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Concept notes
112 concepts · 4 modules · 181 named traps
Mathematics
CSCA Mathematics · Contents
Functions & Sequences
34 concepts · 57 named traps33% of the exam- Function definitiondomain and range; equal functions (same domain and rule)
- Determining the domaindenominator nonzero, even-root radicand ≥0, log argument >0, non-integer-power base >0; intersect conditions
- Finding the rangerange-to-domain reverse problems (incl. substitution t=2x reducing to a quadratic)
- Recovering f(x) from f(g(x)) by substitution / completing the squarewith induced domain
- Recovering f(x) from a functional equatione.g. f(x)+2f(1-x)=x2; Cauchy-type f(x1+x2)=f(x1)+f(x2)+1
- Monotonicityincreasing/decreasing, monotonic intervals, difference method, composite monotonicity
- Parity/symmetryeven, odd, neither; domain-symmetry requirement; parity of shifted functions
- Power functionsdefinition (coefficient 1), graphs/properties by exponent sign, passing-through-origin conditions, fit through a point
- Exponential functions axgraphs/properties by base, domain/range/asymptote; natural exponential e
- Logarithmic functions loga xinverse of exponential, graphs/properties; ln and lg
- Logarithm rulesproduct, quotient, power, change-of-base, identities aloga N=N
- Comparing log/exponential expressions by magnitudebase conversion, monotonicity, estimating ln near 1
- Piecewise functionsbranch selection, nested evaluation, zeros/symmetry across branches
- Inverse functionse.g. inverse of a restricted quadratic with correct branch and domain
- Arithmetic sequencesgeneral term an=a1+(n-1)d, sum Sn=n(a1+an)/2, sign/extremal-sum reasoning
- Geometric sequencesgeneral term a1·qn-1, sum, geometric mean, partial-sum form Sn=a·bn+c constraints, monotonicity conditions
- Sequences from recurrences(auxiliary/transformed sequence) and an=Sn-Sn-1; recovering an from a sum-of-roots form
- Summation techniquesgrouping/pairing alternating signs, splitting into arithmetic+geometric parts, periodic (sine-defined) sequences
- Derivative conceptaverage rate of change, limit definition, derivative as tangent slope; tangent-line equations
- Standard derivative table(xn, sin, cos, ex, ax, ln x, loga x, sqrt x, 1/x) and rules (sum/product/quotient/constant-multiple)
- Chain rule for composite functions
- Applications of derivativesmonotonicity from sign of f', extrema/local max-min, closed-interval max/min, no-critical-point parameter ranges
- Derivative-based comparison via auxiliary functionse.g. g(x)=f(x)/ex when f'(x)<f(x)
- Second derivative / convexity-concavity conditions
- Velocity/kinematics interpretationaverage vs instantaneous velocity; position-time and distance-time graph reading
- Modelingcompound-interest/annuity geometric series; exponential decay (half-life) and ln-table evaluation; qualitative water-height-vs-time graphs
- Domain, range, equal functions, and recovering f(x)9 named traps
- Monotonicity and parity7 named traps
- Power, exponential, logarithmic functions7 named traps
- Piecewise and inverse functions5 named traps
- Arithmetic and geometric sequences9 named traps
- Recurrences, summation techniques, and an from Sn5 named traps
- Derivatives, tangents, and applications12 named traps
- Modeling (interest, decay, graphs)3 named traps
Geometry & Algebra
36 concepts · 56 named traps29% of the exam- Straight lineslope from two points, undefined slope for vertical lines; inclination angle to slope
- Line equation formspoint-slope, slope-intercept, two-point, intercept, general Ax+By+C=0
- Parallel and perpendicular line conditionsreflection of a point across a line; slope range so a line meets a segment
- Distance from a point to a lineminimum point-to-line distance; points equidistant from a line
- Circle: standard and general equations(center/radius); condition for a 2nd-degree equation to be a circle; tangency to an axis
- Two circles' positional relationship via center distancecommon chord length
- Tangents from an external pointchord of contact through a fixed point
- Ellipse: definition(sum of focal distances=2a), standard equation, a2=b2+c2, focal-triangle area with angle condition
- Hyperboladefinition (|difference|=2a), standard equation, asymptotes y=±(b/a)x, eccentricity e>1, midpoint-chord (point-difference) method
- Paraboladefinition (focus/directrix), standard forms, focal distance to x-coordinate, weighted-distance minimization via directrix
- Conic eccentricity problemsfinding/ranging e from focal, asymptote, or inequality conditions
- Trigonometryradian-degree conversion; unit-circle sin/cos/tan; terminal side through a point; signs by quadrant
- Trig identitiessin2+cos2=1, tan=sin/cos, reduction/phase-shift simplification, evaluating via case analysis
- Sum/difference, double-angle, half-angle, product-to-sum/sum-to-product formulas; tan(A+B) to find a triangle angle
- Solving trianglesSine Rule, Law of Cosines, area=(1/2)ab sinC; angle from a parallel-vector/collinearity condition
- Graph of y=A sin(wx+φ) via translation/scalinghorizontal stretch/compression of trig graphs
- Plane vectorsmagnitude, coordinate representation, addition/subtraction/scalar multiplication, vectors in polygons (hexagon)
- Vector parallel/collinearity conditions to solve for a parametervectors with points dividing sides
- Dot productdefinition, coordinate form, magnitude of a sum via law of cosines, angle between vectors, perpendicularity, projection vector
