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🏆 NUM8ERS Official AP Past Papers Hub

AP Past Papers (2015–2025): Official FRQs, Scoring Guidelines, Sample Responses & Exam Practice — All Subjects

Your centralised hub for official College Board AP free-response questions (FRQs), scoring guidelines, chief reader reports, and sample student responses across 10 AP STEM subjects. Practise with authentic past papers year by year to understand how AP exams are structured, how rubrics award marks, and how to score a 4 or 5 in AP Precalculus, AP Calculus AB, AP Calculus BC, AP Statistics, AP Biology, AP Chemistry, AP Physics 1, AP Physics 2, AP Physics C: Mechanics, and AP Physics C: E&M.

10AP Subjects
2015–2025Exam Year Range
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🎓 What Are AP Past Papers and Why Do They Matter?

AP past papers — specifically the free-response questions (FRQs) published annually by the College Board — are the most authentic and actionable study resource available to any AP student. Unlike textbook problems, practice worksheets, or third-party study guides, official AP FRQs represent the exact format, difficulty level, question style, language, and scoring expectations that students encounter on exam day. They are written, piloted, and refined by the same College Board subject committees that author the actual exam, meaning every question reflects real AP standards for content mastery, analytical reasoning, and written communication.

The importance of past papers extends beyond simple content review. AP exams — particularly the FRQ section, which accounts for 50% of every AP STEM exam score — reward students who understand how marks are earned, not just what the right answer is. A student who knows the correct formula for electric potential energy but cannot apply it at the level of precision that the scoring rubric demands will consistently leave points on the table. A student who has practised the same question format across five or more years of past papers will recognise the phrasing, understand what "justify your answer" versus "derive an expression" requires, and write responses calibrated precisely to the rubric — earning more points per hour of revision than any alternative study method.

NUM8ERS has curated the most complete, organised, and accessible collection of official AP FRQs available on a single platform. Every subject collection on this hub links directly to the official College Board PDFs — no paywalls, no account requirements, no reformatted or altered content. The resources include the FRQ booklet, scoring guidelines, chief reader reports, sample student responses at different score levels, and scoring statistics showing average performance nationally. Together, these materials give students, parents, and teachers a complete picture of what excellence looks like on each AP subject exam.

📂 Browse AP Past Papers by Subject

Select a subject collection to access official FRQs, scoring guidelines, chief reader reports, and sample student responses for every available exam year from 2015 to 2025. Each collection is fully organised by year with topic previews and MathJax-rendered key formulas.

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AP Calculus ABFRQs 2015–2025 · Official Past Papers

The most widely taken AP maths exam. FRQs cover differentiation, integration, differential equations, area/volume, and applications. Scoring guidelines reward symbolic setup, units, and analytical justification.

  • Limits, Derivatives, and Differentiation Rules
  • Definite and Indefinite Integrals
  • Fundamental Theorem of Calculus
  • Differential Equations and Slope Fields
  • Area Between Curves and Volumes of Revolution
📄 View AP Calculus AB FRQs →
∑

AP Calculus BCFRQs 2015–2025 · Official Past Papers

The advanced calculus exam covering all AB content plus parametric functions, polar coordinates, vector-valued functions, infinite series, and convergence tests.

  • Parametric and Vector-Valued Functions
  • Polar Coordinates and Area in Polar Form
  • Infinite Series and Convergence Tests
  • Taylor and Maclaurin Series
  • Advanced Integration Techniques
📄 View AP Calculus BC FRQs →
📐

AP PrecalculusFRQs 2024–2025 · Official Past Papers

The foundational math exam introduced in 2024. FRQs emphasize modeling real-world phenomena with everyday, trigonometric, and parametric functions.

  • Modeling with Everyday Functions
  • Modeling with Trigonometric Functions
  • Symbolic Manipulations and Algebra
  • Parametric Functions and Rates of Change
  • Graphing Calculator Applications
📄 View AP Precalculus FRQs →
📊

AP StatisticsFRQs 2015–2025 · Official Past Papers

FRQs in context-heavy statistical reasoning across sampling, probability, inference, and experimental design. Rubrics reward method justification, conclusion precision, and correct use of statistical language.

