Maths Extension 1 flashcards

What this deck covers

• Calculus (ME-C1, C2, C3) (56 cards, 11 dot points)

  • Derivatives and integrals of inverse trigonometric functions (5)
  • Derivatives of functions defined parametrically: dy/dx = (dy/dt) / (dx/dt) by the chain rule, horizontal and vertical tangents, tangents and normals to parametric curves (new in the 2024 Extension 1 syllabus) (4)
  • Differentiating an inverse function: the rule dy/dx = 1/(dx/dy), gradients of an inverse at a point, and the second derivative of an inverse (6)
  • Exponential growth and decay: $\frac{dN}{dt} = k N$, $N = N_0 e^{kt}$, doubling and half-life (5)
  • Integrals giving inverse trig functions: $\arcsin$, $\arctan$ and the patterns to recognise (5)
  • Integration by substitution in HSC Maths Extension 1: choosing $u$, transforming the integral and changing limits (5)
  • Motion as a vector function: position, velocity and acceleration vectors, speed, and dot-product analysis of the motion (6)
  • Projectile motion: parametric equations, range, maximum height and time of flight (5)
  • Related rates of change: linking changing quantities via implicit differentiation (5)
  • Separable differential equations: separating variables, integrating both sides, and initial conditions (5)
  • Volumes of revolution: discs about the x-axis and y-axis (5)
• Combinatorics (ME-A1) (22 cards, 4 dot points)

  • Combinations: counting unordered selections with $\binom{n}{r}$ (5)
  • Permutations: counting ordered arrangements with the multiplication principle (7)
  • The binomial theorem and Pascal's triangle: expansion of $(a + b)^n$ and the general term (4)
  • The pigeonhole principle: guaranteed coincidences in counting problems (6)
• Functions (ME-F1, ME-F2) (21 cards, 5 dot points)

  • Graphing polynomials: leading-term behaviour, intercepts and root multiplicity (2)
  • Parametric equations: parameter elimination, sketches, and standard curves (5)
  • Polynomial and rational inequalities: sign analysis, critical points and excluded values (3)
  • Polynomial division and the remainder and factor theorems (6)
  • Roots and coefficients of polynomials: Vieta's formulas for cubics and quartics (5)
• Proof (ME-P1) (22 cards, 4 dot points)

  • Mathematical induction for divisibility: standard technique and algebraic restructuring (7)
  • Mathematical induction for general statements: recurrence relations and properties (4)
  • Mathematical induction for inequalities: the technique and the algebraic care (5)
  • Mathematical induction for series identities (6)
• Statistical Analysis (ME-S1) (27 cards, 6 dot points)

  • Bernoulli trials: definition, parameters, mean and variance (4)
  • Binomial probability calculations: exact values, cumulative probabilities and complements (5)
  • Normal approximation of the binomial distribution: continuity, validity and z-scores (4)
  • The binomial distribution: definition, probability mass function, mean and variance (4)
  • The sample proportion p-hat as a random variable: mean p, variance pq/n, and its normal approximation (5)
  • The sampling distribution of the mean and the central limit theorem: the mean and variance of a sample mean, and probabilities that it lies within given bounds (new in the 2024 Extension 1 syllabus) (5)
• Trigonometric Functions (ME-T1, T2, T3) (31 cards, 6 dot points)

  • Auxiliary angle: writing $a \sin x + b \cos x$ as $R \sin(x + \alpha)$ (4)
  • General solutions of trigonometric equations: $\sin$, $\cos$ and $\tan$ (5)
  • Inverse trigonometric functions: definitions, principal branches, domains, ranges and graphs (5)
  • Product-to-sum and sum-to-product identities for trigonometric expressions (7)
  • Sum and difference identities for sin, cos and tan: expansions, simplifications and exact values (5)
  • The t-formula: rational expressions for $\sin \theta$, $\cos \theta$ and $\tan \theta$ via $t = \tan(\theta/2)$ (5)
• Vectors (ME-V1) (36 cards, 6 dot points)

  • Geometric proofs with vectors: parallel, perpendicular, midpoint and ratio properties (7)
  • Parametric vector equations of lines: point and direction form, parameter elimination, intersections (6)
  • Relative velocity with a constant crosswind or current: adding velocity vectors, resultant speed and direction, the heading needed to hold a course, and bearings (new in the 2024 Extension 1 syllabus) (6)
  • The scalar (dot) product: component formula, geometric formula, angle between vectors and orthogonality (5)
  • Vector arithmetic: addition, scalar multiplication, magnitude and unit vectors (7)
  • Vector projection: scalar projection and vector projection of one vector onto another (5)
• Combinatorics (ME-A1) (43 cards, 7 dot points)

  • Circular arrangements: why n distinct objects around a circle give (n-1)!, the block method for groups and couples, alternating patterns, 'not together' via the complement, and necklaces/bracelets dividing by 2 for reflection (7)
  • Combinations: counting unordered selections with nCr, the 2^n subset count, the symmetry nCr = nC(n-r), and restriction techniques (at least, complement, conditional, geometry) (7)
  • Distinct arrangements of objects with repeats: the n! over r1! ... rk! formula and the binary two-type special case (6)
  • Factorial notation: definition, the convention 0! = 1, the recursive rule, and simplifying factorial fractions (6)
  • The multiplication principle and ordered selections: counting in stages, n^r with repetition, nPr without repetition, and restriction techniques (5)
  • Three counting principles for restricted arrangements: grouping into blocks, counting the complement, and splitting into cases with inclusion-exclusion (6)
  • Using counting in probability: P = favourable/total with both counts from permutations and combinations, plus complement, cases and inclusion-exclusion for committees, card hands, digit strings and replacement draws (6)
• Functions (ME-F1) (34 cards, 7 dot points)

