Quick study package · prep review

Mathematics · quick study package

49 content pages · 60–90s per stem · flip or print

Notes hub

Page 01 / 49Package← → / Space · content review (not directions-only)

Study package

Mathematics · quick study package

Depth-upgraded prep review · 60–90s methods · printable / PPT flip

Do not memorize isolated tricks. Automate 4–6 universal moves that work on every paper type (SC, MC, NUM, MATCH).
First 20 seconds decide the method; the next 40–70 seconds only execute.
Write one-line reasons on rough sheet — that is what converts accuracy into 100%.
Open the matching topic page for full methods, samples, and families — this hub is the routing layer.
This deck pulls live hub banks and topic triggers — re-print after notes depth updates.
Flip page-by-page under timer, or print the full sheet set for desk revision.

Full package (49 pages). Use Print package for a desk handout.

01 / 49 · Package

Study package

Mathematics · quick study package

Depth-upgraded prep review · 60–90s methods · printable / PPT flip

Do not memorize isolated tricks. Automate 4–6 universal moves that work on every paper type (SC, MC, NUM, MATCH).
First 20 seconds decide the method; the next 40–70 seconds only execute.
Write one-line reasons on rough sheet — that is what converts accuracy into 100%.
Open the matching topic page for full methods, samples, and families — this hub is the routing layer.
This deck pulls live hub banks and topic triggers — re-print after notes depth updates.
Flip page-by-page under timer, or print the full sheet set for desk revision.

02 / 49 · Package

How to use

How to use this package (30–60 min)

Pages are ordered: subject protocol → each hub (overview → methods → bank → topics) → close checklist.
Speak the method name before reading formulas. Formulas without a named move do not convert.
Mark weak pages with a pencil — tomorrow’s opener is your marked set only.
After the deck: 2 timed samples from topic pages or mixed joins — do not reread full hubs.

03 / 49 · Subject

Protocol

Math · master 60–90s protocol

Use on every stem before opening a hub page

0–20s: name the method family (Matrix / Complex / Bayes / Conic / Limit / DE…).
20–80s: execute only that template — no competing plan.
Last 10s: MC = independent T/F; NUM = units/limits; MATCH = P,Q,R,S alone.
Skip rule: unclassified in 20s → mark and return.

04 / 49 · Algebra

Hub review

Algebra · hub overview

Bank ~35–40 /100 Math (~full Algebra share) · ~37% of Math tags (2020–2026)

Treat Algebra as a scoring bank: every stem should start from a named method within 20 seconds, then finish in 60–90 seconds — with enough depth that the method still works when the paper twists the wording.
In a full paper: clear 3–4 ‘easy bank’ Algebra stems first (standard matrix/complex/Bayes) to lock 8–12 marks in under 8 minutes.
Never leave matching for last if you are calm — independent T/F is faster than a tired guess.
If a stem is not classified in 20s, mark and skip. Returning with a clear head is better than a forced wrong method.
Single correct: often a clean numerical identity (matrix power, Arg value, conditional probability) with close distractors that match common mistakes.
Multi-correct: 2–3 true statements; each must be proved separately. One wrong tick can wipe the question under some marking schemes — treat every option as its own mini-question.

05 / 49 · Algebra

Protocol

Algebra · 60–90s protocol

0–15s: Name the family: Matrix / Complex / Prob / P&C / Relation / Quadratic-Sequence. Circle the ask (value / count / which options / code). — Say the family out loud on rough sheet (one word). If you cannot name it, you do not have a method yet — do not start expanding. Also underline constraints: ‘order matters’, ‘with replacement’, ‘principal Arg’, ‘det = 0 allowed?’.
15–30s: Pick ONE universal move (see list below). Write the first equation or sample-space partition on rough sheet — do not expand everything. — This is the highest-ROI window. Example commits: ‘CH reduce Mⁿ’, ‘tree for Bayes’, ‘model sentence for P&C’, ‘∑α,∑αβ first’. If you write two competing plans, you are already late — pick one.
30–75s: Execute only the necessary algebra. For MC: check each option as independent T/F. For MATCH: solve P,Q,R,S independently. — Keep fractions exact; avoid decimal rounding until the end. For MC, physically tick T/F beside each option on rough sheet. For MATCH, write four one-line verdicts before opening the option codes.
75–90s: Sanity check: dimension/order, probability ≤1, Arg principal value, matrix size, overcount. Bubble and move. — Quick gates: is the matrix product size legal? Is P between 0 and 1? Did Arg land in (−π,π]? Did counting double-count ordered pairs? If a check fails, fix once — do not restart from chapter memory.

06 / 49 · Algebra

Methods

Algebra · methods

Reduce before expand · Partition the sample space

Reduce before expand

Applies: Matrices, complex powers, binomial sums
Move: Find a relation that shrinks the object (char poly / N²=0 / z³=1 / recurrence) before writing full expansion.
Why: Turns 4 minutes of mess into 40 seconds of bookkeeping.
Worked: If M = I + N with N² = 0, then Mᵏ = I + kN. Summing powers becomes arithmetic on the scalar k — you never multiply matrices 26 times. Write N²=0 before any entry-wise expansion.
Avoid: If the matrix is already 2×2 with tiny entries and only M² is asked, direct multiply can be faster — still compute det/tr first in case CH helps later options.

Partition the sample space

Applies: Probability, Bayes, conditional
Move: Draw a 2–3 branch tree or write P(A)=∑P(A|Bi)P(Bi) before any formula recall.
Why: Stops random formula hunting; options become checks on one tree.
Worked: Two urns / two factories / defective vs good: write P(Bi), P(A|Bi), then P(Bi|A) ∝ P(A|Bi)P(Bi). Compute the denominator once as total probability. Options that forget the prior die immediately.
Avoid: Pure counting probability with equally likely outcomes may need nCr models instead of Bayes trees — but you still write the sample-space description first.

07 / 49 · Algebra

Methods

Algebra · methods

Independent T/F · Count with a model sentence

Independent T/F

Applies: Multi-correct + Matching
Move: Never force a jigsaw. Decide each claim/list item alone with a one-line proof or reject.
Why: Protects partial marks and matching clusters.
Worked: Wrong: ‘P looks like (2), so Q must be (3)’. Right: ‘P ⇒ entry (1). Q independently ⇒ (2). R ⇒ (4). S ⇒ (5). Then pick the option listing 1,2,4,5’.
Avoid: Never skip this in MATCH/MC. The only nuance is when two list items share an intermediate — still write both conclusions separately.

Count with a model sentence

Applies: P&C, stars-and-bars, relations
Move: Write one sentence: ‘Choose k from n with/without order, with/without replacement, with constraint X’ — then compute.
Why: Kills overcount/undercount before arithmetic.
Worked: ‘Number of positive integer solutions of x+y+z=10’ → set yi=xi−1 ≥ 0 → C(9,2). The sentence ‘positive ⇒ shift by 1’ is the method; C(9,2) is only arithmetic.
Avoid: If the stem is clearly probability with unequal likelihoods, do not force nCr — go back to partition/tree.