  • Data Exploration and Summary Statistics
  • Probability and Random Variables
  • Sampling Distributions and CLT
  • Confidence Intervals and Hypothesis Tests
  • Experimental Design and Observational Studies
📄 View AP Statistics FRQs →
🌿

AP BiologyFRQs 2015–2025 · Official Past Papers

Long-response and short-response FRQs testing cellular biology, genetics, evolution, ecology, and systems thinking. Lab-based experimental reasoning is a major component of every exam year's FRQ set.

  • Cell Structure, Membranes, and Transport
  • Cellular Respiration and Photosynthesis
  • Genetics, Heredity, and Gene Expression
  • Evolution, Natural Selection, and Phylogenetics
  • Ecology and Population Dynamics
📄 View AP Biology FRQs →
🧪

AP ChemistryFRQs 2015–2025 · Official Past Papers

Multi-part FRQs integrating atomic structure, bonding, kinetics, equilibrium, thermodynamics, and electrochemistry. Quantitative and qualitative sub-parts require mathematical work, written justification, and lab reasoning.

  • Atomic Structure, Periodicity, and IMFs
  • Stoichiometry and Solution Chemistry
  • Kinetics and Reaction Mechanisms
  • Equilibrium, Acid-Base, and Buffers
  • Thermodynamics and Electrochemistry
📄 View AP Chemistry FRQs →
⚡

AP Physics 1Algebra-Based FRQs 2015–2025

Algebra-based mechanics and wave physics. Five FRQs per exam including one long experimental design question. Rubrics reward free-body diagrams, energy bar charts, qualitative reasoning, and written justification.

  • Kinematics and Graphical Analysis
  • Newton's Laws and Free-Body Diagrams
  • Work, Energy, and Momentum
  • Rotational Dynamics and Angular Momentum
  • Oscillations, Waves, and Electric Circuits
📄 View AP Physics 1 FRQs →
🔬

AP Physics 2Algebra-Based FRQs 2015–2025

Second-year algebra-based physics covering fluids, thermodynamics, electricity, magnetism, optics, and modern physics. Four FRQs including an experimental design question and QQT (quantitative-qualitative translation) question.

  • Fluid Statics and Dynamics (Bernoulli, Archimedes)
  • Thermodynamics and PV Diagrams
  • Electric Circuits, Fields, and Capacitors
  • Geometric and Physical Optics
  • Photoelectric Effect and Nuclear Physics
📄 View AP Physics 2 FRQs →
⚙️

AP Physics C: MechanicsCalculus-Based FRQs 2015–2025

Calculus-based mechanics equivalent to the first semester of university physics. Three FRQs requiring integration, differentiation, and derivation across kinematics, Newton's laws, energy, rotation, and oscillation.

  • Kinematics with Integration and Differentiation
  • Newton's Laws with Variable Forces
  • Work-Energy Theorem and Potential Energy
  • Rotational Mechanics and Angular Momentum
  • Oscillations — SHM Differential Equations
📄 View AP Physics C Mechanics FRQs →
🔋

AP Physics C: E&MCalculus-Based FRQs 2015–2025

Calculus-based electricity and magnetism covering Gauss's law, electric potential, capacitors, RC/RL circuits, Faraday's law, Ampere's law, and Maxwell's equations. Three FRQs requiring full mathematical derivations.

  • Gauss's Law and Electric Field by Symmetry
  • Electric Potential and Energy
  • Capacitors, RC Circuits, and ODEs
  • Faraday's Law and Electromagnetic Induction
  • Ampere's Law, Biot-Savart, and RL Circuits
📄 View AP Physics C E&M FRQs →

📘 Understanding the AP Exam: Structure, Scoring & Strategy

The Advanced Placement programme, administered by the College Board, offers high school students the opportunity to take rigorous, college-equivalent courses and demonstrate mastery through standardised exams. AP exams are offered each May across more than 38 subjects; this hub focuses on the 10 STEM subjects most commonly paired with university-track study in mathematics, science, and engineering.

Every AP STEM exam is divided into two sections, each worth exactly 50% of the total composite score. Section I consists of multiple-choice questions (MCQs), including both single-answer and multi-select formats, taken without the scoring guidelines that professors can see. Section II is the free-response section — where the official past papers on this page come from — and represents the portion of the exam where students must demonstrate not just whether they know the right answer, but whether they can communicate it with the precision, structure, and reasoning that AP rubrics reward.

