Notation & membership
Roster form lists elements; set-builder form describes them: {x : property}.
01
Set Theory
Collections of objects and the algebra of combining them — the language every later chapter is written in.
Union & intersection
Union = either; intersection = both. A ∩ B = ∅ means disjoint.
Complement & difference
Complement is relative to the universal set U.
Subset & equality
Prove A = B by showing A ⊆ B and B ⊆ A.
Subset test shortcuts
Three equivalent statements.
Inclusion–exclusion (2 sets)
Add sizes, subtract the overlap counted twice.
Inclusion–exclusion (3 sets)
Subtract pairwise overlaps, add back the triple.
De Morgan's laws
Complement flips ∪ ↔ ∩.
Distributive laws
Intersection distributes over union — and vice-versa.
Identity & complement laws
∅ is neutral for ∪; U for ∩.
Power set
n elements → 2ⁿ subsets, counting ∅ and A itself.
Cartesian product
Ordered pairs; in general A × B ≠ B × A. Functions are built from these.
Example 1 · inclusion–exclusion
Class of 40: 25 like algebra, 20 geometry, 8 both. How many like neither?
- Neither = 40 − 37.
Answer
Example 2 · power set
List all subsets of {a, b}.
- 2² = 4 subsets expected.
- ∅, {a}, {b}, {a, b} — count matches.
Answer
Avoid
- vs : 0 elements vs 1.
- Reading A ∪ B as “in both” — that's ∩.
- Writing A ∪ B ∩ C without parentheses.
02
Logic & Reasoning
Statements, implications and quantifiers — the grammar of proof.
Implication
False only when p true, q false.
Contrapositive
Logically identical — often the easiest proof route.
Biconditional
“iff” = two proofs.
De Morgan (logic)
Push NOT through: AND ↔ OR flips.
Negating quantifiers
Swap ∀ ↔ ∃, negate inside.
Rules of inference
Affirm the hypothesis; deny the conclusion.
Chaining & eliminating
Implications chain; an “or” plus one denial settles the other.
Tautology & contradiction
True in every truth-table row, or in none.
Example 1 · negation
Negate: “Every prime greater than 2 is odd.”
- Formalize
- ∀ → ∃, negate inside.
Answer“There exists a prime > 2 that is not odd.”
Example 2 · contrapositive
Write the contrapositive of “if x² is even, then x is even.”
- Negate both parts, reverse direction.
Answer“If x is odd, then x² is odd.” (This is how the theorem is actually proved.)
Avoid
- Affirming the consequent: p ⇒ q and q do not give p.
- Negating “all” as “none” — it's “at least one isn't”.
- Treating converse as equivalent to the original.
03
Number Systems & Absolute Value
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ — and the distance function that makes inequalities behave.
The number ladder
Rationals are ratios p/q; √2, π are irrational.
Definition of |x|
Distance from zero — always ≥ 0.
Absolute value inequalities
“less thAND”, “greater thOR”.
Distance interpretation
|x − a| = distance from x to a.
Triangle inequality
A shortcut never beats both sides added.
Inequality rules
Multiply by negative → reverse the sign.
Laws of exponents
Fractional = root + power; negative = reciprocal.
Radical rules
The last one needs the absolute value.
Interval notation
Round = excluded; square = included.
Example 1 · |inequality|
Solve
- Add 1, divide by 2.
Answer
Example 2 · surds
Simplify
Answer
Avoid
- Forgetting to flip the sign when multiplying by a negative.
- Writing |a + b| = |a| + |b|.
- Dropping the absolute value in .
04
Complex / Imaginary Numbers
Adjoining √(−1) — every polynomial gets roots, and rotation becomes multiplication.
The imaginary unit
Powers cycle: 1, i, −1, −i.
Standard form
Point (a, b) in the Argand plane.
Conjugate & modulus
Conjugate reflects across the real axis.