08 / 49 · Algebra

Methods

Algebra · methods

Symmetric functions first · Principal value lock

Symmetric functions first

Applies: Quadratic / polynomial roots
Move: Write ∑α, ∑αβ, αβ… before expanding (x−α)(x−β).
Why: Most Advanced asks are already in elementary symmetric form.
Worked: ∑α² = (∑α)² − 2∑αβ. Write that identity before substituting numbers. For new roots β=α+1/α, compute ∑β and product from old symmetric sums.
Avoid: If roots are needed individually and discriminant is clean, solving the quadratic can be fine — still keep ∑ and product written for checks.

Principal value lock

Applies: Complex Arg, inverse trig joins
Move: State the branch (Arg ∈ (−π,π]) before comparing options.
Why: Eliminates 2–3 lookalike options instantly.
Worked: If z is in quadrant II, Arg is between π/2 and π (positive), not a careless ‘π + θ’ without reducing to principal form. Compute atan(|y|/|x|) then adjust quadrant, then clamp to (−π,π].
Avoid: If the question asks only |z| or Re/Im with no argument, do not waste time on Arg — but still keep conjugate rules clean.

09 / 49 · Algebra

Mental bank

Algebra · mental bank

Cube roots of unity: 1+ω+ω²=0, ω³=1; 1+ω²=−ω, etc.
|z₁z₂|=|z₁||z₂|, Arg(z₁z₂)=Arg z₁+Arg z₂ (then reduce to principal value)
2×2 Cayley–Hamilton: M² − (tr M)M + (det M)I = 0
Adj A · A = (det A)I; A⁻¹ = (1/det A) Adj A when det≠0
Bayes: P(Bi|A)=P(A|Bi)P(Bi)/∑P(A|Bj)P(Bj)
P(A∪B)=P(A)+P(B)−P(A∩B); independence ≠ mutually exclusive
nCr = n!/(r!(n−r)!); stars-and-bars non-neg: C(n+k−1, k−1)
∑α²=(∑α)²−2∑αβ; quadratic ax²+bx+c=0: ∑α=−b/a, αβ=c/a
AP sum = n/2 [2a+(n−1)d]; infinite GP sum = a/(1−r) for |r|<1
Equivalence relation ⇒ reflexive + symmetric + transitive (all three)
Hub route: name the Algebra family before opening formulae
Reduction > expansion on every Matrices/Complex stem
Model sentence before nPr/nCr or Bayes
Flash this bank for 8 minutes before mocks — not during learning of new ideas.
Each line should be writable from memory without looking at a book. If not, that line is a leak in your 100% plan.
When a stem uses a bank line, tick it on rough sheet (‘used CH’, ‘used Bayes’) so you build method awareness.

10 / 49 · Algebra

Topic cards

Algebra · topic cards

Matrices & determinants · Complex numbers

Matrices & determinants (High every-year)

Trigger: See Mⁿ, adj, inverse, system, or orthogonal-type identity → reduce using char poly / structure.
Step: Note order. If square, compute tr and det early.
Step: If powers/sums of powers: write characteristic equation; reduce.
Step: If system: form AX=B; check det; discuss unique/infinite/none as needed.
Automate: Cayley–Hamilton for 2×2 fluently
Automate: Adj A · A = (det A) I
Formula: For 2×2 [[a,b],[c,d]]: tr=a+d, det=ad−bc
Formula: A⁻¹ = (1/det A) Adj A
Formula: det(AB)=det(A)det(B)
Formula: (AB)⁻¹ = B⁻¹A⁻¹ when invertible
Trap: Assuming AB=BA
Trap: Forgetting det=0 cases in inverse questions
Drill: Daily: 1 power-reduction NUM + 1 system SC/MC. Weekly: 1 matching on matrix properties. After each, write ‘CH / direct / identity-test’ as the method used. Open full Matrices notes for complete process + formulas.

Complex numbers (High · independent page)

Trigger: See modulus/arg/geometry/roots of unity → choose polar or ω toolkit immediately.
Step: Convert to the form matching the ask (x+iy vs re^{iθ}).
Step: For geometry: rewrite in |z−a| / Arg language and recall the locus.
Step: For roots of unity: reduce powers mod 3; use 1+ω+ω²=0.
Automate: De Moivre for powers/roots
Automate: Triangle inequality sense-checks
Formula: z=r(cosθ+i sinθ), zⁿ=rⁿ(cos nθ+i sin nθ)
Formula: |z|² = z z̄
Formula: ω = e^{2πi/3}, ω² + ω + 1 = 0
Formula: Arg(z₁/z₂)=Arg z₁−Arg z₂ (then principalise)
Trap: Wrong Arg branch
Trap: Treating conjugates incorrectly in equations
Drill: Daily: 1 polar NUM + 1 geometry/locus MC. Twice a week: 5 rapid ω simplifications (60s each). Open full Complex notes for complete process + formulas.

11 / 49 · Algebra

Topic cards

Algebra · topic cards

Probability & statistics · Quadratic Equations

Probability & statistics (High · independent page)

Trigger: See conditional/Bayes/random variable → tree + definition before any formula name.
Step: Define events in one line each.
Step: Draw partition/tree (or equally likely sample space).
Step: Apply definition / Bayes / counting model; keep exact fractions.
Automate: P(A|B)=P(A∩B)/P(B)
Automate: Total probability + Bayes
Formula: P(A∪B)=P(A)+P(B)−P(A∩B)
Formula: Bayes formula with explicit denominator
Formula: Equally likely: n(favourable)/n(total)
Formula: Binomial mean np, variance npq (when distribution is binomial)
Trap: Independence assumed without check
Trap: Confusing with mutually exclusive
Drill: Daily: 1 Bayes + 1 mean/variance or counting-probability. Weekly: 1 MC set where each option is independence/ME true-false. Open full Probability notes for complete process + formulas.

Quadratic Equations (Core · independent page)

Trigger: See roots/nature/symmetric functions → write ∑α, αβ (Vieta) before expanding.
Step: Write ax²+bx+c and Vieta.
Step: For nature use D; for transforms compute new ∑/∏.
Step: Form monic equation; verify.
Automate: Vieta α+β=−b/a, αβ=c/a
Automate: ∑α²=(∑α)²−2αβ
Formula: α+β=−b/a; αβ=c/a
Formula: D=b²−4ac
Formula: α²+β²=(α+β)²−2αβ
Trap: Expanding early
Trap: Ignoring a=0 parameter edge
Drill: Daily: 2 symmetric + 1 nature. Open full Quadratic notes.

