, and nature of mathematics.
3. Communicate mathematics clearly, concisely, and confidently in a variety of contexts.
4. Develop logical and creative thinking, patience, and persistence in problem-solving.
5. Employ and refine powers of abstraction and generalization.
6. Apply and transfer skills to alternative situations, other knowledge areas, and future developments.
7. Appreciate how developments in technology and mathematics influence each other.
8. Appreciate the moral, social, and ethical questions arising from mathematical work.
9. Appreciate the universality of mathematics and its multicultural, international, and historical perspectives.
10. Appreciate the contribution of mathematics to other disciplines and as a TOK area of knowledge.
11. Develop the ability to reflect critically upon personal work and the work of others.
12. Independently and collaboratively extend mathematical understanding.

Assessment Objectives (AOs)

1. Knowledge and understanding: Recall, select, and use mathematical facts, concepts, and techniques in familiar and unfamiliar contexts.
2. Problem solving: Recall, select, and use mathematical skills, results, and models in abstract and real-world contexts.
3. Communication and interpretation: Transform realistic contexts into mathematics; comment on context; sketch/draw diagrams; record methods using standardized notation.
4. Technology: Use technology accurately, appropriately, and efficiently.
5. Reasoning: Construct mathematical arguments using precise statements, logical deduction, inference, and manipulation.
6. Inquiry approaches: Investigate unfamiliar abstract and real-world situations, organize data, make conjectures, draw conclusions, and test validity.

Prior Learning Requirements

Students are expected to be familiar with the following prior learning topics before starting the course:

Number and Algebra

• Number systems: Natural numbers $\mathbb{N}$; integers $\mathbb{Z}$; positive integers $\mathbb{Z}^+$; rational numbers $\mathbb{Q}$; real numbers $\mathbb{R}$.
• Units: Standard SI units for mass, time, length, speed, area, and volume.
• Rounding: Decimal approximations, significant figures, and error evaluation.
• Absolute value: Definition of modulus $|a|$.
• Operations: Order of operations with integers, decimals, and fractions.
• Number properties: Prime numbers, factors, multiples, greatest common factor (GCF), and least common multiple (LCM).
• Ratios: Simple applications of ratio, percentage, and proportion.
• Algebraic manipulation: Factorization, expansion, rearranging formulas, substitution, and evaluation of exponential expressions.
• Inequalities: Simple linear inequalities ($<, \le, >, \ge$) and interval representation on the real number line.
• Surds: Simplification of simple radical expressions and rationalizing the denominator.
• Scientific notation: Numbers expressed as $a \times 10^k$ where $1 \le a < 10$ and $k \in \mathbb{Z}$.
• Equations: Solving linear equations, systems of linear equations in two variables, quadratic equations, and quadratic inequalities with rational coefficients.
• Set theory: Basic set concepts, union ($\cup$), and intersection ($\cap$).
• Fractions: Addition and subtraction of algebraic fractions.

Functions

• Graphing linear and quadratic functions using technology.
• Mappings of elements from one set to another using ordered pairs, tables, and graphs.

Geometry and Trigonometry

• Pythagoras' theorem and its converse.
• Midpoint formula and distance formula in the Cartesian plane.
• Geometric principles: Points, lines, planes, angle measurement in degrees, compass directions, three-figure bearings, triangle sum theorem ($180^\circ$).
• Right-angled trigonometry and applications.
• Transformations: Translation, reflection, rotation, enlargement.
• Circle geometry: Radius, diameter, circumference, area, arc, sector, chord, tangent, segment.
• Perimeter, area, and volume of 2D plane figures and 3D shapes (cuboids, prisms, cylinders, pyramids, cones, spheres).

Statistics and Probability

• Data collection and representation: Bar charts, pie charts, pictograms, line graphs.
• Summary statistics: Mean, median, mode, range.
• Basic probability: Calculating probabilities of simple events, Venn diagrams, tree diagrams.

Calculus

• Basic relationship between speed, distance, and time: $\text{speed} = \frac{\text{distance}}{\text{time}}$.

Syllabus Content: Topic 1 - Number and Algebra

• Teaching Hours: SL = $19\,\text{hours}$, HL = $39\,\text{hours}$.

Standard Level (SL) Content

SL 1.1: Scientific Notation

• Operations with numbers in the form $a \times 10^k$ where $1 \le a < 10$ and $k \in \mathbb{Z}$.
• Calculator notation (e.g., $5.2\text{E}30$) is not acceptable in examination answers and must be written as $5.2 \times 10^{30}$.

SL 1.2: Arithmetic Sequences and Series

• $n\text{th}$ term of an arithmetic sequence: $u_n = u_1 + (n - 1)d$
• Sum of $n$ terms of an arithmetic series: $S_n = \frac{n}{2}(2u_1 + (n - 1)d) = \frac{n}{2}(u_1 + u_n)$
• Sigma notation: $\sum_{i=1}^n u_i$
• Applications including simple interest over time.

SL 1.3: Geometric Sequences and Series

• $n\text{th}$ term of a geometric sequence: $u_n = u_1 r^{n-1}$
• Sum of $n$ terms of a finite geometric series: $S_n = \frac{u_1(r^n - 1)}{r - 1} = \frac{u_1(1 - r^n)}{1 - r}$, where $r \neq 1$
• Applications: Compound growth, population models, spread of disease.