Multiplication
FOIL, then i² → −1.
Division
Multiply top and bottom by the conjugate of the denominator.
Modulus & conjugate rules
Modulus respects products.
Polar form
r = |z|, tan θ = b/a (fix quadrant). Multiplying = scale + rotate.
Euler's formula
The exponential unifies trigonometry.
De Moivre's theorem
Modulus to the n, argument times n.
n-th roots
A regular n-gon on a circle of radius ⁿ√r.
Complex roots of quadratics
Δ < 0 → conjugate pair.
Example 1 · De Moivre
Compute
Answer
Example 2 · division
Simplify
- × conjugate:
- Numerator:
Answer
Avoid
- Losing the sign when replacing i² by −1.
- Writing |a + bi| = a + b.
- Dividing without the conjugate trick.
05
Functions & Transformations
One output per input — composing, inverting, sliding and stretching graphs.
Composition
Inside-out. In general f ∘ g ≠ g ∘ f.
Inverse function
Graphs mirror across y = x; only one-to-one functions invert.
General transformation
h right, k up, |a| vertical stretch, b compresses horizontally by 1/b.
Even & odd
cos, x² even; sin, tan, x³ odd.
Average rate of change
Secant slope — the warm-up for the derivative.
Difference quotient
Let h → 0 and this becomes f′(x).
Function library — domains & ranges
| f(x) | ||||||
|---|---|---|---|---|---|---|
| domain | ||||||
| range |
Every graph this semester is a transformation of these six parents.
Line tests
Vertical test → function? Horizontal test → invertible?
One y per x; one x per y.
Inverse recipe
- Write y = f(x).
- Swap x and y.
- Solve for y.
- Restrict domain first if needed.
Verify with f(f⁻¹(x)) = x.
Example 1 · composition
: find (g ∘ f)(x) and its domain.
- ; need — always.
Answer, domain
Example 2 · inverse
Find the inverse of
- Swap:
- Solve:
Answer
Avoid
- Reading f⁻¹ as a reciprocal.
- Wrong transformation order — inside shifts apply before outside scaling.
- Writing √(x² + 1) = x + 1.
06
Polynomials & Roots
Where polynomials cross the axis, and what the coefficients whisper about those crossings.
Quadratic formula
Works every time.
Discriminant
Reads the roots before you solve.
Vieta's relations
Cubic: α+β+γ = −b/a, αβγ = −d/a.
Factor & remainder theorems
Division by (x − a) leaves remainder f(a).
Vertex
Axis of symmetry x = h.
Rational root theorem
Factors of the constant over factors of the leading coefficient.
Factoring identities
Difference of squares; sum/difference of cubes.
Quadratic inequalities
Sketch: a > 0 smiles, a < 0 frowns.
Fundamental theorem of algebra
Degree n ⟹ exactly n complex roots, counting multiplicity.
Why complex numbers exist.
Example 1 · cubic by trial
Solve
- x = 2 works: 8 − 16 + 2 + 6 = 0 ✓
- Divide:
Answer
Example 2 · discriminant
For what k does have a double root?
- Double root ⟺ Δ = 0:
Answer (root x = −2)
Avoid
- Dropping the ± in the quadratic formula.
- Forgetting multiplicity when counting roots.
- Vieta sign slips: sum = −b/a.
07
Exponentials & Logarithms
Inverse operations: exponents grow, logarithms ask “to what power?”
Definition
A logarithm is an exponent in disguise. Domain x > 0.
Laws of logs
Multiplication becomes addition.
Change of base
Any base via ln or log₁₀.
Cancellation
Inverses undo each other — the key solving move.
The number e
The base that grows at its own rate.
Growth & decay
Discrete vs continuous compounding.
Doubling & half-life
Set A = 2P or P/2 and solve.
Solving exponentials
Isolate the exponential, then log both sides.
