BGCSE Additional Mathematics 4037 study plan

Classify expressions into polynomials, rational functions, and composite functions, and state the domain and range of each from its algebraic form.

Sketch quadratic, cubic, and reciprocal graphs by hand, labelling intercepts, turning points, and asymptotes without a graphing tool.

Complete the square on ten quadratic expressions and use the completed form to read off vertex coordinates and maximum or minimum values.

Solve function composition and inverse-function problems, verifying each inverse by checking that f(f⁻¹(x)) = x.

Solve quadratic equations by factorising, completing the square, and the quadratic formula, choosing the most efficient method for each.

Determine the nature of roots of a quadratic from its discriminant and construct equations with prescribed root conditions.

Solve simultaneous equations where one is linear and one is quadratic, interpreting the geometric meaning of zero, one, or two solutions.

Solve linear and quadratic inequalities, expressing solution sets on a number line and in set notation.

Sketch modulus graphs and solve equations and inequalities involving modulus signs.

Simplify rational expressions and solve equations involving algebraic fractions, stating excluded values of the variable.

Convert freely between degrees and radians and use arc length and sector area formulas in multi-step problems.

Derive and apply the three Pythagorean identities to simplify trigonometric expressions and prove short identities.

Solve trigonometric equations over a given interval, generating all solutions from the general solution and from symmetry of the graphs.

Sketch and interpret the sine, cosine, and tangent graphs, including amplitude, period, and phase shifts for transformed forms.

Apply the sine and cosine rules and the area formula to solve non-right triangles, including the ambiguous case.

Calculate the gradient, length, and midpoint of a line segment and derive the equation of a line in several forms.

Find the intersection of a line and a curve algebraically and interpret the discriminant of the resulting quadratic.

Solve problems involving parallel and perpendicular lines and the distance from a point to a line.

Differentiate powers, sums, and constant multiples from first principles for at least two functions and state the rule in general form.

Apply the chain, product, and quotient rules to composite expressions and simplify derivatives fully.

Find equations of tangents and normals and locate stationary points, classifying each by the second derivative or a sign test.

Integrate standard powers and definite integrals, evaluating areas under curves and between a curve and a line.

Set up and solve a simple optimisation or rate-of-change problem from a written context.

Generate terms of arithmetic and geometric sequences and derive the nth term and sum formulas from first principles.

Solve problems involving sum to infinity and determine convergence conditions for a geometric series.

Expand binomial expressions with positive integer indices and find a specified term or coefficient without full expansion.

Convert between exponential and logarithmic forms and apply the laws of logarithms to simplify and solve equations.

Solve exponential equations by taking logarithms and sketch exponential and logarithmic graphs showing key features.

Perform vector addition, subtraction, and scalar multiplication in two dimensions and interpret results geometrically.

Use position vectors to prove collinearity and to find points dividing a segment in a given ratio.

Carry out matrix addition, multiplication, and inversion for 2×2 matrices and verify inverse products.

Solve 2×2 systems of linear equations using inverse matrices and interpret singular matrices geometrically.

Apply a transformation matrix to a shape and describe the transformation it represents.

Complete two full timed papers under exam conditions, allocating time across sections and showing all working.

Diagnose recurring error patterns from marked scripts and classify them as conceptual, procedural, or careless.

Re-teach the three weakest topics from earlier weeks using condensed notes and a short set of targeted questions.

Practise multi-step reasoning problems by writing out a plan before starting each solution.

Build a final revision checklist mapping every syllabus domain to a confidence rating and a specific action.