12 / 49 · Algebra

Topic cards

Algebra · topic cards

Binomial Theorem · Permutations & Combinations

Binomial Theorem (Medium–High · independent page)

Trigger: See coefficient / middle term / identity → write T_{r+1}=nCr a^{n−r}b^r.
Step: Write T_{r+1}.
Step: Equate power / apply identity.
Step: Validate r in 0…n; small-n check for MATCH.
Automate: T_{r+1} general term
Automate: nCr=nC(n−r)
Formula: (a+b)ⁿ=∑ nCr a^{n−r}b^r
Formula: T_{r+1}=nCr a^{n−r}b^r
Formula: ∑nCr=2ⁿ
Trap: Index errors on general term
Trap: Sign errors with (−x)
Drill: Daily: 2 coeffs + 1 identity. Open full Binomial notes.

Permutations & Combinations (Medium–High · independent page)

Trigger: See arrange/choose/distribute/solutions → model sentence before nPr/nCr.
Step: Name object: arrange / choose / distribute / solutions.
Step: Write model sentence (order? replacement? constraints?).
Step: Translate to nPr / nCr / stars-and-bars / complement / cases.
Automate: P vs C classification
Automate: Stars-and-bars with positivity shift
Formula: nPr = n!/(n−r)! ; nCr = n!/(r!(n−r)!)
Formula: xi≥0: C(n+k−1,k−1) ; xi≥1: C(n−1,k−1)
Formula: Identical objects: n!/(n1!n2!…)
Trap: Overcount when order matters inconsistently
Trap: Stars-and-bars without shifting for positive variables
Drill: Daily: 3 timed counts with model sentence by 20s. Open full P&C notes for process + formula bank.

13 / 49 · Algebra

Topic cards

Algebra · topic cards

Sets · Relations

Sets (Medium · independent page)

Trigger: See Venn/cardinality/De Morgan/identity → algebra or region table first.
Step: Parse: identity vs cardinality vs word-Venn.
Step: Apply De Morgan / IE / region fill.
Step: Verify with tiny universe or |U| sum.
Automate: De Morgan pair
Automate: 2-set and 3-set inclusion-exclusion
Formula: (A∪B)′=A′∩B′ ; (A∩B)′=A′∪B′
Formula: |A∪B|=|A|+|B|−|A∩B|
Formula: |A∪B∪C| = singles − pairs + triple
Trap: Missing inclusion-exclusion terms
Trap: Venn regions not summing to |U|
Drill: Daily: 1 IE + 1 identity T/F. Open full Sets notes for process + formula bank.

Relations (Medium–High · independent page)

Trigger: See relation on a set → three lines: reflexive / symmetric / transitive.
Step: Write R rule.
Step: Three T/F lines: R / S / T (counterexample kills).
Step: Conclude type; if counting, use grid free-cell model.
Automate: Equivalence = R+S+T
Automate: Total relations 2^(n²)
Formula: Total: 2^(n²)
Formula: Reflexive: 2^(n²−n)
Formula: Symmetric: 2^(n(n+1)/2)
Formula: Equivalence ⇔ partition into classes
Trap: Calling equivalence after checking only two properties
Trap: Confusing symmetric with antisymmetric
Drill: Daily: 3 property audits. Open full Relations notes for process + formula bank.

14 / 49 · Algebra

Topic cards

Algebra · topic cards

Sequences & Series

Sequences & Series (Medium–High · independent page)

Trigger: See AP/GP/HP/sum → name type, write a,d or a,r, then Tn/Sn.
Step: Name AP/GP/HP (or ∑ polynomial).
Step: Write a,d or a,r from given terms.
Step: Apply Tn/Sn (finite vs infinite gate).
Automate: AP Tn/Sn
Automate: GP Tn/Sn and S∞ with |r|<1
Formula: AP: Tn=a+(n−1)d ; Sn=n/2[2a+(n−1)d]
Formula: GP: Tn=ar^{n−1} ; S∞=a/(1−r) (|r|<1)
Formula: ∑k=n(n+1)/2 ; ∑k²=n(n+1)(2n+1)/6
Trap: Using finite/infinite GP formulae interchangeably
Trap: Treating HP without reciprocal transform
Drill: Daily: 2 AP + 1 GP with parameter lock by 20s. Open full Sequences notes for process + formula bank.

15 / 49 · Calculus

Hub review

Calculus · hub overview

Bank ~28–32 /100 Math (~full Calculus share) · ~28–30% of Math tags (2020–2026)

Calculus full marks come from a fixed pipeline: domain → continuity/diff → tool → evaluate. Run the same pipeline for limits, AOD, integrals, DE, and area so most stems finish in 60–90 seconds with controlled accuracy.
Bank easy continuity and property-definite stems early — they are fast if you gate correctly.
Do not start a long area integral before writing intersection points; wrong limits waste the whole attempt.
If classification (Limit vs DE vs Area) is unclear in 20s, skip and return — forced tools create wrong answers.
Single correct: continuity/differentiability at a point, a clean definite value, tangent slope, or DE with IC — distractors match skipped LHL/RHL or wrong IF.
Multi-correct: several statements about monotonicity, continuity, or integral inequalities; each statement is an independent T/F.

16 / 49 · Calculus

Protocol

Calculus · 60–90s protocol

0–15s: Identify type: Limit / Continuity-Diff / AOD / Indefinite / Definite+property / DE / Area. Mark domain issues (floor, | |, piecewise). — Write the type in one word on rough sheet. Underline break points. If you cannot name the type, do not start differentiating. Also note the ask: value, interval, which options, or matching code.
15–35s: Apply the universal gate: simplify structure (sub, identity, property) BEFORE differentiating/integrating. — This window prevents chain-rule explosions. Commit to one gate: Domain check, Property ∫, DE class, or Intersection solve. Writing two plans means you are already late.
35–75s: Execute the standard tool (L’Hôpital only after 0/0 form is clear; DE: classify first; Area: find intersections first). — Keep algebra exact. For MC, tick T/F beside each option. For area, integrate interval-wise with |f−g| when curves cross. For DE, use only the recipe matching your classification.
75–90s: Check endpoints/sign of derivative/definite limits. For MC verify each statement separately. — Quick gates: did you state indeterminate form before L’Hôpital? Are definite limits in the correct order? Does f′ sign chart support your max/min claim? Fix once if a check fails — do not restart from chapter memory.

17 / 49 · Calculus

Methods

Calculus · methods

Domain & continuity gate · Simplify then differentiate/integrate

Domain & continuity gate

Applies: All Calculus, especially floor/| |/piecewise
Move: Write domain and check LHL/RHL/derivative existence at breaks before any formula.
Why: Prevents solving a smooth problem that is actually discontinuous.
Worked: For f(x)=[x] on [0,2], continuity fails at integers from the right/left pattern you must check — write LHL/RHL/f(n) in a tiny table before talking about differentiability.
Avoid: On an obviously smooth elementary function with no piecewise mark, still glance domain (logs, even roots) — but do not invent break points.

