GCE A/L Combined Mathematics study plan

Derive and apply the factor and remainder theorems to factorise cubic and quartic polynomials with integer and complex roots.

Solve polynomial equations over the complex numbers, expressing all roots in the form a + bi and verifying conjugate-pair behaviour for real coefficients.

Convert complex numbers between Cartesian, polar and exponential forms, and compute products, quotients and integer powers using De Moivre's theorem.

Sketch loci such as |z - z0| = r, |z - a| = |z - b| and arg(z - z0) = theta on the Argand diagram and translate inequalities into shaded regions.

Solve linear and quadratic equations and inequalities involving modulus, expressing solution sets as intervals or unions of intervals.

Expand and simplify partial fractions for proper and improper rational expressions, including repeated linear and irreducible quadratic factors.

Prove identities and inequalities using the binomial theorem, summation notation and mathematical induction.

Apply the binomial theorem to find a specified term, coefficient or approximation in an expansion.

Differentiate products, quotients, composite functions and implicit relations, and state the domain where each derivative is valid.

Differentiate exponential, logarithmic, trigonometric and inverse trigonometric functions and combine them in chain-rule problems.

Find equations of tangents and normals, and locate stationary points and points of inflection using first and second derivative tests.

Solve connected-rates and small-increment problems by setting up a rate equation and differentiating with respect to time.

Integrate standard functions and apply substitution, integration by parts and partial fractions to evaluate indefinite integrals.

Evaluate definite integrals and use them to compute areas between curves, volumes of revolution and mean values.

Differentiate inverse trigonometric functions and integrate the resulting standard forms, including completing the square for quadratic denominators.

Set up and evaluate reduction formulae for integrals such as the integral of sin^n(x) or x^n e^x.

Prove and apply compound-angle, double-angle, sum-to-product and product-to-sum identities to simplify expressions and evaluate exact values.

Solve trigonometric equations over a stated interval, including equations reducible to quadratics in sin or cos and those requiring the R-form a cos x + b sin x.

Derive the standard equations of the circle, parabola, ellipse and rectangular hyperbola, and identify centre, foci, directrix, eccentricity and asymptotes.

Find tangents, normals and chords for conic sections, and solve problems involving intersection of a line with a conic using the discriminant.

Convert between Cartesian and parametric forms for lines, circles and conics, and find the equation of a locus from a geometric condition.

Perform vector addition, scalar multiplication, dot and cross products, and interpret each geometrically.

Find equations of lines and planes in vector form, and compute angles between lines, between planes and between a line and a plane.

Apply vectors to distance problems, including shortest distance from a point to a line or plane and between skew lines.

Resolve forces and apply equilibrium conditions to particles and rigid bodies, including problems with friction and limiting equilibrium on an inclined plane.

Apply Newton's second law to motion in a straight line and on an incline, including connected particles over a pulley and motion with variable acceleration.

Analyse projectiles by resolving initial velocity, finding time of flight, range and maximum height, and handling launch from a height.

Apply conservation of momentum and energy to collisions and impulsive tensions, distinguishing elastic and inelastic cases.

Compute probabilities using the addition and multiplication rules, conditional probability, independence and tree diagrams.

Apply the binomial and normal distributions, including continuity correction and standardisation, to word problems.

Compute mean and variance of discrete random variables and of raw and grouped data, and interpret standard deviation.

Re-solve the highest-error topics from earlier weeks and complete two full timed papers under exam conditions.