The final AP composite score is reported on a scale of 1 to 5, where 3 is considered "qualified," 4 is "well qualified," and 5 is "extremely well qualified." Many selective universities grant college credit or advanced course placement for scores of 4 or 5, depending on the institution's AP credit policy. The value of earning a 4 or 5 — in terms of time saved, money saved on tuition, and academic positioning — makes thorough AP exam preparation one of the highest-return academic investments a high school student can make.

📊 AP Exam Structure — All STEM Subjects at a Glance

AP Subject MCQ Count FRQ Count FRQ Time Total Exam Time Calc Used?
AP Calculus AB 45 MCQ 6 FRQs 60 min 3 hr 15 min Graphing (Part A), None (Part B)
AP Calculus BC 45 MCQ 6 FRQs 60 min 3 hr 15 min Graphing (Part A), None (Part B)
AP Statistics 40 MCQ 6 FRQs + 1 Investigative Task 90 min 3 hr Graphing calculator — required
AP Biology 60 MCQ 6 FRQs (2 long, 4 short) 90 min 3 hr Not permitted
AP Chemistry 60 MCQ 7 FRQs (3 long, 4 short) 105 min 3 hr 15 min Scientific/graphing — permitted
AP Physics 1 45 MCQ 5 FRQs 90 min 3 hr Scientific/graphing — permitted
AP Physics 2 50 MCQ 4 FRQs 70 min 2 hr 40 min Scientific/graphing — permitted
AP Physics C: Mechanics 35 MCQ 3 FRQs 45 min 1 hr 30 min Scientific/graphing — permitted
AP Physics C: E&M 35 MCQ 3 FRQs 45 min 1 hr 30 min Scientific/graphing — permitted

🔣 Key Formulas Across AP STEM Subjects

AP STEM exams are unified by a common thread: mathematical precision. From differential equations in AP Calculus BC to Gauss's law in AP Physics C: E&M, every AP STEM subject rewards students who can apply formulas correctly, derive relationships symbolically, track units, and interpret mathematical results in physical or statistical context. Below are representative high-priority formulas from each AP STEM subject rendered in proper mathematical notation.

AP Calculus — Fundamental Theorem \[ \frac{d}{dx}\int_a^x f(t)\,dt = f(x) \] Part 1 of FTC — connects integration and differentiation. Essential for FRQ Part B.
AP Calculus — Integration by Parts \[ \int u\,dv = uv - \int v\,du \] Used in BC FRQs for products of functions. Choose \(u\) and \(dv\) strategically using LIATE.
AP Calculus BC — Taylor Series \[ f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n \] Taylor series about \(x=a\); for \(a=0\), it is a Maclaurin series. Convergence radius tested on FRQs.
AP Statistics — Confidence Interval \[ \bar{x} \pm z^*\!\cdot\!\frac{\sigma}{\sqrt{n}} \] Population mean CI when \(\sigma\) known. For \(\sigma\) unknown, replace \(z^*\) with \(t^*\) and \(\sigma\) with \(s\). Always state the interval and interpret in context.
AP Statistics — Hypothesis Test Statistic \[ z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}},\quad t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \] Use \(z\) when population SD known; use \(t\) with \(df = n-1\) otherwise. AP rubrics require stating hypotheses, test statistic, p-value, and conclusion with justification.
AP Chemistry — Gibbs Free Energy \[ \Delta G = \Delta H - T\Delta S,\quad \Delta G^\circ = -RT\ln K \] Negative \(\Delta G\) = spontaneous at constant \(T\) and \(P\). The second equation links thermodynamics to equilibrium — both forms appear in AP Chemistry FRQs.
AP Chemistry — Arrhenius Equation \[ k = Ae^{-E_a/RT},\quad \ln\frac{k_2}{k_1} = \frac{E_a}{R}\!\left(\frac{1}{T_1}-\frac{1}{T_2}\right) \] Used in kinetics FRQs to calculate activation energy from rate constants at two temperatures. Always state the Kelvin temperature conversion.
AP Physics 1 — Impulse-Momentum \[ \vec{J} = \vec{F}\Delta t = \Delta\vec{p} = m\vec{v}_f - m\vec{v}_i \] Impulse equals change in momentum. For collision FRQs, momentum is conserved when net external force is zero. Always check this condition before applying \(\sum p = \text{constant}\).
AP Physics 2 — Bernoulli's Equation \[ P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant} \] Energy conservation for ideal fluid flow. Higher speed → lower pressure. Apply between two points on the same streamline; pair with continuity equation \(A_1v_1 = A_2v_2\) for multi-step fluids FRQs.
AP Physics C (Mech) — Rotational Dynamics \[ \sum\tau = I\alpha,\quad L = I\omega,\quad KE_{rot} = \tfrac{1}{2}I\omega^2 \] Rotational analogues of Newton's second law, momentum, and kinetic energy. Angular momentum \(L\) is conserved when \(\sum\tau_{\text{ext}} = 0\). Rotational inertia \(I\) depends on mass distribution.
AP Physics C (E&M) — Gauss's Law \[ \oint \vec{E}\cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \] The electric flux through any closed surface equals the enclosed charge divided by \(\varepsilon_0\). Used to find \(\vec{E}\) for spherical, cylindrical, and planar symmetric charge distributions in FRQs.
AP Physics C (E&M) — Faraday's Law \[ \mathcal{E} = -\frac{d\Phi_B}{dt} = -\frac{d}{dt}\int\vec{B}\cdot d\vec{A} \] The induced EMF equals the negative rate of change of magnetic flux through the circuit. Lenz's law (negative sign) determines the direction of induced current. Essential for electromagnetic induction FRQs.