Example 1 · exponential equation
Solve
- ln:
Answer
Example 2 · log equation
Solve
- Combine:
- ; reject −2 (domain).
Answer
Avoid
- Splitting .
- Logging only one side of an equation.
- Mixing ln with log₁₀.
08
Sequences & Series
Patterns indexed by n — adding them up, sometimes forever.
AP n-th term
Constant step d.
AP sum
Gauss: pair first with last.
GP n-th term
Constant ratio r.
GP sum
r = 1 gives Sₙ = n·a₁.
Infinite GP
Converges only for |r| < 1.
Famous sums
Σc = nc; constants factor out.
Means
Always G.M. ≤ A.M.
Repeating decimals
Every repeating decimal is a GP.
Example 1 · AP
a₃ = 7, a₈ = 22. Find S₁₀.
- ; ;
Answer
Example 2 · infinite GP
Sum
- — |r| < 1 ✓
Answer
Avoid
- AP formula on a GP (or vice versa).
- Writing aₙ = a₁ + nd.
- Using when |r| ≥ 1.
09
Binomial Theorem
Expanding (a + b)ⁿ without multiplying n times.
The theorem
Exponents in each term add to n.
General term
Grab any single term without expanding.
Pascal identities
Symmetry and the triangle rule.
Counting the expansion
Set a = b = 1 for the 2ⁿ sum.
Even / odd coefficient sums
Set a = 1, b = −1: the two halves are equal, each 2ⁿ⁻¹.
Middle terms
Parity of n decides.
Strategy: specific terms
For “the term in x²”: write the general term, match powers, solve for k.
Never expand the whole thing.
Example 1 · a coefficient
Coefficient of x² in
- :
Answer
Example 2 · middle term
Find the middle term of
- n = 6 even → T₄, i.e. k = 3.
Answer — it's also the term independent of x.
Avoid
- Dropping the sign of the second term.
- Confusing term number with k (Tk+1, k from 0).
- Writing (a + b)² = a² + b².
10
Permutations & Combinations
Counting arrangements — the whole chapter turns on: does order matter?
Factorial
Ways to order n distinct objects.
Multiplication principle
Count slot by slot, multiply.
Permutations
Order matters: podiums, passwords.
Combinations
Order doesn't: committees, card hands.
The link
Each group hides r! orderings.
Repeats: multinomial
Divide by each run of identical items.
Circular arrangements
Pin one person, arrange the rest.
Restrictions
Glue for “together”; complement for “apart”.
Complement principle
The universal shortcut for “at least” questions.
Example 1 · repeats
Arrangements of BANANA?
- 3 A's, 2 N's, 1 B:
Answer
Example 2 · committee
Choose a 3-person committee from 8 people.
- Order irrelevant → C:
Answer
Avoid
- P where order is irrelevant (or C where it matters).
- Treating identical items as distinct.
- Double-counting circular arrangements.
11
Trigonometric Functions
Angles, the unit circle, and the six functions that turn geometry into algebra.
SOH · CAH · TOA
Right-triangle definitions.
Unit circle
Extends trig to every angle.
Radians
Arc length and sector area need radians.
Reciprocals
csc↔sin, sec↔cos, cot↔tan.
Periods
For y = sin(bx) the period becomes 2π/b; tan gives π/b.
Sinusoid form
Amplitude |a|, period 2π/b, shift c, midline d.
Reference angles
Reduce to Q1, fix the sign with ASTC.
Inverse trig ranges
Principal values only.
Signs — ASTC
All · Sin · Tan · Cos
Positive functions in quadrants I–IV.
Special angle values
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | |||||
| cos θ | |||||
| tan θ | undef. |
Memorize it — it powers half of Ch 12 and the calculus trig work.
Example 1 · equation
Solve on
- Reference angle ; sin positive in Q1, Q2.
Answer
Example 2 · reference angle
Find cos 120° exactly.
- Reference angle 60°; Q2 → cos negative.