Simplify then differentiate/integrate

Applies: AOD, limits, integrals
Move: Algebraic rewrite, trig identity, or substitution first; calculus second.
Why: Cuts chain-rule explosions.
Worked: Before differentiating ln((1+x)/(1−x)), rewrite as ln(1+x)−ln(1−x). The derivative becomes obvious and safer than quotient-inside-log.
Avoid: If the expression is already minimal and the ask is a simple derivative, do not over-rewrite — but still scan for an easy identity.

18 / 49 · Calculus

Methods

Calculus · methods

Property before expansion · Classify DE first

Property before expansion

Applies: Definite integrals
Move: Try ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx and related even/odd style properties before expanding integrand.
Why: Many Advanced integrals collapse in 20 seconds.
Worked: For I=∫₀^{π/2} sin x/(sin x+cos x) dx, use x→π/2−x to get I=∫ cos/(sin+cos), add to get 2I=π/2 ⇒ I=π/4. No antiderivative struggle.
Avoid: If limits are indefinite or property clearly fails (no symmetry), go to substitution/parts — but you should still have checked property in 5–10s.

Classify DE first

Applies: Differential equations
Move: Label: separable / homogeneous / linear first-order — then apply the matching recipe only.
Why: Stops mixing methods mid-solution.
Worked: dy/dx + (2/x)y = x: linear with P=2/x, IF=x², d/dx(x²y)=x³ ⇒ x²y=x⁴/4+C. Classification took 5s; the rest is mechanics.
Avoid: If the DE is not first-order as in syllabus scope, do not invent higher-order methods — re-read the stem; Advanced sticks to listed first-order types.

19 / 49 · Calculus

Methods

Calculus · methods

Intersections before area · Sign chart for AOD

Intersections before area

Applies: Area between curves
Move: Solve f=g for bounds; sketch only the needed intervals; integrate |f−g| on each.
Why: Avoids wrong limits — the #1 area mark killer.
Worked: Area between y=x² and y=x: intersections at 0 and 1; on [0,1], x≥x² ⇒ ∫₀¹(x−x²)dx=1/6. Writing ‘above = x’ prevents inverted integrand.
Avoid: Never skip intersections. Even ‘obvious’ sketches must be confirmed by solving f=g.

Sign chart for AOD

Applies: Maxima/minima, monotonicity
Move: f′=0 critical points → sign chart of f′ → conclude increase/decrease/extrema.
Why: Replaces vague graphing with a 30-second table.
Worked: f(x)=x³−3x: f′=3(x−1)(x+1); sign chart ⇒ local max at −1, local min at 1. The chart is the answer engine; memorising ‘cubic shape’ is optional.
Avoid: On closed intervals, also compare endpoint values — critical points alone are incomplete.

20 / 49 · Calculus

Mental bank

Calculus · mental bank

L’Hôpital only for 0/0 or ∞/∞ indeterminate forms (state it first)
Rolle / Lagrange MVT: hypotheses + geometric meaning
∫₀ᵃ f(x)/(f(x)+f(a−x)) dx = a/2 (when defined)
King: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx
Linear DE: dy/dx + Py = Q → IF = e^{∫P dx}
Homogeneous first-order: y = vx, then separate
Separable: dy/g(y) = f(x)dx
Area: bounds from intersections; use |f−g| if needed
d/dx |x| = sgn(x) for x≠0; floor breaks at integers
Integration by parts: ∫u dv = uv − ∫v du
Table method before L’Hôpital
Sign chart before extrema claims
Classify DE before integrating
Flash IF, King property, and DE classes for 5–7 minutes before mocks.
Each line must be writable without a book. If not, it is a 100% leak.
When you use a bank line in practice, tick it (‘used King’, ‘used IF’) to build method awareness.

21 / 49 · Calculus

Topic cards

Calculus · topic cards

Limits · continuity · differentiability · AOD · tangents · extrema

Limits · continuity · differentiability (Core every-year)

Trigger: See piecewise/floor/| |/0/0 → LHL/RHL table or indeterminate check first.
Step: Identify problematic points and domain.
Step: For continuity: LHL, RHL, f(a).
Step: For differentiability: LHD, RHD (only after continuity if you use that logic carefully).
Automate: Floor/greatest-integer continuity checklist
Automate: Composite continuity intuition
Formula: lim θ→0 sinθ/θ = 1 (θ in radians)
Formula: L’Hôpital: only 0/0 or ∞/∞
Formula: Definition: f′(a)=lim h→0 [f(a+h)−f(a)]/h
Formula: Rolle: f cont [a,b], diff (a,b), f(a)=f(b) ⇒ f′(c)=0
Trap: Applying L’Hôpital without indeterminate form
Trap: Ignoring endpoint behaviour on closed intervals
Drill: Daily: 2 piecewise continuity/diff + 1 limit. Weekly: 1 Rolle/MVT MC set with hypothesis checklist written. Open full Limits notes.

AOD · tangents · extrema (High · independent page)

Trigger: See max/min/increasing/tangent/normal → simplify, then f′ + sign chart.
Step: Simplify f if needed.
Step: Compute f′ (and f′′ only if needed).
Step: Critical points + sign chart / second derivative test.
Automate: Chain/product/quotient fluency
Automate: Implicit differentiation (first order always; second when required)
Formula: (uv)′=u′v+uv′; (u/v)′=(u′v−uv′)/v²
Formula: d/dx f(g(x))=f′(g(x))g′(x)
Formula: For F(x,y)=0: dy/dx = −F_x/F_y (when F_y≠0)
Formula: Second derivative test: f′′>0 local min, f′′<0 local max (if f′=0)
Trap: Missing critical points from domain edges
Trap: Confusing increasing with f′≥0 vs f′>0 wording
Drill: Daily: 1 extrema (with sign chart) + 1 tangent/normal NUM. Weekly: 1 implicit differentiation MC. Open full AOD notes.

22 / 49 · Calculus

Topic cards

Calculus · topic cards

Integrals · properties · area · Differential equations

Integrals · properties · area (High · independent page)

Trigger: See definite integral → property glance; see area → intersections first.
Step: Definite: check symmetry/King/even-odd before expanding.
Step: If technique needed: pick sub / parts / PF deliberately.
Step: Area: solve intersections; split intervals; integrate |f−g|.
Automate: King property and a/2 classic form
Automate: By parts / sub / partial fractions quick selection
Formula: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx
Formula: ∫₀ᵃ f/(f(x)+f(a−x)) dx = a/2
Formula: ∫u dv = uv − ∫v du
Formula: Area = ∫_a^b |f−g| dx on proper bounds
Trap: Wrong limits for area
Trap: Dropping absolute value when curves cross
Drill: Daily: 1 property definite + 1 area with written intersections. Weekly: 8-question property blitz in 12 minutes. Open full Integrals notes.