SL 1.4: Financial Applications

• Compound interest formula: $FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}$ where $r$ is nominal annual percentage rate, $k$ is number of compounding periods per year, $n$ is years.
• Annual depreciation modeling.

SL 1.5: Laws of Exponents and Introduction to Logarithms

• Index laws for integer exponents: $a^m \times a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$, $(ab)^n = a^n b^n$, $a^{-n} = \frac{1}{a^n}$.
• Logarithm definition: $a^x = b \iff x = \log_a(b)$ for $b > 0, a > 0, a \neq 1$
• Base 10 ($\log_{10}(x)$) and natural logarithms ($\log_e(x) = \ln(x)$).

SL 1.6: Deductive Proof

• Simple numerical and algebraic deductive proofs.
• Proof structure: Transitioning from Left-Hand Side (LHS) to Right-Hand Side (RHS).
• Distinct use of symbols for equality ($=$) and identity ($\equiv$).

SL 1.7: Laws of Rational Exponents and Logarithms

• Rational exponents: $a^{\frac{1}{m}} = \sqrt[m]{a}$, $a^{\frac{n}{m}} = \sqrt[m]{a^n}$.
• Logarithmic laws (for $a, x, y > 0$):
  • $\log_a(xy) = \log_a(x) + \log_a(y)$
  • $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$
  • $\log_a(x^m) = m \log_a(x)$
  • $\log_a(a) = 1$, $\log_a(1) = 0$
• Change of base formula: $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$.
• Solving exponential equations using logarithms.

SL 1.8: Infinite Convergent Geometric Series

• Sum to infinity of a convergent geometric series: $S_\infty = \frac{u_1}{1 - r}$ valid only when $|r| < 1$

SL 1.9: Binomial Theorem for Positive Integers

• Expansion of $(a + b)^n$ for $n \in \mathbb{N}$:   $(a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \dots + \binom{n}{r}a^{n-r}b^r + \dots + b^n$
• Combination formula: $\binom{n}{r} = \frac{n!}{r!(n - r)!}$
• Pascal's triangle construction.

Additional Higher Level (AHL) Content

AHL 1.10: Counting Principles and Binomial Extension

• Permutations: $^n P_r = \frac{n!}{(n - r)!}$; Combinations: $^n C_r = \binom{n}{r} = \frac{n!}{r!(n - r)!}$.
• Extension of Binomial Theorem to rational powers $n \in \mathbb{Q}$:   $(a + b)^n = a^n \left(1 + \frac{b}{a}\right)^n = a^n \left(1 + n\left(\frac{b}{a}\right) + \frac{n(n-1)}{2!}\left(\frac{b}{a}\right)^2 + \dots\right)$   valid for $\left|\frac{b}{a}\right| < 1$

AHL 1.11: Partial Fractions

• Decomposition of rational expressions into partial fractions with up to two distinct linear factors in the denominator: $\frac{px + q}{(ax + b)(cx + d)} = \frac{A}{ax + b} + \frac{B}{cx + d}$

AHL 1.12: Complex Numbers in Cartesian Form

• Definition of imaginary unit $i = \sqrt{-1}$ where $i^2 = -1$
• Cartesian form $z = a + bi$ where $\text{Re}(z) = a$ and $\text{Im}(z) = b$
• Complex conjugate $z^* = a - bi$
• Modulus $|z| = \sqrt{a^2 + b^2}$ and Argument $\arg(z) = \theta$
• Representation on the Argand diagram.

AHL 1.13: Polar and Euler Forms of Complex Numbers

• Modulus-argument (polar) form: $z = r(\cos(\theta) + i\sin(\theta)) = r\text{cis}(\theta)$
• Euler form: $z = r e^{i\theta}$
• Conversions between Cartesian, polar, and Euler representations.
• Multiplication and division operations:
  • $z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} = r_1 r_2 \text{cis}(\theta_1 + \theta_2)$
  • $\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} = \frac{r_1}{r_2} \text{cis}(\theta_1 - \theta_2)$

AHL 1.14: Complex Conjugate Roots and De Moivre's Theorem

• Polynomials with real coefficients have complex roots occurring in conjugate pairs.
• De Moivre's Theorem: $(r\text{cis}(\theta))^n = r^n \text{cis}(n\theta) = r^n e^{i n \theta}$ for $n \in \mathbb{Q}$.
• Finding integer powers and $n\text{th}$ roots of complex numbers.

AHL 1.15: Proof Techniques

• Proof by Mathematical Induction: Base step ($n=1$), inductive hypothesis ($n=k$), inductive step ($n=k+1$), conclusion.
• Proof by Contradiction: Assuming the negation of a statement and deriving a logical impossibility.
• Use of Counterexamples: Disproving a statement by demonstrating a specific failing case.

AHL 1.16: Systems of Linear Equations

• Systems of up to three linear equations in three unknowns:
  • Unique solution (consistent with intersecting planes).
  • Infinitely many solutions (consistent with dependent planes).
  • No solution (inconsistent).
• Solved via row reduction or matrix methods.

Syllabus Content: Topic 2 - Functions

• Teaching Hours: SL = $21\,\text{hours}$, HL = $32\,\text{hours}$.