Answer
Avoid
- Stopping at the first solution of a trig equation.
- Assuming arcsin(sin x) = x outside the principal range.
- Mixing degrees and radians in one equation.
12
Trigonometric Identities & Equations
Rewriting trig into the form the problem demands — and solving non-right triangles.
Pythagorean identities
One identity, three disguises.
Even–odd
sin, tan odd; cos even.
Sum & difference
Cosine flips the sign.
Double angle
Pick the form that cancels.
Power-reducing
Squared trig → linear in cos 2θ. Essential for Ch 19.
Product-to-sum
Products become integrable sums.
Sum-to-product
Great for factoring equations.
Auxiliary angle form
One sinusoid; extrema ±R.
Sine & cosine rules
Sine rule: opposite pairs. Cosine rule: SAS or SSS.
Triangle area
Two sides + included angle.
Example 1 · auxiliary form
Write as R sin(θ + α); state the max.
- ;
Answer, max 5
Example 2 · equation, 4 solutions
Solve on
- All four quadrants contribute.
Answer
Avoid
- Cosine-sum sign: takes a minus.
- Dividing away sin θ and losing solutions.
- Squaring both sides without checking for extraneous roots.
13
Vectors
Magnitude and direction — plus two very different ways to multiply.
Components & magnitude
Pythagoras in 2D/3D.
Operations & unit vectors
Componentwise; normalize to length 1.
Dot product
Result is a scalar.
Angles & perpendicularity
Dot > 0 acute, = 0 right, < 0 obtuse.
Projections
One is a vector, one is a length.
Cross product magnitude
In 2D: |u₁v₂ − u₂v₁|. Half of it is the triangle area.
Work
Only the along-motion component counts.
Example 1 · angle
: dot product and angle.
Answer
Example 2 · triangle area
Area of the triangle spanned by
- 2D cross magnitude:
- Triangle = half the parallelogram.
Answer
Avoid
- Forgetting the square root in |v|.
- Dot product when the question asks an area.
- Skipping normalization in .
14
Matrices & Determinants
Arrays encoding linear systems — with one surprise: multiplication doesn't commute.
Matrix multiplication
Row of A · column of B; inner dims match.
The big warning
In general AB ≠ BA, and AB = 0 doesn't force A = 0 or B = 0.
Never divide by a matrix.
Determinant 2×2
Main diagonal minus anti-diagonal.
Inverse 2×2
Swap diagonal, flip the rest, divide by det.
Identity & singularity
Also — order reverses.
Transpose rules
Transpose also reverses order.
Row ops & det
Determinant behavior under elimination.
Cramer's rule
Replace column with constants, det, divide.
Matrix equations
det = 0 → none or infinitely many solutions.
Example 1 · solve with A⁻¹
Solve
Answer
Example 2 · AB ≠ BA, live
: compute AB and BA.
- (columns of A swapped)
- (rows of A swapped)
AnswerAB ≠ BA — order matters, concretely.
Avoid
- Assuming AB = BA.
- Order slips in and .
- det = ad + bc — it's ad − bc.
15
Lines, Circles & Conics
Equations as pictures: slopes, distances, and the curves from slicing a cone.
Slope
Vertical: undefined; horizontal: 0.
Forms of a line
Point-slope, slope-intercept, general.
Parallel & perpendicular
Negative reciprocals.
Distance & midpoint
Pythagoras; averaging coordinates.
Point-to-line distance
Shortest distance from (x₀, y₀) to Ax + By + C = 0 — the tangency test for circles.
Circle
Center (h, k); signs flip when reading off.
Conic standard forms
+ → ellipse, − → hyperbola, one square → parabola.
Focus, directrix, asymptotes
Conics are focus–directrix distance ratios (eccentricity).
Perpendicular bisector
The standard move for finding circle centers.
Example 1 · complete the square
Center & radius of
Answercenter (3, −2), r = 5
Example 2 · perpendicular bisector
Bisector of the segment from (1, 2) to (5, 6)?