Differential equations (Medium–High · independent page)

Trigger: See dy/dx or Mdx+Ndy → rewrite, then stamp SEP / HOM / LIN.
Step: Rewrite in standard form.
Step: Classify: separable / homogeneous / linear.
Step: Apply matching recipe only.
Automate: Separable integration + constant handling
Automate: y=vx for homogeneous (including dy/dx = v+x dv/dx)
Formula: Linear: dy/dx + P(x)y = Q(x), IF=e^{∫P dx}
Formula: Homogeneous: dy/dx = f(y/x) → y=vx
Formula: Separable: ∫ dy/g(y) = ∫ f(x) dx
Formula: Keep ln|·| unless domain guarantees sign
Trap: Calling a DE linear incorrectly
Trap: Losing absolute value in ln| |
Drill: Daily: 1 DE rotating SEP/HOM/LIN. Weekly: 1 matching DE↔solution + 1 IC particular solution. Open full DE notes.

23 / 49 · Coordinate

Hub review

Coordinate Geometry · hub overview

Bank ~12–16 /100 Math (full Coordinate share + joins) · ~13% of Math tags (2020–2026) — still every-year & high accuracy if templated

Coordinate Geometry becomes 100% when every conic/line/circle stem starts from standard form + one geometric dictionary entry (focus, directrix, tangent slope, chord) — then finish in 60–90 seconds with a point check.
Bank standard circle/parabola/line toolkit stems early — they are fast once standardised.
Never apply a conic formula before writing the standard form on rough sheet.
If the object is unclear in 20s (is it pair of lines or circle?), skip and return — forced dictionary hits create wrong answers.
Single correct: clean numerical ask (eccentricity, tangent slope, distance, length of tangent) with distractors from wrong ‘a’ after shift or swapped e formulae.
Multi-correct: several geometric claims (focus location, director circle, asymptotes); each claim is independent T/F.

24 / 49 · Coordinate

Protocol

Coordinate Geometry · 60–90s protocol

0–15s: Name object: Line / Pair of lines / Circle / Parabola / Ellipse / Hyperbola / Locus. Write the given equation. — Say the object in one word on rough sheet. Underline what is asked (focus, tangent, e, locus, chord). If you cannot name the object, do not open the formula bank yet.
15–35s: Bring to standard form (complete square / shift origin / divide). List known a, e, focus, directrix if conic. — This is the highest-ROI window. Completing the square for a circle, dividing by the constant for an ellipse, or shifting a parabola to y²=4ax unlocks the dictionary. Write a, e, focus, directrix in a one-line list before choosing a tool.
35–75s: Apply the one needed tool: tangent, chord of contact, parametric, distance, angle, director circle, etc. — Pick ONE dictionary entry. For moving points, prefer parametric. For circle–circle, prefer S1−S2. For MC, treat each geometric claim as its own mini-question.
75–90s: Plug a known point or eccentricity check. For MC, test each geometric claim independently. — Quick gates: does the focus satisfy the definition? Is e in (0,1) for ellipse and >1 for hyperbola? Does a known point lie on the tangent? Fix once if a check fails — do not restart from chapter memory.

25 / 49 · Coordinate

Methods

Coordinate Geometry · methods

Standardise first · Dictionary hit

Standardise first

Applies: All conics and circles
Move: Complete square / shift / divide until you recognise y²=4ax, x²/a²+y²/b²=1, etc.
Why: Formula sheets only work on standard form.
Worked: x²+y²−4x+6y−3=0 → (x−2)²+(y+3)²=16 → centre (2,−3), r=4. Only then use tangent length / power. Writing centre/radius before any formula prevents g,f sign mistakes.
Avoid: If the equation is already clearly y²=4ax or x²/a²+y²/b²=1, skip busywork — but still write a, e, focus on rough sheet.

Dictionary hit

Applies: Focus, directrix, tangent, normal, chord
Move: Once standard, recall the single entry you need — do not re-derive from definition unless asked.
Why: Saves 1–2 minutes per stem.
Worked: y²=4ax with a=2 → focus (2,0), directrix x=−2, parametric (2t²,4t). Tangent at t: ty=x+at². Hit the entry matching the ask; ignore the rest.
Avoid: If the stem explicitly asks a definition-based proof, derive carefully — but for value/option questions, dictionary first.

26 / 49 · Coordinate

Methods

Coordinate Geometry · methods

Parametric when points move · Distance & angle toolkit

Parametric when points move

Applies: Chord joining, locus of mid-point, triangle on conic
Move: Use (at²,2at) / (a cosθ,b sinθ) etc. and eliminate parameter for locus.
Why: Cleaner than heavy Cartesian for moving points.
Worked: Mid-point of focal chord of parabola: use ends t₁,t₂ with t₁t₂=−1, mid-point in terms of t₁+t₂, eliminate → locus. Writing t₁t₂=−1 early is the method.
Avoid: Fixed single point with a tangent ask may be faster in Cartesian/slope form — still fine to use parametric if you are fluent.

Distance & angle toolkit

Applies: Lines, normals, pair of lines
Move: Point–line distance, angle between lines, bisectors — write formula with numbers plugged.
Why: Most line stems are pure toolkit.
Worked: Distance from (1,2) to x+y−1=0 is |1+2−1|/√2=√2. Write numerator absolute value and √(a²+b²) explicitly — do not ‘simplify mentally’ under pressure.
Avoid: If the stem is clearly a conic, do not force line toolkit — but concurrency/family-of-lines still uses L1+λL2.

27 / 49 · Coordinate

Methods

Coordinate Geometry · methods

Common chord / intersection · Locus discipline

Common chord / intersection

Applies: Circle–circle, circle–conic
Move: S1−S2=0 for common chord; solve system carefully.
Why: Avoids expanding both circles fully.
Worked: S1−S2 gives the common chord line; drop perpendicular from centre to that line; half-chord from Pythagoras. This is faster than solving quadratic systems entry-wise.
Avoid: If one ‘circle’ is not actually a circle after completing square, re-standardise first — S1−S2 still needs correct S forms.

Locus discipline

Applies: Any ‘locus of P is’
Move: Introduce parameter → write condition → eliminate parameter → recognise curve.
Why: Prevents random guessing among conic options.
Worked: P=(at²,2at) with condition ‘chord joins t and −t is focal’ already known; for mid-point locus write h=…, k=…, eliminate t. Final equation in h,k is the locus.
Avoid: Never skip this for locus. The only shortcut is recognising a standard locus you have drilled — still write one elimination line for safety.

28 / 49 · Coordinate

Mental bank

Coordinate Geometry · mental bank

y²=4ax: focus (a,0), directrix x=−a, parametric (at²,2at)
x²=4ay: focus (0,a), directrix y=−a, parametric (2at,at²)
Ellipse x²/a²+y²/b²=1 (a>b): e²=1−b²/a², foci (±ae,0)
Hyperbola x²/a²−y²/b²=1: e²=1+b²/a², foci (±ae,0)
Circle: x²+y²+2gx+2fy+c=0 → centre (−g,−f), r=√(g²+f²−c)
Tangent slope form parabola: y=mx+a/m
Point–line distance: |ax₀+by₀+c|/√(a²+b²)
Angle between lines: tanθ=|(m₁−m₂)/(1+m₁m₂)|
Length of tangent from (x₁,y₁) to circle: √S₁
Director circle of ellipse: x²+y²=a²+b²
Standard form before focus/directrix
Extract a from 4a on parabola first
Ellipse e²=1−b²/a²; hyperbola e²=1+b²/a²
Flash parabola/ellipse/hyperbola dictionaries for 7 minutes before mocks — focus, directrix, e, parametric, tangent.
Each line must be writable without a book. If not, it is a 100% leak.
When you use a bank line in practice, tick it (‘used S1−S2’, ‘used parametric’) to build method awareness.