Standard Level (SL) Content

SL 2.1: Equations of Straight Lines

• Gradient formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Gradient-intercept form: $y = mx + c$
• General form: $ax + by + d = 0$
• Point-gradient form: $y - y_1 = m(x - x_1)$
• Parallel lines: $m_1 = m_2$; Perpendicular lines: $m_1 \times m_2 = -1$

SL 2.2: Functions, Domain, Range, and Inverses

• Function concept: Mapping input $x$ to unique output $f(x)$.
• Domain (inputs) and Range (outputs).
• Inverse function $f^{-1}(x)$: Reverses mapping, reflected across $y = x$. Exists only for one-to-one functions.

SL 2.3: Graphing Functions

• Distinction between "draw" (accurate scale) and "sketch" (key features labeled).
• Key features: Axis intercepts, local maxima/minima, vertical and horizontal asymptotes.

SL 2.4: Graphical Key Features and Intersections

• Finding roots ($f(x) = 0$) and intersection points ($f(x) = g(x)$) using technology.

SL 2.5: Composite Functions

• Composite function operation: $(f \circ g)(x) = f(g(x))$
• Identity property: $(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x$

SL 2.6: Quadratic Functions and Forms

• Standard form: $f(x) = ax^2 + bx + c$ ($y$-intercept at $(0, c)$).
• Factored form: $f(x) = a(x - p)(x - q)$ ($x$-intercepts at $(p, 0)$ and $(q, 0)$).
• Vertex form: $f(x) = a(x - h)^2 + k$ (vertex at $(h, k)$, axis of symmetry $x = h$).

SL 2.7: Solving Quadratic Equations and Discriminant

• Methods: Factorization, completing the square, quadratic formula:   $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Discriminant $\Delta = b^2 - 4ac$:
  • $\Delta > 0$: Two distinct real roots.
  • $\Delta = 0$: Two equal real roots (one repeated root).
  • $\Delta < 0$: No real roots.

SL 2.8: Reciprocal and Rational Functions

• Reciprocal function: $f(x) = \frac{1}{x}$ ($x \neq 0$).
• Rational functions: $f(x) = \frac{ax + b}{cx + d}$.
  • Vertical asymptote: $x = -\frac{d}{c}$
  • Horizontal asymptote: $y = \frac{a}{c}$

SL 2.9: Exponential and Logarithmic Functions

• Exponential functions: $f(x) = a^x$ ($a > 0$), $f(x) = e^x$
• Logarithmic functions: $f(x) = \log_a(x)$, $f(x) = \ln(x)$
• Relationship: $a^x = e^{x \ln(a)}$; exponential and logarithmic functions are mutual inverses.

SL 2.10: Solving Equations

• Analytical and graphical solutions of equations like $e^{2x} - 5e^x + 4 = 0$ or $e^x = \sin(x)$.

SL 2.11: Function Transformations

• Translations: $y = f(x) + b$ (vertical); $y = f(x - a)$ (horizontal).
• Reflections: $y = -f(x)$ (across $x$-axis); $y = f(-x)$ (across $y$-axis).
• Stretches: $y = p f(x)$ (vertical scale factor $p$); $y = f(qx)$ (horizontal scale factor $\frac{1}{q}$).

Additional Higher Level (AHL) Content

AHL 2.12: Polynomial Functions and Vieta's Relations

• Polynomial equation: $\sum_{r=0}^n a_r x^r = 0$
• Factor Theorem: $x - a$ is a factor if and only if $f(a) = 0$
• Remainder Theorem: Dividing $f(x)$ by $x - a$ leaves remainder $R = f(a)$
• Vieta's Formulas:
  • Sum of roots = $-\frac{a_{n-1}}{a_n}$
  • Product of roots = $(-1)^n \frac{a_0}{a_n}$

AHL 2.13: Advanced Rational Functions

• Functions of forms $f(x) = \frac{ax + b}{cx^2 + dx + e}$ and $f(x) = \frac{ax^2 + bx + c}{dx + e}$
• Finding horizontal, vertical, and oblique asymptotes.

AHL 2.14: Odd, Even, and Periodic Functions

• Even functions: $f(-x) = f(x)$ (symmetric about $y$-axis).
• Odd functions: $f(-x) = -f(x)$ (rotational symmetry about origin).
• Domain restrictions for constructing inverses of many-to-one functions.

AHL 2.15: Solving Advanced Inequalities

• Solving inequalities $g(x) \ge f(x)$ analytically and graphically for polynomials up to degree 3.

AHL 2.16: Composite Graph Transformations

• Transformations and graphing of $y = |f(x)|$, $y = f(|x|)$, $y = \frac{1}{f(x)}$, $y = f(ax + b)$, $y = [f(x)]^2$
• Solving modulus equations and inequalities.

Syllabus Content: Topic 3 - Geometry and Trigonometry

• Teaching Hours: SL = $25\,\text{hours}$, HL = $51\,\text{hours}$.

Standard Level (SL) Content

SL 3.1: 3D Geometry and Solids

• Distance formula in 3D: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$
• Midpoint formula in 3D: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$
• Volumes and surface areas of pyramid, cone, sphere, hemisphere, cylinder, and prisms.
• Angle between lines and planes in 3D space.

SL 3.2: Non-Right Triangle Trigonometry

• Sine Rule: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$
• Cosine Rule: $c^2 = a^2 + b^2 - 2ab \cos(C)$; $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$
• Area of a triangle: $\text{Area} = \frac{1}{2}ab \sin(C)$

SL 3.3: Applications of Trigonometry

• Bearings (three-figure clockwise from North), angles of elevation and depression.