- Midpoint (3, 4); segment slope → perp slope −1.
Answer
Avoid
- Un-flipped center signs: (x − 3)² means h = +3.
- Plugging a slope in for a vertical line.
- Forgetting the √ after computing d².
16
Limits & Continuity
The doorway to calculus: what a function approaches — even where it isn't defined.
Limit laws
Limits pass through +, −, ×, ÷.
Famous limit #1
Radians; for small x, sin x ≈ x.
Famous limit #2
Multiply by (1 + cos x) first.
Famous limits: exp & log
eˣ ≈ 1 + x and ln(1 + x) ≈ x, made rigorous.
The number e
Compound interest pushed to continuous.
ε–δ definition
Within ε of L by staying within δ of a.
Continuity at a point
Fail any one and the graph breaks.
Types of discontinuity
- Removable — limit exists; f(a) missing/wrong (a hole).
- Jump — one-sided limits exist but differ.
- Infinite — blow-up (vertical asymptote).
Classify before describing.
One-sided & infinite
Both sides must agree; blow-up → vertical asymptote.
Limits at ∞ (rationals)
Degree showdown → horizontal asymptote.
Asymptote recipe
- Vertical: zeros of denominator where numerator ≠ 0.
- Horizontal: degree showdown.
- Slant: deg N = deg D + 1 — long-divide.
Check both sides of every vertical candidate.
Squeeze theorem
Standard for oscillating factors like sin(1/x).
Factor-then-substitute
0/0 means factor and cancel.
Example 1 · factoring
- 0/0 →
Answer (removable hole)
Example 2 · conjugate
- × conjugate:
Answer
Example 3 · continuity
Find k so is continuous at 1.
- Left limit 2; right value k — match them.
Answer
Avoid
- Declaring “undefined” on 0/0 — factor instead.
- Calling ∞ a value; one-sided signs can disagree.
- only at x → 0.
- Cancelling across sums: .
- Forgetting continuity needs all three conditions.
17
Differentiation Rules
One limit definition, a complete toolbox of rules, and four techniques for every shape of function.
Definition
Tangent slope as the secant tightens.
Power rule
Any real n — negatives and fractions included.
Linearity
Term by term; constants die.
Product rule
E.g. .
Quotient rule
low·D-high − high·D-low, over low².
Chain rule
Outside-in; multiply by the inner derivative.
Trig derivatives
Co-functions carry a minus.
Exp & log derivatives
e^x equals its own derivative.
Inverse trig derivatives
arccos is the negative twin.
Implicit differentiation
Each y-term picks up a dy/dx; collect and solve.
Logarithmic differentiation
- Take ln of both sides.
- Differentiate:
- Solve for y′; substitute y back.
The weapon for xˣ and product towers.
Derivative of an inverse
Reciprocal slope at the mirrored point.
Differentials
Powers substitution in Ch 19.
Master derivative table
| f(x) | f′(x) | f(x) | f′(x) |
|---|---|---|---|
Everything else is this table plus product, quotient and chain.
Example 1 · chain
Differentiate
- Outside × inside .
Answer
Example 2 · logarithmic
Differentiate
Answer
Example 3 · implicit
Slope of at (1, 2).
Answer
Avoid
- Missing inner derivatives: .
- — the 3 belongs in front.
- Product vs chain mix-ups: x·sin x = product; sin(x²) = chain.
- , not 5.
- Dropping the minus in and .
18
Applications of Derivatives
Tangents, motion, extrema, concavity — plus related rates and optimization.
Tangent line
Normal line: slope −1/f′(a).
Motion
Speeding up ⟺ v, a same sign.
Increase / decrease
Sign of f′ = direction of the graph.
Critical points
Extrema can only occur here.
First derivative test
Sign chart of f′ — always reliable.