29 / 49 · Coordinate

Topic cards

Coordinate Geometry · topic cards

Straight lines · Circle

Straight lines (Base layer · every-year toolkit)

Trigger: See angle/bisector/distance/concurrency/family → write form, then toolkit formula.
Step: Write line in convenient form (slope or ax+by+c=0).
Step: Apply distance/angle/bisector formula with absolute values and signs careful.
Step: For family of lines: use L1+λL2=0; fix λ from the extra condition.
Automate: Angle between two lines (slope and coefficient forms)
Automate: Distance from point to line
Formula: Distance: |ax₀+by₀+c|/√(a²+b²)
Formula: tanθ=|(m₁−m₂)/(1+m₁m₂)| (if 1+m₁m₂≠0)
Formula: Bisectors of a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0: (L1)/√(a₁²+b₁²)=±(L2)/√(a₂²+b₂²)
Formula: Family: L1+λL2=0
Trap: Sign errors in bisector formula
Trap: Mixing slope angle with line–pair conditions
Drill: Alt days: 2 line toolkit NUM (distance/angle/bisector). Weekly: 1 pair-of-lines + 1 L1+λL2 family. Open full Lines notes.

Circle (High · independent page)

Trigger: See tangent/chord/power/intersection → complete square, then S₁ / S1−S2.
Step: Complete square → centre, radius (or read g,f,c).
Step: Tangent condition / chord of contact / √S₁ as needed.
Step: Intersection: S1−S2 for common chord when useful.
Automate: Centre (−g,−f), r=√(g²+f²−c)
Automate: Length of tangent √S₁
Formula: x²+y²+2gx+2fy+c=0 → centre (−g,−f), r=√(g²+f²−c)
Formula: Length of tangent from (x₁,y₁): √S₁
Formula: Chord of contact from (x₁,y₁): xx₁+yy₁+g(x+x₁)+f(y+y₁)+c=0
Formula: S1−S2=0 → common chord / radical axis
Trap: Wrong radius after completing square
Trap: Forgetting condition of tangency forms
Drill: Daily: 1 circle toolkit + 1 circle–circle or circle–conic mix. Weekly: 1 multi-correct on tangent/power claims. Open full Circle notes.

30 / 49 · Coordinate

Topic cards

Coordinate Geometry · topic cards

Parabola · Ellipse

Parabola (High · independent page)

Trigger: See y²=… / x²=… / shifted parabola → standardise, then focus/parametric/tangent.
Step: Standardise to y²=4ax (or x²=4ay).
Step: Write a, focus, directrix.
Step: Use parametric or tangent slope form as needed.
Automate: Parametric (at²,2at) and (2at,at²)
Automate: Tangent y=mx+a/m
Formula: y²=4ax: focus (a,0), directrix x=−a
Formula: Point (at²,2at); tangent: ty=x+at²
Formula: Slope form: y=mx+a/m
Formula: Focal chord ends t₁,t₂ ⇒ t₁t₂=−1
Trap: Using ellipse formulae on parabola
Trap: Wrong ‘a’ after shifting
Drill: Daily: 1 parabola tangent/chord + alternate day 1 locus eliminate-t. Weekly: 1 focal-chord property set. Open full Parabola notes.

Ellipse (High · independent page)

Trigger: See a,b,e / foci → standardise, e²=1−b²/a², then dictionary.
Step: Standardise; identify major axis.
Step: Compute e²=1−(minor²)/(major²); write foci.
Step: Apply tangent/director-circle as asked.
Automate: e²=1−b²/a² (a>b)
Automate: Foci (±ae,0) or (0,±be)
Formula: Ellipse (a>b): e²=1−b²/a², foci (±ae,0)
Formula: Tangent: xx₁/a²+yy₁/b²=1
Formula: y=mx±√(a²m²+b²)
Formula: Director circle: x²+y²=a²+b²
Trap: Swapping ellipse/hyperbola eccentricity formulae
Trap: Using a<b case without rewriting
Drill: Daily: 1 e/foci + 1 tangent. Open full Ellipse notes.

31 / 49 · Coordinate

Topic cards

Coordinate Geometry · topic cards

Hyperbola

Hyperbola (High · independent page)

Trigger: See a,b,e / asymptotes → standardise, e²=1+b²/a², then asymptotes.
Step: Standardise; identify transverse axis.
Step: e²=1+b²/a²; foci; asymptotes y=±(b/a)x.
Step: Apply tangent tool as asked.
Automate: e²=1+b²/a²
Automate: Asymptotes y=±(b/a)x
Formula: Hyperbola: e²=1+b²/a², foci (±ae,0)
Formula: Asymptotes: y=±(b/a)x
Formula: Tangent: xx₁/a²−yy₁/b²=1
Formula: Rectangular: a=b, e=√2
Trap: Swapping ellipse/hyperbola eccentricity formulae
Trap: Mixing parametric forms with ellipse
Drill: Daily: 1 e/asymptotes + 1 tangent. Open full Hyperbola notes.

32 / 49 · Trig

Hub review

Trigonometry · hub overview

Bank formulas + equations + inverse; keep SOT ready for joins · Official Trig unit + frequent Calculus/Complex joins; SOT as join toolkit

Trigonometry becomes exam-ready when you open the right topic page: Formulas for reduce, Equations for general solutions, Inverse for principal values, Triangles for sine/cosine-rule joins — each with its own 60–90s process.
Bank clean formula exact-values and standard general solutions early.
Never ship inverse without a written principal range.
If the stem is clearly SOT-in-disguise, sketch the triangle — do not force pure identity algebra.
Exact values and identity simplifies (Formulas page).
General solutions on ℝ / counts in an interval (Equations page).

33 / 49 · Trig

Protocol

Trigonometry · 60–90s protocol

0–15s: Route the stem: Formulas / Equations / Inverse / Triangles. Circle the ask. — Inverse → write range now. Equation → note ℝ vs interval. Triangle → sketch. Formula → name expand/compress/double.
15–35s: Apply that page’s first method (identity job / reduce / range lock / case name). — Do not mix pages. Formula reduce feeds Equations — finish reduce before template.
35–75s: Execute: expand/factor, general template, principal value, or Law of Sines/Cosines. — Write n∈ℤ on equation answers. Check ab vs 1 on arctan sums. Use (1/2)ab sinC on SAS area.
75–90s: Verify: value ranges, extraneous roots, triangle inequality, principal interval. — MC: tick each option as independent T/F against your template/range.

