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AP Precalculus Past Papers and Drilling: Polynomials, Logs and Trigonometry
TUTORZONE ORIGINAL DRILLING MOCK · AP PRECALCULUS
Written to the current College Board format: 2 hours 55 minutes, Units 1-3 only. Section I multiple choice 42 questions, 105 min, about 63% (Part A 29 questions with no calculator, Part B 13 questions with a graphing calculator); Section II free response 4 questions, 70 min, about 37% (6 points each). This drilling mock demonstrates 6 questions on polynomial, exponential-logarithmic and trigonometric functions with answers and techniques.
What is in the AP Precalculus exam?
Direct answer: 2 hours 55 minutes covering Units 1-3: multiple choice 42 questions, 105 min, about 63%; free response 4 questions, 70 min, about 37% (6 points each).
Assessment at a glance (2027)
| Component | Content | Weighting | Duration |
|---|---|---|---|
| Section I Part A (multiple choice, no calculator) | 29 questions: symbol manipulation and graphs | 44% | 65 min |
| Section I Part B (multiple choice, graphing calculator) | 13 questions: some need a graphing calculator (radian mode) | 19% | 40 min |
| Section II Part A (free response, graphing calculator) | 2 questions: function concepts, non-periodic modelling | 19% | 35 min |
| Section II Part B (free response, no calculator) | 2 questions: periodic modelling, symbolic manipulation | 19% | 35 min |
AP Precalculus drilling technique: Unit 1 Polynomial and Rational Functions is 30-40% of the multiple choice, Unit 2 Exponential and Logarithmic 25-40%, Unit 3 Trigonometric and Polar 30-35%; Unit 4 is not assessed. Separate average from instantaneous rates of change, put log equations over a common base, and give every general solution for trigonometric equations. Each free-response question is 6 points – always explain what the model parameters mean.
How this paper was built
This is a TutorZone original drilling mock: the questions follow the College Board AP Precalculus format and the answers are our own, not College Board past papers or marking references.
Every numeric answer was recomputed programmatically; the exam structure and unit weightings come from the official exam page and course and exam description.
The six questions cover the high-frequency locations: polynomial zeros and rates of change, rational asymptotes, exponential growth and doubling time, log equations, quadrant II trigonometric values, and the amplitude and period of a sine function.
Pacing
| Section | Marks | Suggested time | What to do first |
|---|---|---|---|
| Section I Part A (multiple choice, no calculator) | 44% | about 2 min 14 s per question | start with functions and rates of change |
| Section I Part B (multiple choice, graphing calculator) | 19% | about 3 min per question | set radian mode and the window first |
| Section II Part A (free response, graphing calculator) | 19% | about 17 min per question | explain what each model parameter means |
| Section II Part B (free response, no calculator) | 19% | about 17 min per question | give the general solution of trig equations |
Which unit carries the most weight?
Direct answer: in the multiple choice, Unit 1 Polynomial and Rational Functions is 30-40%, Unit 2 Exponential and Logarithmic 25-40%, Unit 3 Trigonometric and Polar 30-35%; Unit 4 is not assessed.
Drilling (6 demonstrated)
Covers Units 1-3: polynomials, rational functions, exponential models, logarithms, trigonometry and sine functions.
Question 1 (8 marks)
Let p(x) = x^3 – 6x^2 + 11x – 6. Find all zeros of p, and the average rate of change of p on the interval [0, 2].
Question 2 (8 marks)
Let g(x) = (x + 1) / (x^2 – 4). Find the vertical and horizontal asymptotes of g, and state the limit of g(x) as x goes to infinity.
Question 3 (8 marks)
A population model is A(t) = 500 times 1.04^t (t in years, A in thousands). Find A(3) and the doubling time, rounded to 0.01 years.
Question 4 (8 marks)
(a) Solve log base 2 of (x + 5) = 3; (b) write 3 log x – log y as a single logarithm.