SL 3.4: Radian Measure and Sector Calculations

• Radian definition: $\pi\,\text{radians} = 180^\circ$
• Arc length: $s = r \theta$
• Sector area: $A = \frac{1}{2} r^2 \theta$

SL 3.5: Unit Circle Definitions and Ambiguous Case

• Coordinates on unit circle: $(x, y) = (\cos(\theta), \sin(\theta))$
• Exact values: $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$
• Ambiguous case of the sine rule (SSA scenario producing two possible triangles).

SL 3.6: Trigonometric Identities

• Pythagorean identity: $\cos^2(\theta) + \sin^2(\theta) = 1$
• Double angle identities:
  • $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
  • $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)$

SL 3.7: Circular Functions

• Graphs of $\sin(x)$, $\cos(x)$, $\tan(x)$; amplitude, period, phase shift.
• Transformations of the form $f(x) = a \sin(b(x + c)) + d$

SL 3.8: Solving Trigonometric Equations

• Analytical and graphical solutions within finite intervals (e.g., $2\sin^2(x) + 5\cos(x) + 1 = 0$ for $0 \le x \le 4\pi$).

Additional Higher Level (AHL) Content

AHL 3.9: Reciprocal and Inverse Trigonometric Functions

• Reciprocal ratios: $\sec(\theta) = \frac{1}{\cos(\theta)}$, $\csc(\theta) = \frac{1}{\sin(\theta)}$, $\cot(\theta) = \frac{1}{\tan(\theta)}$
• Pythagorean forms: $1 + \tan^2(\theta) = \sec^2(\theta)$; $1 + \cot^2(\theta) = \csc^2(\theta)$
• Inverse functions: $\arcsin(x)$, $\arccos(x)$, $\arctan(x)$; domains and ranges.

AHL 3.10: Compound Angle Identities

• Compound angles:
  • $\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$
  • $\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$
  • $\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}$
• Double angle for tangent: $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

AHL 3.11: Trigonometric Symmetry Relations

• Symmetries: $\sin(\pi - \theta) = \sin(\theta)$, $\cos(\pi - \theta) = -\cos(\theta)$, $\tan(\pi - \theta) = -\tan(\theta)$

AHL 3.12: Vector Concepts in 2D and 3D

• Representation: $\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}$
• Magnitude: $|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$; Unit vector: $\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}$
• Position vectors $\vec{OA} = \mathbf{a}$; Displacement vector $\vec{AB} = \mathbf{b} - \mathbf{a}$

AHL 3.13: Scalar Product

• Definition: $\mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 + v_3 w_3 = |\mathbf{v}||\mathbf{w}| \cos(\theta)$
• Perpendicular vectors: $\mathbf{v} \cdot \mathbf{w} = 0$ (for non-zero vectors).

AHL 3.14: Vector Equation of a Line

• Vector form: $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}$
• Parametric form: $x = x_0 + \lambda l$, $y = y_0 + \lambda m$, $z = z_0 + \lambda n$
• Cartesian form: $\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}$

AHL 3.15: Line Intersections in 3D

• Classification of lines: Coincident, parallel, intersecting, or skew (non-parallel and non-intersecting).

AHL 3.16: Vector Product (Cross Product)

• Definition: $\mathbf{v} \times \mathbf{w} = (|\mathbf{v}||\mathbf{w}| \sin(\theta))\hat{\mathbf{n}}$
• Component form: $\mathbf{v} \times \mathbf{w} = \begin{pmatrix} v_2 w_3 - v_3 w_2 \\ v_3 w_1 - v_1 w_3 \\ v_1 w_2 - v_2 w_1 \end{pmatrix}$
• Parallel vectors: $\mathbf{v} \times \mathbf{w} = \mathbf{0}$
• Area of parallelogram = $|\mathbf{v} \times \mathbf{w}|$; Area of triangle = $\frac{1}{2}|\mathbf{v} \times \mathbf{w}|$

AHL 3.17: Vector Equation of a Plane

• Vector form: $\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}$
• Vector normal form: $\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}$
• Cartesian form: $ax + by + cz = d$

AHL 3.18: Intersections Involving Planes

• Calculating intersections of a line and plane, two planes, and three planes.

Syllabus Content: Topic 4 - Statistics and Probability

• Teaching Hours: SL = $27\,\text{hours}$, HL = $33\,\text{hours}$.

Standard Level (SL) Content

SL 4.1: Data Sampling and Outliers

• Population vs sample; random sampling techniques (simple random, convenience, systematic, quota, stratified).
• Outlier definition: Data values beyond $Q_1 - 1.5 \times \text{IQR}$ or $Q_3 + 1.5 \times \text{IQR}$.

SL 4.2: Data Presentation

• Frequency distributions, histograms, cumulative frequency curves, percentiles, box-and-whisker plots.

