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Mathematical Methods
Solving equations
Chapter 1
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RESTS ON
Recognise whether an equation is linear from the power on its unknowns
FOUNDATIONS
Signed numbers
CH 3
Substitute known values into a formula and evaluate it correctly
CH 4
Determine the sign of a quadratic on intervals separated by its roots, accounting for concavity and even or odd root multiplicity
CH 2
State a polynomial's degree, leading coefficient, constant term, monic status and coefficient set after arranging its terms by descending power
CH 6
Solve a one-step linear equation
CH 7
Remove the greatest common numerical and algebraic factor from every term before considering any further factorisation
CH 6
Factor numerators and denominators, cancel common factors rather than terms, retain the original excluded values, and present the remaining numerator and denominator in a consistent form
CH 3
Generate and recognise angles with the same terminal point by adding integer multiples of one full revolution in the active angle unit
FOUNDATIONS
Multiplication and division
CH 9
Identify the domain shown, range, equilibrium, amplitude, period, intercepts and turning points of the basic sine and cosine functions, and relate their horizontal offset to complementary values
CH 2
Read the point of intersection of two graphed lines
CH 2
Evaluate sine, cosine or tangent at a stated degree or radian argument to a stated decimal precision, with the calculator mode matching the written angle unit
FOUNDATIONS
Factor a simple quadratic with integer coefficients by finding factor data whose product and sum reproduce the constant and linear coefficients
CH 3
Solve a one-variable linear inequation, reversing the order relation exactly when multiplication or division by a negative value occurs
CH 3
Identify the intercept and slope in y = a + bx, including its edge shapes
CH 2
Use equal gradients for parallel oblique lines and product negative one for perpendicular oblique lines, with explicit horizontal–vertical edge cases
CH 5
Compute a slope from two coordinate pairs with the slope formula
CH 2
Solve equations with a scaled bracket by expanding or dividing first
FOUNDATIONS
Write a line's equation from its y-intercept and slope
CH 5
Polynomial equations and factorisation
Polynomial identities
Circular functions and exact values
Solving circular equations
Literal and simultaneous equations
Parameterised linear systems
Recognise a quadratic expression or equation by a nonzero coefficient of the squared term and no higher power after simplification
Expand two linear factors and factorise a simple quadratic with rational coefficients, preserving the leading coefficient and middle term
Connect roots of an equation, zeros of its polynomial, linear factors and horizontal-axis intercepts, including a repeated root
Rearrange to zero, factorise, apply the null factor law and solve the resulting linear equations
Substitute exact coefficients into the quadratic formula, keep the whole denominator 2a, take both sign branches and simplify
Represent irrational real roots in a canonical exact surd form rather than prematurely converting them to decimals
Identify a, b, c in zero form and calculate b squared minus 4ac with signs intact
Identify a repeated subexpression, replace it with a temporary variable, solve the resulting quadratic and substitute back
Reject values excluded by the substitution domain or introduced by a non-equivalent operation such as squaring, by checking the original equation exactly
Express a discriminant in a parameter, impose the sign condition matching the required number of roots or intersections, and solve the resulting parameter equation or inequality
Solve a quadratic inequality and report the correct between/outside/all/none interval pattern with endpoints included only for a non-strict sign
Divide one polynomial by a lower-degree polynomial, maintaining descending-power columns and zero placeholders, to obtain an exact quotient and a remainder of lower degree
Verify and use dividend = divisor × quotient + remainder, including inspection cases
Find the remainder on division by a linear divisor by evaluating the polynomial at the divisor's zero
Connect zero remainder, P(a)=0, the linear factor corresponding to a, and the zero a, in both directions
Evaluate P(a) by substituting an integer, exact rational or simple exact irrational value into every occurrence of the variable
Add or subtract polynomials by aligning equal powers and combining their coefficients, including zero coefficients for missing powers
Multiply polynomials using complete distribution and exponent laws, then collect equal powers
Completely factorise a low-degree polynomial by taking out common factors, using familiar quadratic patterns, or finding one linear factor and dividing to obtain a lower-degree factor
Rearrange a polynomial equation to zero, factorise it, and use the null factor law to obtain all real solutions
Solve two-step linear equations by inverse operations in reverse order
Simplify a fraction
Recognise that an identity holds for every input and determine unknown parameters by expanding and equating coefficients of like powers
Solve a pair of simultaneous linear equations for the shared (x, y)