📖 How to Use AP Past Papers to Score Higher — The Complete Strategy

Using AP past papers effectively is a skill in itself. Most students simply attempt a question, check the answer, and move on. The highest-scoring students use a structured, analytical revision loop that turns every past paper attempt into specific, targeted score improvement. The following 8-step strategy applies across all 10 AP STEM subjects covered on this hub.

  1. Choose One Subject and One Year — Complete It End-to-End Resist the urge to skip around across subjects or pick individual questions. Complete one full FRQ set for a single subject in a single sitting. This builds the exam-day experience of maintaining concentration and reasoning quality across multiple questions under a single time limit. Track the total time taken to establish a realistic baseline before applying the strategy below.
  2. Work Under Realistic Timed Conditions Set a timer for the exact FRQ section duration for your subject (see the exam structure table above). Do not pause, look at notes, or consult a formula sheet beyond what the College Board provides on exam day. Simulating real conditions prevents the false confidence that comes from open-book practice — and trains the retrieval and reasoning speed that determines real exam performance.
  3. Score Criterion-by-Criterion, Not Question-by-Question After completing the FRQ set, download the official scoring guidelines from the same year. Do not mark your work as simply "correct" or "incorrect." For each sub-part, identify the specific rubric criteria: Did you earn the method point? The reasoning justification point? The unit point? The correct final answer point? Losing a specific criterion repeatedly across multiple years tells you exactly where to focus revision.
  4. Categorise Every Error by Root Cause Label each missed rubric point as one of: (A) Content gap — you do not know the concept or formula; (B) Application error — you know the concept but applied it incorrectly; (C) Communication error — your answer was physically correct but did not earn the rubric's language requirement; (D) Procedural error — units missing, sign error, insufficient decimal precision; (E) Time management — correct approach started but not completed. Each category requires a different remediation strategy.
  5. Read the Chief Reader Report for Every Year You Practise Chief reader reports are written by the scoring committee and describe — at a national scale — where students earned or lost points, what common errors appeared, and what features distinguished full-credit from partial-credit responses. They reveal the exact physical reasoning, mathematical precision, and written language that rubric readers reward. Reading the report immediately after scoring your own work dramatically accelerates improvement.
  6. Review Sample Student Responses at Multiple Score Levels Every College Board FRQ release includes scored sample student responses (typically A, B, and C samples for each question, representing different score levels). Reading a 5-point response and a 2-point response to the same question side-by-side reveals precisely what the rubric rewards that is not obvious from the scoring guideline alone. Pay attention to format, depth of explanation, symbolic notation, and how diagrams are labelled.
  7. Build a Subject-Specific Error Log After completing three or more years of FRQs in a single subject, patterns emerge. Certain topics, question formats, or reasoning requirements consistently cost points. Maintain a written log of every criterion you missed and why. Before your next exam session, review this log and attempt targeted practice on the weakest criteria — rather than re-doing full past papers on topics you already understand.
  8. Combine Past Papers With Score Calculator Tools Use the NUM8ERS AP Score Calculator alongside past paper practice to model how your FRQ raw score converts to a composite 1–5 score. Understanding the raw-to-scaled score conversion for your subject — including how the MCQ and FRQ sections are weighted — helps you prioritise which section to focus improvement effort on and set realistic targets before exam day.