  • Defining curves parametrically and eliminating the parameter (5)
  • Graphing y = |f(x)| and y = f(|x|) (3)
  • Inequations with absolute values and with the unknown in the denominator (6)
  • Inverse relations, the horizontal line test and inverse functions (3)
  • Sketching y = 1/f(x) from the graph of y = f(x) (5)
  • Sketching y = f(x) +/- g(x) by adding ordinates (5)
  • The sign of a function and sign tables (7)
• Year 11: Polynomials (43 cards, 7 dot points)

  • Consequences of the factor theorem: distinct zeroes give distinct factors, a degree-n polynomial has at most n zeroes, agreement at n + 1 points forces identity, a graph is determined by n + 1 points, intersection counts are bounded by degree via P(x) - Q(x), and re-expressing a polynomial in powers of (x - a) (7)
  • Division of polynomials and the division algorithm: quotient, remainder, missing terms and the identity P = DQ + R (7)
  • Geometry using polynomial techniques: tangency as a double root of P(x) minus Q(x), chord midpoints from the sum of roots, tangents from an external point, a line through a fixed point on a cubic meeting it twice more, and a common tangent to two curves, all without calculus (5)
  • Graphs of polynomials: end behaviour from the leading term and the multiplicity of each zero (5)
  • Sums and products of zeroes: relating coefficients to the elementary symmetric functions of the roots for quadratics, cubics and quartics, and using them to find missing zeroes, evaluate symmetric expressions, find unknown coefficients, and handle roots in arithmetic or geometric progression (6)
  • The language of polynomials: degree, leading term, monic, sums and products, and equating coefficients (6)
  • The remainder and factor theorems: the remainder on division by (x - a) is P(a), the factor test P(a) = 0, the integer-zero test, finding unknown coefficients and fully factoring a polynomial (7)

Sample cards from this deck

1. Definition

   Differentiating an inverse function: the rule dy/dx = 1/(dx/dy), gradients of an inverse at a point, and the second derivative of an inverse

   Define Reflection intuition

   The graph of $f^{-1}$ is the graph of $f$ reflected in the line $y = x$. Reflection in $y = x$ swaps the horizontal and vertical directions, so it swaps the rise and the run of any tangent line. A tangent of gradient $\frac{\text{rise}}{\text{run}}$ on $f$ becomes a tangent of gradient $\frac{\text{run}}{\text{rise}}$ on $f^{-1}$, that is, the reciprocal. This is why the rule has a reciprocal in it, and it is the picture to draw if a question asks you to explain geometrically.

2. Definition

   Combinations: counting unordered selections with $\binom{n}{r}$

   Define Boundary values

   $\binom{n}{0} = \binom{n}{n} = 1$ (one way to choose nothing, one way to choose everything), and $\binom{n}{1} = n$.

3. Definition

   Parametric equations: parameter elimination, sketches, and standard curves

   Define The domain of $t$ matters

   A restricted $t$ traces only part of the full Cartesian curve, so always state the range of $x$ (or $y$) the curve actually covers, and note the direction of travel as $t$ increases.

4. Definition

   Mathematical induction for divisibility: standard technique and algebraic restructuring

   Define Composite divisors

   For divisibility by a composite such as $6$ or $12$, you can either reach the factor directly, or, if the algebra resists, prove divisibility by the coprime factors separately (for $6$, by $2$ and by $3$) and combine. In HSC the direct factor is almost always reachable.

5. Definition

   The sampling distribution of the mean and the central limit theorem: the mean and variance of a sample mean, and probabilities that it lies within given bounds (new in the 2024 Extension 1 syllabus)

   Define Check the conditions

   Say that $n \geq 30$, so the central limit theorem applies.

6. Definition

   Product-to-sum and sum-to-product identities for trigonometric expressions

   Define Equation-solving calls for sum-to-product

   $\sin x + \sin 3x = 0$ is awkward as a sum, but becomes $2\sin 2x\cos x = 0$, which is zero exactly when $\sin 2x = 0$ or $\cos x = 0$, two easy equations.

7. Definition

   Geometric proofs with vectors: parallel, perpendicular, midpoint and ratio properties

   Define Equal and parallel

   Equal and parallel ($\overrightarrow{XY} = \overrightarrow{UV}$): same direction and length, the test for a parallelogram.

8. Definition

   Circular arrangements: why n distinct objects around a circle give (n-1)!, the block method for groups and couples, alternating patterns, 'not together' via the complement, and necklaces/bracelets dividing by 2 for reflection

   Define Alternating

   Alternating (equal numbers $n$ of two types): $(n-1)! \times n!$ (seat one type as a circle, slot the other into the gaps).

9. Definition

   Inequations with absolute values and with the unknown in the denominator

   Define Method 1, graph it

   Solve the equation $|ax + b| = k$ first, draw $y = |ax + b|$ (a V shape) and the horizontal line $y = k$ on one set of axes, and read the solution off the picture. Where the V is below the line solves $<$; where it is above solves $>$.

10. Definition

    Consequences of the factor theorem: distinct zeroes give distinct factors, a degree-n polynomial has at most n zeroes, agreement at n + 1 points forces identity, a graph is determined by n + 1 points, intersection counts are bounded by degree via P(x) - Q(x), and re-expressing a polynomial in powers of (x - a)

    Define List the candidates

    Write the divisors of the constant term (both signs); these are the only possible integer zeroes.