34 / 49 · Trig

Methods

Trigonometry · methods

Route to the right Trig page · Name the identity job

Route to the right Trig page

Applies: All Trig stems
Move: In 10s choose Formulas / Equations / Inverse / Triangles — then open that habit set only.
Why: Stops using inverse tricks on equation stems and vice versa.
Worked: ‘Solve sin x = … on ℝ’ → Equations. ‘sin⁻¹(−1/2)’ → Inverse. ‘sin75°’ → Formulas.

Name the identity job

Applies: Simplifies and exact values
Move: Say expand / compress / double / evaluate before writing.
Why: Ends random formula shopping.
Worked: sin75° → expand via 45+30.

35 / 49 · Trig

Methods

Trigonometry · methods

General solution template · Principal inverse lock

General solution template

Applies: Trig equations on ℝ
Move: Find α, stamp sin/cos/tan general form, write n∈ℤ.
Why: Template is half the scored answer.
Worked: sinx=1/2 → x=nπ+(−1)ⁿ(π/6).

Principal inverse lock

Applies: Inverse values and identities
Move: Write ranges before evaluating or adding.
Why: Kills wrong-branch MC options.
Worked: cos⁻¹(−1/2)=2π/3 not −2π/3.

36 / 49 · Trig

Methods

Trigonometry · methods

Extraneous check · Triangle case name

Extraneous check

Applies: Squared equations / Weierstrass
Move: Plug every candidate back into the original.
Why: Ghost roots are designed distractors.

Triangle case name

Applies: SOT / area joins
Move: Sketch and name SAS/ASA/SSS/SSA before choosing a law.
Why: Chooses Cosines vs Sines in seconds.
Worked: Two sides + included angle → (1/2)ab sinC or Cosines.
Avoid: Pure inverse/equation stems — do not force a triangle sketch.

37 / 49 · Trig

Mental bank

Trigonometry · mental bank

sin(A±B), cos(A±B), tan(A±B)
sin2A, cos2A (three forms), power-reduce
sinx=k → nπ+(−1)ⁿα; cos → 2nπ±α; tan → nπ+α
sin⁻¹∈[−π/2,π/2]; cos⁻¹∈[0,π]; tan⁻¹∈(−π/2,π/2)
sin⁻¹x+cos⁻¹x=π/2 (|x|≤1)
a/sinA=2R; Δ=(1/2)bc sinA
15°=45°−30°; factor before expand-all
Route page first: Formulas / Equations / Inverse / Triangles
Name the identity job before shopping formulae
General solution template with correct period
Principal inverse range before value
Formulas page owns A±B and factor; Equations owns templates; Inverse owns ranges; Triangles owns Sines/Cosines.
SOT is join insurance — revise weekly, not as a promise of direct Advanced weight.

38 / 49 · Trig

Topic cards

Trigonometry · topic cards

Sin / Cos / Tan formulas & applications · Trigonometric equations (general solutions)

Sin / Cos / Tan formulas & applications (Foundation + Calculus joins)

Trigger: Simplify / exact value → name expand/compress/double → identity → evaluate.
Step: Name the identity job.
Step: Apply A±B, double, or factor.
Step: Finish value or feed Calculus.
Automate: A±B and tan(A±B)
Automate: Double / power-reduce
Formula: sin(A±B), cos(A±B), tan(A±B)
Formula: sin2A, cos2A, sin²A=(1−cos2A)/2
Formula: sinC+sinD=2sin((C+D)/2)cos((C−D)/2)
Trap: Wrong cos(A±B) signs
Trap: Expand-all when factor compresses
Drill: Daily: 2 exact values + 1 factor simplify.

Trigonometric equations (general solutions) (Official Advanced core)

Trigger: Equation on ℝ → reduce → α → sin/cos/tan template → n∈ℤ.
Step: Isolate one function (or R-method).
Step: Find α.
Step: Stamp general form.
Automate: Three general templates
Automate: R-method for a sin+b cos
Formula: sinθ=sinα ⇒ nπ+(−1)ⁿα
Formula: cosθ=cosα ⇒ 2nπ±α
Formula: tanθ=tanα ⇒ nπ+α
Trap: Only [0,2π] when ℝ asked
Trap: Wrong period
Drill: Daily: 2 full general solutions written out.

39 / 49 · Trig

Topic cards

Trigonometry · topic cards

Inverse trigonometry (principal values) · Solution of triangles (join toolkit)

Inverse trigonometry (principal values) (Official Advanced — high trap density)

Trigger: Inverse ask → write principal range → evaluate → identity only if domain-safe.
Step: Range lock.
Step: Principal value.
Step: Composition triangle if needed.
Automate: Three principal ranges
Automate: sin⁻¹x+cos⁻¹x=π/2
Formula: sin⁻¹∈[−π/2,π/2]; cos⁻¹∈[0,π]; tan⁻¹∈(−π/2,π/2)
Formula: sin⁻¹x+cos⁻¹x=π/2
Formula: sin(cos⁻¹x)=√(1−x²)
Trap: Negative cos⁻¹ values
Trap: π/2 identity outside |x|≤1
Drill: Daily: 3 principal values (include negatives).

Solution of triangles (join toolkit) (Not a direct Advanced unit — keep for joins)

Trigger: Triangle data → sketch + SAS/ASA/SSS/SSA → Sines/Cosines/area.
Step: Sketch and name case.
Step: Pick Law of Sines or Cosines.
Step: Area with matching formula.
Automate: a/sinA=2R
Automate: Cosine law
Formula: a/sinA=b/sinB=c/sinC=2R
Formula: c²=a²+b²−2ab cosC
Formula: Δ=(1/2)bc sinA
Trap: Heron on SAS
Trap: Ignoring ambiguous SSA
Drill: Daily: 1 SAS area + 1 Sines/2R.

40 / 49 · Vectors · 3D

Hub review

Vectors & 3D Geometry · hub overview

Bank Basics → Dot/Cross → Triple → Lines → Planes as five topic pages · Official Vectors + Analytical Geometry (3D) — high accuracy if templated

Vectors & 3D become exam-ready when you open the right page: Basics for components, Dot/Cross for meaning, Triple for coplanar/volume, Lines for skew SD, Planes for n/distance/foot — each with its own 60–90s process.
Bank clean ·/×/STP and standard plane-distance stems early.
Never quote skew SD before the coplanar gate fails.
Write b and n before any 3D formula sheet.
Unit vector / section / linear combo (Basics).
Angle, ⊥/∥, area (Dot/Cross).

