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150 multiple-choice questions and 165 flashcards, organised into 10 chapters, written to the Common Core HS Pre-Calculus / typical course standards blueprint. Every question carries a full rationale.
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Common Core HS Pre-Calculus / typical course standards — covers: functions and inverses, polynomial/rational/exponential/logarithmic functions, trigonometric functions and identities, analytic trigonometry, sequences and series, conic sections, vectors, polar coordinates, parametric equations, limits introduction, complex numbers. Preparation for AP Calculus. MCQs reference public mathematical pre-calculus standards.
CoStudy's High School Pre-Calculus bank holds 315 items organised into 10 chapters that follow the published blueprint. Every multiple-choice question carries a written rationale explaining why the correct answer is correct and why each distractor is tempting but wrong.
Each chapter follows a domain of the published exam outline. Practise one on its own:
A sample of 12 multiple-choice questions from the bank, with the full rationale shown.
Limit: lim(x→0) sin(x)/x = ?
Answer: D — 1 (fundamental trig limit; foundation of derivative of sin)
Famous limit: lim(x→0) sin(x)/x = 1 (with x in radians). Proven by squeeze theorem. Foundation for d/dx[sin x] = cos x. Related: lim(x→0) (1-cos x)/x = 0.
Convert polar (r, θ) = (4, 60°) to rectangular:
Answer: E — (2, 2√3) (x = r·cos(θ) = 4·(1/2) = 2; y = r·sin(θ) = 4·(√3/2) = 2√3)
Polar-to-rectangular: x = r cos θ, y = r sin θ. r=4, θ=60°. cos 60° = 1/2, sin 60° = √3/2. x = 4·(1/2) = 2; y = 4·(√3/2) = 2√3. Result: (2, 2√3).
Unit circle: angle π/2 corresponds to:
Answer: C — (0, 1)
C) Top of circle. A/B/D) Each is at a different angle.
lim(x→∞) (1 + 1/x)^x =
Answer: C — e
C) Classical limit defining e. A/B/D) Each is incorrect.
Solve 4·e^(2x) = 100 for x (exact form).
Answer: D — x = ln(25)/2
D) Divide: e^(2x) = 25. Take ln: 2x = ln(25). x = ln(25)/2. A) Forgot to divide by 2 after taking the log. C) Used log₁₀ instead of ln (inverse of e is ln, not log). B) Mixed exponential and log operations.
What is the magnitude of vector v = ⟨3, 4⟩?
Answer: D — 5 (|v| = √(3² + 4²) = √25 = 5)
Vector magnitude: |v| = √(v_x² + v_y²). |⟨3,4⟩| = √(9+16) = √25 = 5. Generalizes distance formula. 3-4-5 Pythagorean triple again. Pre-Calc vectors.
1 + 2 + 4 + 8 + ... + 2^n is a:
Answer: A — Geometric series with ratio 2 — sum = 2^(n+1) − 1
A) Standard geometric. B/C/D) Each is incorrect.
Half-angle: if cos(θ) = 1/3 and θ is in QI, then cos(θ/2) =
Answer: D — √(2/3)
D) cos(θ/2) = ±√((1 + cos θ)/2) = ±√((1 + 1/3)/2) = ±√(2/3). θ ∈ QI means θ/2 ∈ QI (positive cosine). A = √(2/3). B) Used (1 − cos θ)/2 — that gives sin(θ/2). C) Wrong sign (θ/2 is QI). A) Lost the factor of 2.
If f(x) = x² and g(x) = x + 1, find (f ∘ g)(2).
Answer: C — 9 (g(2) = 3, then f(3) = 9)
Composition (f∘g)(2) = f(g(2)). First g(2) = 2+1 = 3. Then f(3) = 3² = 9. Order matters: (f∘g)(x) means g first, then f. Compare (g∘f)(2) = g(f(2)) = g(4) = 5.
Eliminate the parameter: x = t + 1, y = t².
Answer: D — y = x² − 2x + 1
D) From x = t + 1 we get t = x − 1, so y = t² = (x − 1)² = x² − 2x + 1. B) Substituted t = x without solving. C) Used the linear equation alone. A) Inverted the wrong way.
ln(x^n) =
Answer: A — n · ln(x)
A) Power rule of logs. B/C/D) Each is incorrect.
cos(2x) =
Answer: B — cos²(x) − sin²(x) = 1 − 2sin²(x) = 2cos²(x) − 1
B) Three equivalent forms. A) sin(2x). C/D) Not the identity.
6 sample cards from the 165 in the bank.
What are the inverse trig functions?
arcsin (sin⁻¹), arccos (cos⁻¹), arctan (tan⁻¹) — return the angle for a given trig value, on restricted ranges.
What test on a graph determines whether a function has an inverse function?
The horizontal line test — if every horizontal line crosses the graph at most once, f is one-to-one and has an inverse.
What is an odd function? Give a graphical property.
f(−x) = −f(x). Symmetric about the origin. Example: x³, sin x.
Worked Example: Find sin(7π/6) using reference angle and unit circle
Step 1: Identify the quadrant. 7π/6 is between π (= 6π/6) and 3π/2 (= 9π/6). So 7π/6 is in Quadrant III. Step 2: Find the reference angle. Reference angle = 7π/6 − π = 7π/6 − 6π/6 = π/6 Step 3: Recall sin(π/6). sin(π/6) = 1/2 Step 4: Apply the quadrant's sign. In Quadrant III, sine is NEGATIVE (only tangent is positive). Final answer: sin(7π/6) = −1/2
What does log(x) mean (without subscript)?
Common log: log₁₀(x).
What kinds of discontinuities are there?
Removable (hole), jump, infinite (asymptote).
These samples are a small slice. The full bank runs flashcards, multiple choice and timed mock exams with per-chapter progress tracking, on the web and in the iOS app.
Open High School Pre-Calculus →
The High School Pre-Calculus bank holds 315 items: 150 multiple-choice questions, 165 flashcards. 18 of them are on this page to read free, with no signup.
Yes. Every multiple-choice item carries a written rationale that states the controlling principle behind the correct answer and then addresses each wrong option in turn — why it tempts and precisely where it fails. Knowing why the plausible answer was wrong is worth more than knowing which letter was right.
It is organised into 10 chapters that follow the published exam blueprint: Functions and Their Graphs; Polynomial and Rational Functions; Exponential and Logarithmic Functions; Trigonometric Functions; Analytic Trigonometry: Identities and Equations; Applications of Trigonometry: Laws, Vectors, Polar; Systems of Equations and Matrices; Conic Sections and Parametric Equations; Sequences, Series, and Probability; Limits and Introduction to Calculus. The number of questions in each chapter is proportional to that domain's published weight, so working through the bank exposes you to roughly the mix the real exam uses.
Common Core HS Pre-Calculus / typical course standards — covers: functions and inverses, polynomial/rational/exponential/logarithmic functions, trigonometric functions and identities, analytic trigonometry, sequences and series, conic sections, vectors, polar coordinates, parametric equations, limits introduction, complex numbers. Preparation for AP Calculus. MCQs reference public mathematical pre-calculus standards.
The samples on this page are free to read in full, rationales included, with no account. The complete 315-item bank, the timed mock exams and per-chapter progress tracking are part of CoStudy on the web and in the iOS app.
Last reviewed 2026-08-22. Banks are written against the certifying body's published exam outline and re-checked when that outline changes — exams get renumbered, retired and reweighted, and a bank written to a superseded outline teaches the wrong proportions. Figures that are re-indexed annually are deliberately not asserted as rules; the questions test the governing principle instead.