Grade 12 Mathematics

Algebra Equations Inequalities

• Solving a quadratic equation: "Solve for x", "Find the roots"
• Solving using the substitution (k-) method: An equation that is quadratic in some repeated expression, not in x directly
• Equations involving surds: A square root sign in the equation
• Simultaneous equations: "Solve simultaneously", two equations are given
• Inequalities in the context of functions: "Use the graph to determine...", "for which values of x is f(x) ≥ 0"
• Nature of roots, read from a function or graph: "For which value(s) of k does the graph touch the x-axis", "...does not intersect the x-axis", "...has two x-intercepts"

Patterns Sequences Series

• Arithmetic sequences and series: "Arithmetic sequence", "linear pattern", "Sn =", "sum of the first n terms"
• Geometric sequences and series: "Geometric sequence", "common ratio", "Sn =", "r ="
• Sigma notation: "Σ" (sigma), "Write in sigma notation", "Calculate the value of"
• Mixed sequence: identify the type first: "Determine whether ... is arithmetic or geometric", a sequence is given without a label

Finance Growth Decay

• Compound interest and decay (revision foundation): "Compound interest", "depreciated on a reducing balance", "A ="
• Future value annuity: "Regular deposits", "saving plan", "how much will be in the account", "Fv ="
• Present value annuity (loan repayments): "Monthly repayments", "loan", "bond", "how much is still owed", "Pv ="

Functions Graphs

• Determining the equation of a function from its graph: "Determine the equation", "Find a, p, q", "Write down the values of a and b"
• Transformations of functions: "Describe the transformation", "Write down the equation of g if g(x) = f(x − 2)", "Reflect"
• Comparing and combining graphs: "For which values of x is f(x) > g(x)", "For which values of x is f(x) ⋅ g(x) > 0", "Find the x-coordinates of the intersection"

Differential Calculus

• Finding the derivative from first principles: "From first principles", "using the definition of the derivative"
• Differentiation rules (power rule): "Determine f'(x)", "Find dy/dx", "Differentiate"
• Gradient of a tangent and equation of a tangent: "Determine the gradient of the tangent", "Find the equation of the tangent at", "At which point is the gradient equal to"
• Cubic graph: stationary points and sketching: "Sketch the graph of", "Find the coordinates of the turning points", "Determine the x-values of the stationary points"
• Applied calculus (optimisation and rate of change): "Maximum volume", "minimum cost", "rate of change", "At what time is the velocity zero"

Probability

• Venn diagrams and probability rules (revision and extension): "Draw a Venn diagram", "P(A and B)", "P(A or B)", "are the events independent"
• The fundamental counting principle: "How many ways", "In how many different ways can", "arrangements"
• Permutations: "Permutations", "arrangements where order matters", "P(n, r)" or "nPr"
• Combinations: "Combinations", "how many ways can ... be chosen", "selected", "C(n, r)" or "nCr"

Euclidean Geometry

• Cyclic quadrilateral theorems: "ABCD is a cyclic quadrilateral", "opposite angles", "exterior angle of a cyclic quad"
• Tangent-chord angle (tan-chord theorem): "Tangent at", "angle between the tangent and the chord", "alternate segment"
• Proving riders using circle theorems: "Prove that", "Show that", "Hence determine"

Analytical Geometry

• Equation of a circle: "Equation of the circle", "centre and radius", "(x − a)² + (y − b)² = r²"
• Equation of the tangent to a circle: "Tangent at the point", "equation of the tangent", "line that touches the circle"

Trigonometry

• Compound and double angle identities: "Prove", "Simplify", an expression containing sin(A ± B), cos(A ± B), sin 2A, cos 2A
• Solving trigonometric equations (general solutions): "Solve for x", "General solution", "for x ∈ [0° ; 360°]"
• Proving trigonometric identities: "Prove that", "Show that LHS = RHS"

Statistics

• Scatter plots and correlation: "Draw a scatter plot", "describe the correlation", "positive correlation", "r ="
• Regression line (line of best fit): "Least squares regression line", "ŷ = a + bx", "equation of the regression line"