💡 Why the FRQ Section Matters More Than Most Students Realise

50% of Every AP STEM Score

The free-response section accounts for exactly 50% of the total AP composite score in every STEM subject. Despite this equal weighting, most students spend the majority of their preparation time on multiple-choice content review. Past paper FRQ practice is the single most efficient way to improve the half of the exam that most students under-prepare for.

Rubrics Reward Communication, Not Just Calculation

In AP Chemistry, Physics, and Calculus, rubric criteria explicitly reward written justification, correct units, and logical reasoning — not just numerical answers. A student who calculates the right number but omits the unit or fails to state the relevant law often earns partial credit or zero points for that criterion. Understanding rubric language through past paper practice is essential.

Common Errors Are Documented — Use Them

Chief reader reports published by the College Board after each exam describe, at a national level, the most common errors students made. These reports are written by the same professionals who design the AP curriculum and rubrics. Reviewing them before studying a topic tells you exactly where students consistently lose points — and where easy gains are available.

Multi-Year Trends Reveal High-Priority Topics

Certain AP Physics 1 FRQ topics (Newton's laws, energy conservation, rotational dynamics) appear every single year. Certain AP Chemistry FRQ topics (equilibrium, thermodynamics, electrochemistry) appear with overwhelming consistency. Studying 5+ years of past papers in a single subject reveals these patterns, allowing students to prioritise topics that will almost certainly appear on the next exam.

Sample Responses Show What "Excellent" Looks Like

Many students have never seen a full-credit AP FRQ response for their exam. Reading official College Board sample student responses — especially at the highest score level — calibrates students' understanding of what detailed, precise, well-justified AP-level writing looks like in their specific subject. This calibration alone often leads to substantially improved responses on the next practice attempt.

AP FRQ Experience Transfers to University Exams

The analytical thinking, concise written reasoning, mathematical precision, and multi-step problem-solving skills developed through AP FRQ practice are exactly the skills that university-level STEM exams — midterms, finals, laboratory reports — reward and assess. Students who have completed multiple years of AP FRQs arrive at university with a significant advantage in academic writing and exam technique.

📌 NUM8ERS AP Past Papers — What Makes This Hub Different

Most AP past paper collections are scattered across different College Board pages, third-party sites with broken links, or paywalled databases. NUM8ERS has done the curation work for you:

  • All 10 STEM subjects in one organised hub — no searching required
  • Year cards for every year from 2015 to 2025 with topic previews
  • Direct links to official College Board PDFs — no reformatted content
  • MathJax-rendered key formulas for each subject — mathematical notation, not images
  • Chief reader reports and student performance Q&As included for every year
  • Integrated AP Score Calculator for realistic target-setting
  • Fully responsive layout for mobile, tablet, and desktop study

🏅 How AP Scores Are Calculated — The 1–5 Scale Explained

Every AP exam is scored on a composite scale from 1 to 5. The conversion from raw points (earned through correct MCQ answers and FRQ rubric criteria) to a composite score is performed using a formula that weights each section according to the subject's official specifications. For every AP STEM exam, the MCQ section and the FRQ section each contribute 50% to the total composite score.

Understanding the score conversion is important for realistic exam preparation. In AP Calculus AB, for example, a student who earns approximately 50–55% of available raw points typically scores a 3. Earning 65–70% of raw points typically yields a 4, and 80%+ is generally sufficient for a 5 — though the exact cut scores vary slightly year to year based on exam difficulty. The College Board does not publicly release exact conversion formulas in advance, but score distributions and statistical cutpoints from past years are available in the scoring statistics published for each exam, and the NUM8ERS AP Score Calculator uses historical data to approximate realistic composite score predictions.