Second derivative test
f″(c) = 0 → test silent; use the sign chart.
Concavity & inflection
Inflection = f″ changes sign.
Closed-interval method
Global extrema live among these values.
Mean Value Theorem
Rolle: f(a) = f(b) ⟹ f′(c) = 0.
L'Hôpital's rule
Confirm the form first; repeatable.
Linear approximation & error
Tangent as estimator; differential as error propagator.
Related rates — method
- Draw it; label changing quantities.
- Link them with a geometric equation.
- Differentiate wrt t.
- Substitute after differentiating.
Every time-varying variable earns a d/dt term.
Curve-sketching checklist
- Domain, intercepts, symmetry.
- Asymptotes (Ch 16 recipe).
- f′ sign chart → extrema.
- f″ sign chart → concavity.
- Plot key points; sketch.
Run in order — each step narrows the picture.
Example 1 · optimization
100 m fencing, three sides (barn is the fourth). Maximize area.
- ;
- ; ✓
Answer
Example 2 · related rates
Ripple expands at dr/dt = 2 m/s. dA/dt when r = 5?
Answer
Example 3 · L'Hôpital twice
- 0/0 → , still 0/0 →
Answer
Example 4 · linear approximation
Estimate by hand.
Answer
Avoid
- Skipping endpoints on closed intervals.
- L'Hôpital before verifying 0/0 or ∞/∞.
- “Critical point” ≠ “extremum” (y = x³ at 0).
- Substituting numbers before differentiating in related rates.
- Reading concavity from f′ instead of f″.
19
Integration
Differentiation run backwards — and, via the Fundamental Theorem, exact areas.
Definition (Riemann)
Area as the limit of finer rectangles.
Power rule (reverse)
n = −1 is the log case.
Basic antiderivatives
Ch 17's table read right-to-left.
More trig integrals
sin²/cos²: power-reduce first.
Reverse chain-rule patterns
| Integral | Antiderivative | Integral | Antiderivative |
|---|---|---|---|
Linear inside? Divide by its slope a. The pattern behind u-substitution.
Linearity
Split sums, pull out constants.
FTC, Part 2
Definite integral = antiderivative at the ends.
FTC, Part 1
Upper limit g(x) → chain: f(g(x))·g′(x). Exam favorite.
Integral properties
Splitting and reversing intervals.
u-substitution
Chain rule in reverse.
Substitution with limits
Change bounds with the variable — no back-substitution.
Area between curves
Top minus bottom; intersections give the bounds.
Average value
Some c satisfies f(c) = f_avg (MVT for integrals).
Which technique?
- Power / table / linear inside → integrate directly.
- Inside function and its derivative present → u-sub.
- Trig powers/products → identities first.
- Ugly back-substitution? Change the limits instead.
Scan for the derivative of the inside — the #1 tell.
Example 1 · FTC
Answer
Example 2 · u-sub with limits
- ; bounds
Answer
Example 3 · area between curves
Area enclosed by and .
- Meet at 0, 1; line on top.
Answer
Example 4 · FTC + chain
Differentiate
- Upper limit x² → chain:
Answer
Avoid
- Forgetting + C on indefinite integrals.
- , not +cos x.
- Missing reverse-chain factors: .
- Keeping x-bounds after switching to u.
- Quoting signed area as “the area”.
20
Probability & Statistics
Quantifying chance and summarizing data — where Ch 1's sets get measured.
Classical probability
Equally likely outcomes assumed.
Complement
“At least one” = 1 − P(none).
Addition rule
Mutually exclusive → just add.
Conditional
Sample space shrinks to B.
Independence
No influence ⇔ multiply.
Total probability
Bayes' denominator machinery.
Bayes' theorem
Update belief with evidence; P(A|B) ≠ P(B|A).
Binomial distribution
k successes in n independent trials.
Binomial mean & variance
Expected successes = n·p.