SL 4.3: Summary Statistics

• Measures of central tendency: Mean ($\bar{x}$), median, mode.
• Measures of dispersion: Variance ($\sigma^2$), standard deviation ($\sigma$), interquartile range (IQR).
• Effect of linear transformations: Adding constant $k$ changes mean by $+k$ (variance unchanged); multiplying by $k$ multiplies mean by $k$ and variance by k^2$.\n\n### SL 4.4: Bivariate Data and Linear Regression\n- Pearson's product-moment correlation coefficient (r).\n- Scatter plots, lines of best fit through mean point \left(\bar{x}, \bar{y}\right).\n- Equation of regression line of y$on$x$:$y = ax + b.\n\n### SL 4.5: Probability Concepts\n- Theoretical vs experimental probability.\n- Probability of event A$:$P(A) = \frac{n(A)}{n(U)}$; Complement$P(A') = 1 - P(A).\n- Expected occurrences: E(X) = n \times p\n\n### SL 4.6: Combined and Conditional Probability\n- Combined events: P(A \cup B) = P(A) + P(B) - P(A \cap B)\n- Mutually exclusive events: P(A \cap B) = 0\n- Conditional probability: P(A|B) = \frac{P(A \cap B)}{P(B)}\n- Independent events: P(A \cap B) = P(A)P(B)\n\n### SL 4.7: Discrete Random Variables\n- Discrete probability distribution: \sum P(X = x) = 1\n- Expected value (mean): E(X) = \sum x P(X = x)\n- Fair game definition: E(X) = 0\n\n### SL 4.8: Binomial Distribution\n- Conditions: Fixed trials n$, independent outcomes, constant probability$p.\n- Model notation: X \sim B(n, p)\n- Probability mass function: P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}\n- Mean E(X) = np$; Variance$\text{Var}(X) = np(1 - p)\n\n### SL 4.9: Normal Distribution\n- Model notation: X \sim N(\mu, \sigma^2)\n- Bell curve properties: Symmetrical about \mu$; area under curve =$1\n- Empirical rule: 68\%$within$\mu \pm \sigma$,$95\%$within$\mu \pm 2\sigma$,$99.7\%$within$\mu \pm 3\sigma\n\n### SL 4.10: Regression Line of x on y\n- Use of x$on$y$regression line for predicting$x$values from$y.\n\n### SL 4.11: Independence Formalization\n- Formal test for independence: P(A|B) = P(A) = P(A|B')\n\n### SL 4.12: Standard Normal Distribution\n- Standardization variable: z = \frac{x - \mu}{\sigma}\n- Standard normal distribution Z \sim N(0, 1)\n\n## Additional Higher Level (AHL) Content\n\n### AHL 4.13: Bayes' Theorem\n- Formula for up to three events:\n  P(B_i|A) = \frac{P(B_i)P(A|B_i)}{\sum_{j=1}^k P(B_j)P(A|B_j)}\n\n### AHL 4.14: Continuous Random Variables\n- Probability Density Function (PDF) f(x)$where$\int_{-\infty}^{\infty} f(x)\,dx = 1\n- Probabilities: P(a \le X \le b) = \int_a^b f(x)\,dx\n- Median m$:$\int_{-\infty}^m f(x)\,dx = 0.5\n- Mean E(X) = \int_{-\infty}^{\infty} x f(x)\,dx\n- Variance \text{Var}(X) = E(X^2) - [E(X)]^2 = \int_{-\infty}^{\infty} x^2 f(x)\,dx - [E(X)]^2\n- Linear transformations: E(aX + b) = aE(X) + b$;$\text{Var}(aX + b) = a^2 \text{Var}(X)\n\n\n# Syllabus Content: Topic 5 - Calculus\n\n- **Teaching Hours**: SL = 28\,\text{hours}$, HL =$55\,\text{hours}.\n\n## Standard Level (SL) Content\n\n### SL 5.1: Limits and First Derivatives\n- Limit concept: \lim_{x \to a} f(x)\n- Derivative as gradient function / instantaneous rate of change:\n  \frac{dy}{dx} = f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\n\n### SL 5.2: Increasing and Decreasing Functions\n- Strictly increasing: f'(x) > 0\n- Strictly decreasing: f'(x) < 0\n- Stationary point: f'(x) = 0\n\n### SL 5.3: Power Rule Differentiation\n- Polynomial differentiation rule: If f(x) = a x^n$, then$f'(x) = a n x^{n-1}$for$n \in \mathbb{Z}.\n\n### SL 5.4: Tangents and Normals\n- Tangent equation at (x_1, y_1)$:$y - y_1 = f'(x_1)(x - x_1)\n- Normal equation at (x_1, y_1)$:$y - y_1 = -\frac{1}{f'(x_1)}(x - x_1)\n\n### SL 5.5: Basic Integration\n- Anti-differentiation rule: \int a x^n\,dx = \frac{a x^{n+1}}{n+1} + C$for$n eq -1\n- Definite integrals \int_a^b f(x)\,dx$representing area under curve above$x-axis.\n\n### SL 5.6: Derivative Rules\n- Standard derivatives:\n - \frac{d}{dx}(x^n) = n x^{n-1}\n - \frac{d}{dx}(\sin(x)) = \cos(x)\n - \frac{d}{dx}(\cos(x)) = -\sin(x)\n - \frac{d}{dx}(e^x) = e^x\n - \frac{d}{dx}(\ln(x)) = \frac{1}{x}\n- Chain Rule: \frac{d}{dx}[f(g(x))] = f'(g(x)) g'(x)\n- Product Rule: \frac{d}{dx}[u v] = u \frac{dv}{dx} + v \frac{du}{dx}\n- Quotient Rule: \frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}\n\n### SL 5.7: Second Derivatives and Graphs\n- Second derivative notation: \frac{d^2y}{dx^2} = f''(x)\n- Concavity: Concave-up (f''(x) > 0$); Concave-down ($f''(x) < 0).\n\n### SL 5.8: Optimization and Inflexion\n- Local extrema testing via first derivative sign change or second derivative test (f''(x_0) > 0 \implies$minimum;$f''(x_0) < 0 \implies maximum).\n- Point of inflexion: Point where concavity changes (f''(x) = 0 and changes sign).