41 / 49 · Vectors · 3D

Protocol

Vectors & 3D Geometry · 60–90s protocol

0–15s: Route: Basics / Dot·Cross / Triple / Lines / Planes. Circle the ask. — Angle/area → Dot·Cross. Coplanar/volume/nested × → Triple. Skew SD → Lines. Foot/distance/plane equation → Planes. Components-only → Basics.
15–35s: Write components; extract b and/or n; choose · / × / STP / SD / distance. — Points → displacements. Two-point line → b=P₂−P₁. Three-point plane → n=AB×AC.
35–75s: Execute one tool from the routed page. — BAC–CAB for nested cross. Skew gate then SD. λ-template for foot/image.
75–90s: Sanity: |â|=1; ·=0 for ⊥; STP=0 for coplanar; SD≥0; foot on plane; sinφ for line–plane. — MC: each geometric claim is independent T/F.

42 / 49 · Vectors · 3D

Methods

Vectors & 3D Geometry · methods

Route to the right Vectors page · Meaning before matrix

Route to the right Vectors page

Applies: All Vectors/3D stems
Move: In 10s choose Basics / Dot·Cross / Triple / Lines / Planes — then use only that habit set.
Why: Stops using plane tools on skew lines and · on area asks.
Worked: ‘SD between two lines’ → Lines. ‘a×(b×c)’ → Triple. ‘Area of △’ → Dot·Cross.

Meaning before matrix

Applies: Product stems
Move: Angle/⊥ → · ; area/normal/∥ → × ; coplanar/volume → STP.
Why: Wrong product is the usual full-mark killer.
Worked: Perpendicular → a·b=0. Parallelogram area → |a×b|.

43 / 49 · Vectors · 3D

Methods

Vectors & 3D Geometry · methods

Components / edges first · DR / normal first

Components / edges first

Applies: Points and coordinate stems
Move: Form AB=b−a (and other edges) before products or 3D formulae.
Why: Position vs free-vector confusion dies early.

DR / normal first

Applies: 3D lines and planes
Move: Write b for lines and n for planes before any distance/angle formula.
Why: Every 3D formula is built from these two.
Worked: Plane ax+by+cz+d=0 ⇒ n=⟨a,b,c⟩.

44 / 49 · Vectors · 3D

Methods

Vectors & 3D Geometry · methods

Skew gate then SD · Foot / image λ template

Skew gate then SD

Applies: Two lines in space
Move: Check (a₂−a₁)·(b₁×b₂)≠0; only then apply SD.
Why: Prevents skew SD on intersecting lines.

Foot / image λ template

Applies: Reflection and feet on planes
Move: P+μn into plane → foot F; image = 2F−P.
Why: Turns a scary ask into arithmetic.

45 / 49 · Vectors · 3D

Mental bank

Vectors & 3D Geometry · mental bank

AB=b−a; â=a/|a|
a·b=|a||b|cosθ; ⊥ ⇔ ·=0
|a×b|=|a||b|sinθ; ∥ ⇔ ×=0
[a b c]=det; volume=|STP|; BAC–CAB
Skew SD=|(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|
Point–plane |ax+by+cz+d|/√(a²+b²+c²)
Line–plane: sinφ=|b·n|/(|b||n|)
Route page first: Basics / Dot·Cross / Triple / Lines / Planes
AB = b − a from position vectors
Meaning first: · for angle/projection, × for area/perp
Skew SD only after coplanar test fails
Basics owns components; Dot·Cross owns ·/×; Triple owns STP/VTP; Lines owns skew; Planes owns n/foot.
sinφ for line–plane — cosθ for plane–plane. Do not swap.

46 / 49 · Vectors · 3D

Topic cards

Vectors & 3D Geometry · topic cards

Vector Algebra (basics) · Dot & Cross products

Vector Algebra (basics) (Foundation for every Vectors/3D stem)

Trigger: Magnitude / unit / section / combo → components → AB=b−a → tool.
Step: Write components.
Step: Form displacements if points given.
Step: Apply |a|, â, section, or equate components.
Automate: AB=b−a
Automate: Unit vector
Formula: |a|=√(a₁²+a₂²+a₃²)
Formula: â=a/|a|
Formula: AB=b−a
Formula: (na+mb)/(m+n)
Trap: a+b as joining vector
Trap: Swapping m,n
Drill: Daily: 1 unit vector + 1 section.

Dot & Cross products (Highest-frequency Vector toolkit)

Trigger: Angle/⊥/area/normal → meaning → · or × → compute.
Step: Name geometric ask.
Step: Pick · or ×.
Step: Compute (determinant for cross).
Automate: · and × meanings
Automate: ⊥/∥ zero tests
Formula: a·b=|a||b|cosθ
Formula: |a×b|=|a||b|sinθ
Formula: ⊥ ⇔ ·=0; ∥ ⇔ ×=0
Trap: · for area
Trap: Forgetting 1/2
Drill: Daily: 1 angle + 1 area + 1 zero-test.

47 / 49 · Vectors · 3D

Topic cards

Vectors & 3D Geometry · topic cards

Scalar & vector triple products · Lines in three dimensions

Scalar & vector triple products (Coplanarity, volume, nested cross)

Trigger: Coplanar/volume → STP det; nested × → BAC–CAB.
Step: Choose STP or VTP.
Step: Det or BAC–CAB.
Step: Interpret 0 / |det| / simplified vector.
Automate: STP as determinant
Automate: Volume = |STP|
Formula: [a b c]=det
Formula: V=|[a b c]|
Formula: a×(b×c)=(a·c)b−(a·b)c
Trap: Missing | | on volume
Trap: Component expansion of nested cross
Drill: Daily: 1 STP + 1 BAC–CAB.

Lines in three dimensions (Skew SD + line forms — high-accuracy bank)

Trigger: Write b → gate skew/intersect → angle or SD.
Step: Extract a,b for each line.
Step: Parallel / coplanar / skew gate.
Step: Apply angle or SD.
Automate: Symmetric & vector forms
Automate: Skew SD formula
Formula: r=a+λb
Formula: Skew SD formula
Formula: (a₂−a₁)·(b₁×b₂)=0 coplanar
Trap: Skew SD on intersecting lines
Trap: 2D distance intuition
Drill: Daily: 1 line form + alt-day skew gate.

48 / 49 · Vectors · 3D

Topic cards

Vectors & 3D Geometry · topic cards

Planes in three dimensions

Planes in three dimensions (Plane forms, distance, foot/image)

Trigger: Write n → distance/angle/foot λ-template.
Step: Build or read n.
Step: Write plane.
Step: Apply distance, angle, or foot/image.
Automate: Plane from three points via cross
Automate: Point–plane distance
Formula: ax+by+cz+d=0
Formula: |ax+by+cz+d|/√(a²+b²+c²)
Formula: sinφ=|b·n|/(|b||n|)
Trap: cos for line–plane angle
Trap: Wrong d sign
Drill: Daily: 1 distance + alt-day foot/image.

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Checklist

Math package · close checklist

I can name 4–6 universal moves without notes.
I know which hub is today’s bank vs stretch.
I have 2 weak topic pages marked for tomorrow.
I will not expand Algebra / Trig before classification.
Next drill: notes topic page or /mixed — not another full reread.

After this package: open Math notes notes only for weak marked pages, or drill /mixed. Directions without content pages do not move marks.