LearnMathematics
01≈33% of the exam · 34 concepts
Functions & Sequences
- Function definition; domain and range; equal functions (same domain and rule)→
- Determining the domain (denominator nonzero, even-root radicand ≥0, log argument >0, non-integer-power base >0; intersect conditions)→
- Finding the range; range-to-domain reverse problems (incl. substitution t=2x reducing to a quadratic)→
- Recovering f(x) from f(g(x)) by substitution / completing the square (with induced domain)→
- Recovering f(x) from a functional equation (e.g. f(x)+2f(1-x)=x2; Cauchy-type f(x1+x2)=f(x1)+f(x2)+1)→
- Monotonicity: increasing/decreasing, monotonic intervals, difference method, composite monotonicity→
- Parity/symmetry: even, odd, neither; domain-symmetry requirement; parity of shifted functions→
- Power functions: definition (coefficient 1), graphs/properties by exponent sign, passing-through-origin conditions, fit through a point→
- Exponential functions ax: graphs/properties by base, domain/range/asymptote; natural exponential e→
- Logarithmic functions loga x: inverse of exponential, graphs/properties; ln and lg→
- Logarithm rules: product, quotient, power, change-of-base, identities aloga N=N→
- Comparing log/exponential expressions by magnitude (base conversion, monotonicity, estimating ln near 1)→
- Piecewise functions: branch selection, nested evaluation, zeros/symmetry across branches→
- Inverse functions (e.g. inverse of a restricted quadratic with correct branch and domain)→
- Arithmetic sequences: general term an=a1+(n-1)d, sum Sn=n(a1+an)/2, sign/extremal-sum reasoning→
- Geometric sequences: general 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 techniques: grouping/pairing alternating signs, splitting into arithmetic+geometric parts, periodic (sine-defined) sequences→
- Derivative concept: average 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 derivatives: monotonicity from sign of f', extrema/local max-min, closed-interval max/min, no-critical-point parameter ranges→
- Derivative-based comparison via auxiliary functions (e.g. g(x)=f(x)/ex when f'(x)<f(x))→
- Second derivative / convexity-concavity conditions→
- Velocity/kinematics interpretation: average vs instantaneous velocity; position-time and distance-time graph reading→
- Modeling: compound-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)→
- Monotonicity and parity→
- Power, exponential, logarithmic functions→
- Piecewise and inverse functions→
- Arithmetic and geometric sequences→
- Recurrences, summation techniques, and an from Sn→
- Derivatives, tangents, and applications→
- Modeling (interest, decay, graphs)→
02≈29% of the exam · 36 concepts
Geometry & Algebra
- Straight line: slope from two points, undefined slope for vertical lines; inclination angle to slope→
- Line equation forms: point-slope, slope-intercept, two-point, intercept, general Ax+By+C=0→
- Parallel and perpendicular line conditions; reflection of a point across a line; slope range so a line meets a segment→
- Distance from a point to a line; minimum 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 distance; common chord length→
- Tangents from an external point; chord of contact through a fixed point→
- Ellipse: definition (sum of focal distances=2a), standard equation, a2=b2+c2, focal-triangle area with angle condition→
- Hyperbola: definition (|difference|=2a), standard equation, asymptotes y=±(b/a)x, eccentricity e>1, midpoint-chord (point-difference) method→
- Parabola: definition (focus/directrix), standard forms, focal distance to x-coordinate, weighted-distance minimization via directrix→
- Conic eccentricity problems: finding/ranging e from focal, asymptote, or inequality conditions→
- Trigonometry: radian-degree conversion; unit-circle sin/cos/tan; terminal side through a point; signs by quadrant→
- Trig identities: sin2+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 triangles: Sine 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/scaling; horizontal stretch/compression of trig graphs→
- Plane vectors: magnitude, coordinate representation, addition/subtraction/scalar multiplication, vectors in polygons (hexagon)→
- Vector parallel/collinearity conditions to solve for a parameter; vectors with points dividing sides→
- Dot product: definition, coordinate form, magnitude of a sum via law of cosines, angle between vectors, perpendicularity, projection vector→
- Complex numbers: i2=-1, algebraic form, real/imaginary parts, modulus, conjugate; reflection across real axis→
- Complex arithmetic: add/subtract/multiply/divide-by-conjugate; polar form and De Moivre powers; complex numbers as rotated vectors→
- Spatial rectangular coordinates: point location (axes/coordinate planes), 3D distance formula, symmetry across axis/plane/origin→
- Solid figures: surface area & volume of cuboid, cube, cylinder, cone, sphere; lateral vs total surface area→
- Cone/frustum: axial cross-section relations, sector-unrolling (central angle ↔ base radius), slant vs vertical height, volume→
- Inscribed/circumscribed spheres of prisms and tetrahedra (space diagonal, equal-volume inradius method)→
- Cross-sections of solids (which polygons can/can't appear; midpoint-defined sections)→
- 3D analysis in a cube: moving 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 ratio; solids of revolution (annulus/torus)→
- Lines, circles, and distance→
- Conic sections→
- Trigonometry→
- Vectors→
- Complex numbers→
- Spatial coordinates and symmetry→
- Solid geometry: solids, surfaces, volumes→
03≈21% of the exam · 22 concepts
Probability & Statistics
- 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 permutations; ordered vs unordered selection→
- Probability via combinatorial assignment/distribution (surjective '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 tendency: mean (incl. weighted/combined-group mean), median, mode; choosing the right statistic for a question→
- Dispersion: variance and standard deviation; effect of adding a constant or an outlier; trimming extremes before computing→
- Combining strata/groups: pooled mean and pooled variance (between + within), size-ratio weighting→
- Standard deviation under linear vs nonlinear transformations; variance scaling D(aX+b)=a2 D(X)→
- Linear regression: least-squares slope/intercept from sums; reading scatter/regression context→
- Normal distribution: bell curve, symmetry about μ, role of μ and σ, total area = 1→
- 68-95-99.7 empirical rule: interval 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-size (complement of none-exceed)→
- Normal approximation to the binomial (De Moivre-Laplace) using σ-band probabilities→
- Conditional probability and independence; P(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 counting→
- Descriptive statistics: mean, median, mode, variance→
- Combining groups, transformations, regression→
- Normal distribution→
- Conditional probability, independence, distribution types→
04≈17% of the exam · 20 concepts
Sets & Inequalities
- Sets and elements; membership (belongs-to / not-belongs-to)→
- Number sets and notation (N, N*/N+, Z, Q, R); classifying real numbers→
- Finite vs infinite sets; the empty set and {empty-set} distinction→
- Set representation: roster/enumeration, set-builder, interval notation (open/closed/half-open/infinite)→
- Subset and proper subset; subset count 2n and proper-subset count 2n-1; counting sets between a fixed subset and superset→
- Set operations: intersection, union, complement relative to a universal set→
- Parameters from set conditions (equality/complement/intersection) with element-distinctness constraints→
- Properties of inequalities: symmetry, transitivity, additive, multiplicative (sign-flip on negative multiply)→
- Comparing algebraic expressions by the difference method→
- Solving one-variable quadratic inequalities via discriminant/parabola; expressing 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 value (sign-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 negations; sufficient vs necessary conditions→
- Range of a linear combination from interval/sign constraints on variables→
- Set operations and relationships→
- Parameters and counting from set conditions→
- Quadratic and rational/abs-value inequalities→
- Root location, parameters, propositions, conditions→