Use the terminal point (x, y) on the unit circle to obtain cos θ = x and sin θ = y, or recover the coordinate pair from known sine and cosine values
Obtain tan θ = y/x from a unit-circle point and recognise that tangent is undefined when x = 0
State and apply the maximal domains and ranges of the sine, cosine and tangent functions, including tangent's excluded odd half-turn boundaries
Determine which of sine, cosine and tangent are positive, negative, zero or undefined in each quadrant or on a boundary
Reduce an angle to its first-quadrant reference angle measured from the horizontal axis
Obtain exact sine, cosine and tangent values for the standard first-quadrant angles, their symmetric points and axis boundaries
Apply reflection, parity and periodicity relations to rewrite a circular-function value in terms of a base angle and the correct sign
Find all angles in a stated finite domain whose sine, cosine or tangent has a given exact value, using reference angles and periodicity
Use sin²(x)+cos²(x)=1 and tan(x)=sin(x)/cos(x) to replace, simplify or verify an expression on its valid domain
Obtain the other two exact trigonometric ratios from one known ratio and a quadrant or interval, choosing the square-root signs from the stated domain
Apply complementary sine–cosine relations together with quadrant symmetry to rewrite values involving an odd quarter-turn offset with the correct function and sign
Reduce an extended positive or negative rotation and identify its quadrant or axis boundary
Rounding
Multiply and divide fractions
Identify the hypotenuse and compute it from the two shorter sides
Connect intersections of a circular graph and an algebraic graph with solutions, and identify a root count or sign-change bracket from supplied graphical or tabular evidence
Apply a fixed number of bisection iterations to a continuous circular–algebraic difference function, retaining the half-interval whose endpoint values have opposite signs
Isolate sine, cosine or tangent, determine admissible signs and boundary cases, and find every exact or approximate solution in a stated finite domain
Use exact algebra to reduce a relation involving one circular function, a squared circular function, or a sine–cosine ratio to one or more basic value equations, rejecting auxiliary values outside the function's range
Transform the stated domain with an affine circular-function argument, solve for the inner argument, and map every solution back to the original variable without losing endpoints or periods
Express the complete infinite solution family of a basic circular equation for sine, cosine or tangent in terms of an integer parameter, using the correct period and symmetric positions for the function and the applicable base-angle convention for a negative target
Solve a general circular equation whose argument is an affine function of the variable, writing the family for the inner argument first and then dividing or translating at the very end to recover the variable's exact family
Generate consecutive integer-parameter values of a general solution family and retain exactly those solutions that fall inside a stated finite interval, honouring endpoint inclusion and avoiding double-counting of boundary solutions
Isolate a nominated subject in a linear literal equation by applying equivalent operations to the whole of both sides and simplifying only valid common factors
Recognise equivalent point–gradient, gradient–intercept and primitive general forms of the same line, including horizontal and vertical edge cases
Rearrange a linear equation into y = a + bx by inverse operations
Recognise from equal slopes that a system has no solution or every point
Solve a non-singular system of three or four linear equations by exact elimination or substitution, preserving variable order and checking every original equation
Determine a parameter value that makes a point incident, lines parallel or perpendicular, a system non-unique, or several lines concurrent
Determine exactly whether a point lies on, above or below a non-vertical line, or to the left or right of a vertical line
Verify whether three or more points are collinear by equal gradients sharing a common point or by exact substitution into one line equation
Find the exact intersection of two lines and verify whether it lies on every remaining line
Solve a pair of simultaneous linear equations whose coefficients are symbolic parameters, by exact elimination or substitution, expressing each solution as one simplified rational expression in the parameters and stating the nonzero-denominator conditions
Solve equations with unknowns on both sides by collecting them onto one side
Check a claimed solution or point by substituting into the original equation
Find the equation of the line through two points
Determine the full solution classification of a two-equation linear system containing one or more parameters, by comparing exact gradient and intercept or coefficient-ratio conditions and naming the geometric relation as unique, parallel or coincident
Express the infinite solution set of a consistent under-determined linear system by choosing one free variable as a real parameter and giving every other variable as an exact affine function of it, then interpret the set geometrically