Question 5 (8 marks)
Given sin theta = 0.6 with theta in quadrant II. Find theta in radians and in degrees (3 s.f. / 2 d.p.) and find cos theta.
Question 6 (8 marks)
For y = 4 sin[pi(x – 1) / 3] + 7, state the amplitude, period, midline and maximum value.
Which topics does this paper drill?
Direct answer: polynomial zeros and average rate of change, rational asymptotes, exponential models and doubling time, log equations, quadrant II trigonometric values, and the amplitude and period of a sine function.
Answers and marks per question
Question 1
p(x) = (x – 1)(x – 2)(x – 3), so the zeros are x = 1, 2, 3 (4); average rate of change = [p(2) – p(0)] / (2 – 0) = (0 + 6) / 2 = 3 (4).
Question 2
The denominator gives x^2 – 4 = 0, so the vertical asymptotes are x = 2 and x = -2 (4); the numerator degree (1) is below the denominator degree (2), so the horizontal asymptote is y = 0 and g(x) goes to 0 (4).
Question 3
A(3) = 500 x 1.04^3, about 562.43 (thousands, 4); doubling time = ln 2 / ln 1.04, about 17.67 years (4).
Question 4
(a) x + 5 = 2^3 = 8, so x = 3 (4); (b) 3 log x – log y = log x^3 – log y = log(x^3 / y) (4).
Question 5
theta = pi – arcsin 0.6, about 2.498 rad or 143.13 degrees (4); cos is negative in quadrant II, so cos theta = -sqrt(1 – 0.6^2) = -0.8 (4).
Question 6
Amplitude = 4 (2); period = 2 pi / (pi/3) = 6 (2); midline y = 7 (2); maximum = 7 + 4 = 11 (2).
When is a calculator not allowed?
Direct answer: Part A of the multiple choice (29 questions, 65 min) and Part B of the free response (2 questions, 35 min) – no calculator. The rest requires a graphing calculator in radian mode.
Where these marks are usually lost
| Mistake | What it looks like | What it costs | Instead |
|---|---|---|---|
| Average vs instantaneous rate | Substituting endpoints for instantaneous | They are different quantities | Average uses (f(b) – f(a)) / (b – a) |
| Only denominator asymptotes | Miss the horizontal one | Half the marks | Compare the degrees |
| One solution only | Stop at 2.498 | The general solution is missing | Write theta = pi – arcsin 0.6 + 2k pi |
How to use this paper
First drill: all 6 questions on the clock, with the calculator-free parts done by hand.
Mark it: check radian mode, rounding and the meaning of each model parameter.
Second drill: redo three to seven days later, aiming to finish in under 45 minutes.
What should I do after drilling this paper?
Direct answer: log the rate-of-change and general-solution mistakes; redo the paper in three to seven days; then drill the AP Calculus AB mock to turn function skills into limits and derivatives.
FAQ
Is this a College Board past paper?
No. The questions and answers are TutorZone originals, not College Board past papers or marking references.
What is in the AP Precalculus exam?
2 hours 55 minutes covering Units 1-3: 42 multiple-choice questions in 105 min (about 63%) and 4 free-response questions in 70 min (about 37%).
Can I use a calculator throughout?
No. Part A of the multiple choice (29 questions) and Part B of the free response (2 questions) do not permit a calculator; the rest require a graphing calculator in radian mode.
Official references
| Resource | What to check |
|---|---|
| College Board AP Precalculus exam page | Confirm the format, question counts and calculator policy |
| Official course and exam description (CED) | Check the Units 1-3 multiple-choice weightings |
| https://www.tutorzone.com.hk/en-ap-calculus-ab-mock-2027/ | AP Calculus AB drilling mock |
| https://www.tutorzone.com.hk/en-ap-calculus-past-paper-drill-guide/ | AP Calculus drilling guide |
Related: practice area · AP Calculus AB drilling mock · AP Calculus drilling guide
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