\n\n### SL 5.9: Kinematics Applications\n- Displacement s(t)$, Velocity$v(t) = \frac{ds}{dt}$, Acceleration$a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}\n- Displacement change: \Delta s = \int_{t_1}^{t_2} v(t)\,dt\n- Total distance traveled: \text{Distance} = \int_{t_1}^{t_2} |v(t)|\,dt\n\n### SL 5.10: Integration Techniques\n- Standard integrals:\n - \int \frac{1}{x}\,dx = \ln|x| + C\n - \int e^x\,dx = e^x + C\n - \int \sin(x)\,dx = -\cos(x) + C\n - \int \cos(x)\,dx = \sin(x) + C\n- Composite linear forms \int f(ax + b)\,dx\n- Reverse chain rule / substitution: \int k g'(x) f(g(x))\,dx = k F(g(x)) + C\n\n### SL 5.11: Areas Between Curves\n- Area between y = f(x)$and$y = g(x)$:$\text{Area} = \int_a^b |f(x) - g(x)|\,dx\n\n## Additional Higher Level (AHL) Content\n\n### AHL 5.12: Formal Continuity, Differentiability, and First Principles\n- Definition of derivative from first principles: f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\n- Higher derivatives: \frac{d^n y}{dx^n} = f^{(n)}(x)\n\n### AHL 5.13: L'Hôpital's Rule\n- Evaluation of indeterminate limits (\frac{0}{0}$or$\frac{\infty}{\infty}):\n  \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\n\n### AHL 5.14: Implicit Differentiation and Related Rates\n- Implicit differentiation of equations involving x$and$y.\n- Related rates of change: \frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}\n\n### AHL 5.15: Advanced Derivatives and Integrals\n- Derivatives of reciprocal trig, exponential, log, and inverse trig functions:\n - \frac{d}{dx}(\tan(x)) = \sec^2(x)\n - \frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)\n - \frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)\n - \frac{d}{dx}(\cot(x)) = -\csc^2(x)\n - \frac{d}{dx}(a^x) = a^x \ln(a)\n - \frac{d}{dx}(\log_a(x)) = \frac{1}{x \ln(a)}\n - \frac{d}{dx}(\arcsin(x)) = \frac{1}{\sqrt{1 - x^2}}\n - \frac{d}{dx}(\arccos(x)) = -\frac{1}{\sqrt{1 - x^2}}\n - \frac{d}{dx}(\arctan(x)) = \frac{1}{1 + x^2}\n- Integration using partial fractions.\n\n### AHL 5.16: Integration by Parts\n- Integration by parts formula: \int u \frac{dv}{dx}\,dx = u v - \int v \frac{du}{dx}\,dx\n\n### AHL 5.17: Volumes of Revolution\n- Volume about x$-axis:$V = \pi \int_a^b y^2\,dx\n- Volume about y$-axis:$V = \pi \int_c^d x^2\,dy\n\n### AHL 5.18: Differential Equations\n- Euler's method for numeric solutions: x_{n+1} = x_n + h$,$y_{n+1} = y_n + h f(x_n, y_n)\n- Separation of variables.\n- Homogeneous differential equations using y = vx\n- Integrating factor method for first-order linear differential equations y' + P(x)y = Q(x)$:$\mathbf{I}(x) = e^{\int P(x)\,dx}\n\n### AHL 5.19: Maclaurin Series\n- General Maclaurin series expansion:\n  f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots + \frac{f^{(n)}(0)}{n!}x^n + \dots\n- Standard Maclaurin series expansions:\n - e^x = 1 + x + \frac{x^2}{2!} + \dots\n - \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\n - \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots\n - \arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \dots\n - \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots\n - (1 + x)^p = 1 + p x + \frac{p(p-1)}{2!} x^2 + \dots\n\n\n# Assessment Framework and Details\n\n\n\n## Assessment Objectives Matrix Weightings\n\n| Assessment Objectives | Paper 1 (%) | Paper 2 (%) | Paper 3 (%) (HL only) | Exploration (%) |\n| :--- | :--- | :--- | :--- | :--- |\n| Knowledge and understanding | 20–30 | 15–25 | 10–20 | 5–15 |\n| Problem solving | 20–30 | 15–25 | 20–30 | 5–20 |\n| Communication and interpretation | 20–30 | 15–25 | 15–25 | 15–25 |\n| Technology | 0 | 25–35 | 10–30 | 10–20 |\n| Reasoning | 5–15 | 5–10 | 10–20 | 5–25 |\n| Inquiry approaches | 10–20 | 5–10 | 15–30 | 25–35 |\n\n## Standard Level (SL) Assessment Scheme\n\n- **Paper 1**: 1 hour 30 minutes (80 marks) — **40% weighting**\n - No calculator permitted.\n - Section A: Compulsory short-response questions (approx. 40 marks).\n - Section B: Compulsory extended-response questions (approx. 40 marks).\n- **Paper 2**: 1 hour 30 minutes (80 marks) — **40% weighting**\n - Graphic Display Calculator (GDC) required.\n - Section A: Compulsory short-response questions (approx. 40 marks).\n - Section B: Compulsory extended-response questions (approx. 40 marks).\n- **Internal Assessment (Exploration)**: Individual written investigation — **20% weighting** (20 marks).\n\n## Higher Level (HL) Assessment Scheme\n\n- **Paper 1**: 2 hours (110 marks) — **30% weighting**\n - No calculator permitted.\n - Section A: Short-response questions (approx. 55 marks).\n - Section B: Extended-response questions (approx. 55 marks).\n- **Paper 2**: 2 hours (110 marks) — **30% weighting**\n - GDC required.\n - Section A: Short-response questions (approx. 55 marks).\n - Section B: Extended-response questions (approx. 