Mean, variance, σ
σ in the data's units; σ² is not.
Expected value & z-score
Long-run average; σ's from the mean.
Example 1 · dice
Two fair dice: P(sum = 7)?
- 36 outcomes; 6 favorable.
Answer
Example 2 · binomial
5 fair coin flips: P(exactly 3 heads)?
Answer
Avoid
- The gambler's fallacy.
- Swapping P(A|B) with P(B|A).
- Adding probabilities of overlapping events without subtracting.
A1
Symbol Glossary
Every mark you've met this semester, in one table — read it aloud column by column and the course sounds like a sentence.
Notation dictionary
| Symbol | Name | How to read it |
|---|---|---|
| element of | x ∈ A — “x belongs to A” | |
| not element of | y ∉ A — “y is not in A” | |
| subset | A ⊆ B — every element of A is in B | |
| union | in A or B (or both) | |
| intersection | in both A and B | |
| empty set | the set with no elements | |
| power set | the set of all subsets of A | |
| Cartesian product | A × B — all ordered pairs (a, b) | |
| for all | ∀x — “for every x” | |
| there exists | ∃x — “at least one x” | |
| negation | ¬p — “not p” | |
| conjunction | p ∧ q — “p and q” | |
| disjunction | p ∨ q — “p or q (or both)” | |
| implies | p ⇒ q — “if p then q” | |
| iff | p ⇔ q — “p if and only if q” | |
| absolute value / modulus / size | distance from 0; |z|; |A| = number of elements; a | b = divides | |
| square root | the non-negative number whose square is x | |
| infinity | unbounded growth — not a number | |
| constants | π ≈ 3.14159; e ≈ 2.71828 | |
| number sets | naturals · integers · rationals · reals · complex | |
| summation | Σ aₖ — add the terms as k runs its range | |
| integral | ∫ f dx — signed area / antiderivative | |
| limit | the value f approaches as x approaches a | |
| derivative | instantaneous rate of change of f | |
| approximately | ≈ — close to, not exactly | |
| perpendicular | meeting at 90°; u ⊥ v ⟺ u·v = 0 | |
| change / discriminant | Δy = change in y; Δ = b² − 4ac | |
| mean, std deviation | center and spread of data | |
| probability | chance of A; chance of A given B |
If a symbol on an exam question stops you cold, find it here first — notation is half the subject.
A2
Master Cram Sheet
The twenty-four formulas most likely to appear, anywhere, in any form. If you know these cold, you can rebuild the rest.
06 quadratic formula
06 discriminant
07 log laws
07 change of base
04 imaginary unit
04 modulus
04 De Moivre
08 infinite GP
09 binomial theorem
10 permutations
10 combinations
12 Pythagoras trig
12 cosine rule
16 famous limit
17 definition
17 power rule
17 product rule
17 quotient rule
17 chain rule
18 tangent line
18 MVT
19 power rule ∫
19 FTC
20 Bayes
A3
Exam Playbook
The non-mathematical skills that decide 10–15% of the grade: how to run the paper, and how to check the answers.
During the exam
- Scan the whole paper first; do the sure-thing questions first to bank marks.
- Write the formula before substituting numbers — it earns method marks even if arithmetic slips.
- Show every step; partial credit is real and generous.
- Check units and modes: radians vs degrees, cm vs m.
- Stuck for 3+ minutes? Mark it, move on, return later.
- Sketch graphs — a picture often earns the mark algebra can't.
- Re-read what was actually asked: slope vs equation, area vs integral, exact vs decimal.
- Don't scribble out mistakes — one clean line through them keeps working legible.
The checking pass
- Substitute roots back into the original equation.
- Check domains — reject extraneous log/root answers.
- Indefinite integral? Confirm + C is present.
- Sanity checks: area > 0, 0 ≤ P ≤ 1, probabilities sum to 1.
- Final 5 minutes: sweep for unanswered questions — something beats nothing.