- Complex numbersi2=-1, algebraic form, real/imaginary parts, modulus, conjugate; reflection across real axis
- Complex arithmeticadd/subtract/multiply/divide-by-conjugate; polar form and De Moivre powers; complex numbers as rotated vectors
- Spatial rectangular coordinatespoint location (axes/coordinate planes), 3D distance formula, symmetry across axis/plane/origin
- Solid figuressurface area & volume of cuboid, cube, cylinder, cone, sphere; lateral vs total surface area
- Cone/frustumaxial cross-section relations, sector-unrolling (central angle ↔ base radius), slant vs vertical height, volume
- Inscribed/circumscribed spheres of prisms and tetrahedraspace diagonal, equal-volume inradius method
- Cross-sections of solidswhich polygons can/can't appear; midpoint-defined sections
- 3D analysis in a cubemoving points, skew-line angle ranges, perpendicularity, constant-volume pyramids, loci and path lengths
- Volume of a tetrahedron from coordinates(scalar triple product / base x height); corner-cutoff volume ratios
- Oblique (cavalier) projection area ratiosolids of revolution (annulus/torus)
- Lines, circles, and distance9 named traps
- Conic sections8 named traps
- Trigonometry8 named traps
- Vectors8 named traps
- Complex numbers7 named traps
- Spatial coordinates and symmetry4 named traps
- Solid geometrysolids, surfaces, volumes12 named traps
Probability & Statistics
22 concepts · 39 named traps21% of the exam- Random experiments, sample space, events (basic/certain/impossible)identifying the classical (equally-likely) model
- Classical probability P(A)=favorable/total by direct counting
- Counting with combinations C(n,k) and permutationsordered vs unordered selection
- Probability via combinatorial assignment/distributionsurjective 'at least one each', same-group, ends-of-row
- Probability from real data charts(air-quality index, frequency bar/line charts) including joint multi-day conditions
- Central tendencymean (incl. weighted/combined-group mean), median, mode; choosing the right statistic for a question
- Dispersionvariance and standard deviation; effect of adding a constant or an outlier; trimming extremes before computing
- Combining strata/groupspooled mean and pooled variance (between + within), size-ratio weighting
- Standard deviation under linear vs nonlinear transformationsvariance scaling D(aX+b)=a2 D(X)
- Linear regressionleast-squares slope/intercept from sums; reading scatter/regression context
- Normal distributionbell curve, symmetry about μ, role of μ and σ, total area = 1
- 68-95-99.7 empirical ruleinterval probabilities via symmetry; estimating counts = probability x population
- Normal distribution symmetry identities(Φ(x)+Φ(-x)=1), making an interval-probability function even, comparing two normal curves
- Normal combined with at-least-one / minimum-sample-sizecomplement of none-exceed
- Normal approximation to the binomial(De Moivre-Laplace) using σ-band probabilities
- Conditional probability and independenceP(A|B)=P(AB)/P(B); distinguishing P(A|B) from P(B|A)
- Recognizing distribution types(hypergeometric vs binomial; sampling with/without replacement); Weibull CDF evaluation
- Classical probability and counting8 named traps
- Descriptive statisticsmean, median, mode, variance10 named traps
- Combining groups, transformations, regression7 named traps
- Normal distribution9 named traps
- Conditional probability, independence, distribution types5 named traps
Sets & Inequalities
20 concepts · 29 named traps17% of the exam- Sets and elementsmembership (belongs-to / not-belongs-to)
- Number sets and notation (N, N*/N+, Z, Q, R)classifying real numbers
- Finite vs infinite setsthe empty set and {empty-set} distinction
- Set representationroster/enumeration, set-builder, interval notation (open/closed/half-open/infinite)
- Subset and proper subsetsubset count 2n and proper-subset count 2n-1; counting sets between a fixed subset and superset
- Set operationsintersection, union, complement relative to a universal set
- Parameters from set conditions(equality/complement/intersection) with element-distinctness constraints
- Properties of inequalitiessymmetry, transitivity, additive, multiplicative (sign-flip on negative multiply)
- Comparing algebraic expressions by the difference method
- Solving one-variable quadratic inequalities via discriminant/parabolaexpressing solution sets
- Rational/fractional and absolute-value inequalities
- Inequalities always true(solution set = R) and modeling inequalities from word problems
- Root-location / root-distribution of a quadratic relative to a valuesign-of-f(k), discriminant
- Range/count of a parameter from subset, intersection-empty, or integer-solution-count conditions
- Truth of quantified propositions (universal/existential) and negationssufficient vs necessary conditions
- Range of a linear combination from interval/sign constraints on variables
- Set operations and relationships8 named traps
- Parameters and counting from set conditions5 named traps
- Quadratic and rational/abs-value inequalities8 named traps
- Root location, parameters, propositions, conditions8 named traps
In every note
- Key formulasThe rules the concept runs on, set out like a formula sheet.
- Named trapsEvery wrong move the exam is built to catch, and how to get it right.
- PracticeOne button, straight into questions on it.
If the ground under a module is shaky, these outside courses rebuild it before the traps will make sense.
FoundationsAlgebra 2 foundationsPrecalculus bridgeStatistics and probability