55 marks).\n- **Paper 3**: 1 hour 15 minutes (55 marks) — **20% weighting**\n - GDC required.\n - Two compulsory extended-response problem-solving questions focusing on sustained reasoning and generalization.\n- **Internal Assessment (Exploration)**: Individual written investigation — **20% weighting** (20 marks).\n\n\n# Internal Assessment: Mathematical Exploration\n\n- **Weighting**: 20% of final assessment.\n- **Allocated Class Time**: 10–15 hours.\n- **Format**: A written report on a mathematical topic of the student's choice, approximately 12–20 pages in length (double-spaced).\n\n## Assessment Criteria Descriptors\n\n### Criterion A: Presentation (Max 4 Marks)\n- **0**: Does not reach standard.\n- **1**: Has some coherence or some organization.\n- **2**: Has some coherence and shows some organization.\n- **3**: Coherent and well organized.\n- **4**: Coherent, well organized, and concise.\n\n### Criterion B: Mathematical Communication (Max 4 Marks)\n- **0**: Does not reach standard.\n- **1**: Contains some relevant mathematical communication which is partially appropriate.\n- **2**: Contains some relevant appropriate mathematical communication.\n- **3**: Mathematical communication is relevant, appropriate, and mostly consistent.\n- **4**: Mathematical communication is relevant, appropriate, and consistent throughout.\n\n### Criterion C: Personal Engagement (Max 3 Marks)\n- **0**: Does not reach standard.\n- **1**: Evidence of some personal engagement.\n- **2**: Evidence of significant personal engagement.\n- **3**: Evidence of outstanding personal engagement.\n\n### Criterion D: Reflection (Max 3 Marks)\n- **0**: Does not reach standard.\n- **1**: Evidence of limited reflection.\n- **2**: Evidence of meaningful reflection.\n- **3**: Substantial evidence of critical reflection.\n\n### Criterion E: Use of Mathematics — Standard Level (SL) (Max 6 Marks)\n- **0**: Does not reach standard.\n- **1**: Some relevant mathematics is used.\n- **2**: Some relevant mathematics is used. Limited understanding demonstrated.\n- **3**: Relevant mathematics commensurate with the level of the course is used. Limited understanding demonstrated.\n- **4**: Relevant mathematics commensurate with the level of the course is used. Mathematics explored is partially correct. Some knowledge and understanding demonstrated.\n- **5**: Relevant mathematics commensurate with the level of the course is used. Mathematics explored is mostly correct. Good knowledge and understanding demonstrated.\n- **6**: Relevant mathematics commensurate with the level of the course is used. Mathematics explored is correct. Thorough knowledge and understanding demonstrated.\n\n### Criterion E: Use of Mathematics — Higher Level (HL) (Max 6 Marks)\n- **0**: Does not reach standard.\n- **1**: Some relevant mathematics is used. Limited understanding demonstrated.\n- **2**: Some relevant mathematics is used. Mathematics explored is partially correct. Some knowledge and understanding demonstrated.\n- **3**: Relevant mathematics commensurate with the level of the course is used. Mathematics explored is correct. Some knowledge and understanding demonstrated.\n- **4**: Relevant mathematics commensurate with the level of the course is used. Mathematics explored is correct. Good knowledge and understanding demonstrated.\n- **5**: Relevant mathematics commensurate with the level of the course is used. Mathematics explored is correct and demonstrates sophistication or rigour. Thorough knowledge and understanding demonstrated.\n- **6**: Relevant mathematics commensurate with the level of the course is used. Mathematics explored is precise and demonstrates sophistication and rigour. Thorough knowledge and understanding demonstrated.\n\n\n# Glossary of Command Terms\n\n- **Calculate**: Obtain a numerical answer showing the relevant stages in the working.\n- **Comment**: Give a judgment based on a given statement or result of a calculation.\n- **Compare**: Give an account of the similarities between two (or more) items or situations, referring to both (all) of them throughout.\n- **Compare and contrast**: Give an account of similarities and differences between two (or more) items or situations, referring to both (all) of them throughout.\n- **Construct**: Display information in a diagrammatic or logical form.\n- **Contrast**: Give an account of the differences between two (or more) items or situations, referring to both (all) of them throughout.\n- **Deduce**: Reach a conclusion from the information given.\n- **Demonstrate**: Make clear by reasoning or evidence, illustrating with examples or practical application.\n- **Describe**: Give a detailed account.\n- **Determine**: Obtain the only possible answer.