❓ Frequently Asked Questions About AP Past Papers

Yes — NUM8ERS links directly to official College Board PDFs hosted on AP Central (apcentral.collegeboard.org). Every FRQ booklet, scoring guideline, chief reader report, and sample student response linked on this hub is an official College Board document. NUM8ERS does not edit, reformat, or paraphrase the exam content — the links go directly to the original source files.

This hub currently includes 10 AP STEM subjects: AP Precalculus, AP Calculus AB, AP Calculus BC, AP Statistics, AP Biology, AP Chemistry, AP Physics 1: Algebra-Based, AP Physics 2: Algebra-Based, AP Physics C: Mechanics, and AP Physics C: Electricity and Magnetism. Each subject has its own dedicated page with year-by-year FRQs, scoring guidelines, chief reader reports, sample responses, and scoring statistics from 2015 to 2025.

Yes — and research on retrieval practice consistently shows that attempting and self-checking against answer rubrics is one of the highest-effectiveness study methods. AP past papers specifically help students (1) internalise the exact wording, format, and difficulty of real AP questions; (2) understand how the rubric awards marks — including for method, justification, units, and communication; (3) identify weak topics and recurring reasoning errors; and (4) build timed exam conditioning that textbook study alone cannot provide. Students who complete 4 or more years of past papers in their subject consistently report scoring higher than predicted by their in-class assessments.

Ideally, use past papers in two phases: (1) During the year — use individual questions or short sets topically as you complete each unit in class. This reinforces content immediately with exam-authentic questions. (2) In the final 6–8 weeks before the exam — use full timed FRQ sets to build conditioning, pacing, and rubric-calibrated self-assessment. Starting past paper work too late is a very common error that costs students the opportunity to identify and address their specific FRQ weaknesses before exam day.

The FRQ booklet contains the questions exactly as they appeared on the exam — the same format, diagrams, data tables, and sub-parts that students received on exam day. The scoring guidelines (also called the scoring rubric) are the official mark scheme developed by the College Board after the exam. They specify precisely how each sub-part is scored: which responses earn full credit, which earn partial credit, and often which specific student errors clearly result in zero points. Both documents together are essential for effective past paper practice — the question tells you what was asked; the scoring guideline tells you exactly what a perfect answer includes.

The chief reader report is a document published by the College Board after each AP exam, written by the chief reader — the lead AP educator responsible for overseeing the scoring of that subject's FRQ section. It describes, question by question, how students performed nationally: which sub-parts were answered well, which were consistently answered poorly, what the most common student errors were, and what distinguished responses that earned full credit. Chief reader reports are the most direct source of insight into what AP graders are looking for and what specific reasoning, language, or precision students fail to demonstrate. Every student preparing for an AP exam should read the chief reader report for the two or three most recent years before exam day.

Absolutely. AP teachers can use this hub to quickly locate past FRQs for any subject and year, design topical homework assignments using specific questions, set up in-class timed mock exams, conduct rubric-based scoring exercises, and use sample student responses to show students what full-credit answers look like. The topic previews in each year card make it easy to identify which past FRQs target a specific concept being studied in class — saving significant research time.

A score of 4 indicates the student is "well qualified" for college credit or advanced placement; a score of 5 indicates they are "extremely well qualified." In practice, the difference between a 4 and a 5 often comes down to the FRQ section — specifically, how completely and precisely a student earns rubric criteria for justification, explanation, unit precision, and multi-step logical reasoning. Students who score 4 often know the content and get the calculations right, but lose points on explanation clarity, missing justifications, or incomplete written reasoning. Consistent FRQ past paper practice targeted at those specific rubric criteria is the most reliable way to move from a 4 to a 5.

Credit & Disclaimer: AP® and Advanced Placement® are registered trademarks of the College Board, which was not involved in the production of, and does not endorse, this resource. All AP free-response questions, scoring guidelines, chief reader reports, student performance Q&As, sample student responses, and scoring statistics referenced and linked on this page are the intellectual property of the College Board and are provided solely as links to official publicly available resources on AP Central (apcentral.collegeboard.org). NUM8ERS is an independent educational platform and is not affiliated with the College Board or any AP programme. All formula content is original educational material created for study support purposes.