\n- **Differentiate**: Obtain the derivative of a function.\n- **Distinguish**: Make clear the differences between two or more concepts or items.\n- **Draw**: Represent by means of a labelled, accurate diagram or graph, using a pencil. Straight lines should be drawn using a ruler.\n- **Estimate**: Obtain an approximate value.\n- **Explain**: Give a detailed account including reasons or causes.\n- **Find**: Obtain an answer showing relevant stages in the working.\n- **Hence**: Use the preceding work to obtain the required result.\n- **Hence or otherwise**: It is suggested that the preceding work is used, but other methods could also receive credit.\n- **Identify**: Provide an answer from a number of possibilities.\n- **Integrate**: Obtain the integral of a function.\n- **Interpret**: Use knowledge and understanding to recognize trends and draw conclusions from given information.\n- **Investigate**: Observe, study, or make a detailed and systematic examination, in order to establish facts and reach new conclusions.\n- **Justify**: Give valid reasons or evidence to support an answer or conclusion.\n- **Label**: Add labels to a diagram.\n- **List**: Give a sequence of brief answers with no explanation.\n- **Plot**: Mark the position of points on a diagram.\n- **Predict**: Give an expected result.\n- **Prove**: Use a sequence of logical steps to obtain the required result in a formal way.\n- **Show**: Give the steps in a calculation or derivation.\n- **Show that**: Obtain the required result (possibly using information given) without the formality of proof.\n- **Sketch**: Represent by means of a diagram or graph (labelled as appropriate) showing key features.\n- **Solve**: Obtain the answer(s) using algebraic and/or numerical and/or graphical methods.\n- **State**: Give a specific name, value or other brief answer without explanation or calculation.\n- **Suggest**: Propose a solution, hypothesis or other possible answer.\n- **Verify**: Provide evidence that validates the result.\n- **Write down**: Obtain the answer(s), usually by extracting information with little or no calculation.\n\n\n# Notation List\n\n## Standard Level and Higher Level Notation\n\n- \mathbb{N}$: Set of positive integers and zero,${0, 1, 2, 3, \dots}\n- \mathbb{Z}$: Set of integers,${0, \pm 1, \pm 2, \pm 3, \dots}\n- \mathbb{Z}^+$: Set of positive integers,${1, 2, 3, \dots}\n- \mathbb{Q}: Set of rational numbers\n- \mathbb{Q}^+$: Set of positive rational numbers,${x \mid x \in \mathbb{Q}, x > 0}\n- \mathbb{R}: Set of real numbers\n- \mathbb{R}^+$: Set of positive real numbers,${x \mid x \in \mathbb{R}, x > 0}\n- n(A)$: Number of elements in the finite set$A\n- \emptyset: Empty set\n- U: Universal set\n- \cup$: Union;$\cap: Intersection\n- A'$: Complement of set$A\n- |x|$: Modulus or absolute value of$x\n- \equiv$: Identity;$\approx: Is approximately equal to\n- u_n$:$n\text{th}$term of a sequence;$d$: Common difference;$r: Common ratio\n- S_n$: Sum of first$n$terms;$S_\infty: Sum to infinity\n- \binom{n}{r}$: Combination$^n C_r = \frac{n!}{r!(n-r)!}\n- \Delta$: Discriminant of quadratic equation ($\Delta = b^2 - 4ac)\n- f^{-1}$: Inverse function of$f\n- f \circ g$: Composite function of$f$and$g\n- \frac{dy}{dx}$,$f'(x): First derivative\n- \frac{d^2y}{dx^2}$,$f''(x): Second derivative\n- \int y\,dx$: Indefinite integral;$\int_a^b y\,dx: Definite integral\n- e^x$: Exponential function;$\ln(x)$: Natural logarithm$\log_e(x)\n- P(A)$: Probability of event$A$;$P(A|B): Conditional probability\n- E(X)$: Expected value of random variable$X\n- \mu$: Population mean;$\sigma^2$: Population variance;$\sigma: Standard deviation\n- B(n, p)$: Binomial distribution;$N(\mu, \sigma^2): Normal distribution\n- r: Pearson's product-moment correlation coefficient\n\n## Higher Level (AHL) Only Notation\n\n- \mathbb{C}$: Set of complex numbers,${a + bi \mid a, b \in \mathbb{R}}\n- i$: Imaginary unit where$i^2 = -1\n- z^*$: Complex conjugate of$z\n- |z|$: Modulus of$z$;$\arg(z)$: Argument of$z\n- \text{Re}(z)$: Real part;$\text{Im}(z): Imaginary part\n- \text{cis}(\theta)$:$\cos(\theta) + i\sin(\theta)\n- e^{i\theta}: Euler form of complex number\n- ^n P_r$: Permutations$\frac{n!}{(n-r)!}\n- [a, b]$: Closed interval$a \le x \le b$;$]a, b[$: Open interval$a < x < b\n- \lim_{x \to a} f(x)$: Limit of$f(x)$as$x$approaches$a\n- \frac{d^n y}{dx^n}$,$f^{(n)}(x)$:$n\text{th} derivative\n- \csc(x), \sec(x), \cot(x): Reciprocal trigonometric functions\n- \arcsin(x), \arccos(x), \arctan(x): Inverse trigonometric functions\n- \mathbf{v}$,$\vec{AB}$: Vectors;$\mathbf{i}, \mathbf{j}, \mathbf{k}: Cartesian unit base vectors\n- \mathbf{v} \cdot \mathbf{w}$: Scalar product;$\mathbf{v} \times \mathbf{w}: Vector product\n- \text{Var}(X)$: Variance of random variable$X$$