Every student should have a strategic broad picture of the concepts and skills assessed in the Additional Mathematics syllabus. Such knowledge will enable the student to study smartly and to anticipate likely questions that may appear in any examination. The following checklist has been compiled with the aim to help the student learn more effectively and efficiently (minus all the hassles, I hope).

1. Trigonometry
- Prove identities.
- Solve equations (sometimes after proving an identity).
- Find min/max of an expression using the R-sine or R-cosine result.
- Find values of trigonometric expressions, given trigonometric ratios.
- Sketch trigonometric graphs (directly or after some transformations) and show appreciation of period and amplitude.
- Transformations include these forms: y = f(ax), y = f(x) + a, y = |f(x)| and combinations.
- Determine the number of roots of given equations, where their graphs need to be sketched and checked for the number of intersections.
- Find trigonometric ratios of specific angles (30, 45, 60 degrees) without using calculator.
- Know about radian and degree modes.

2. Surds and indices
- Form expressions using surds.
- Rationalise surds.
- Solve equations involving indices.

3. Logarithmic and exponential functions
- Solve logarithmic equations.
- Solve real-life problems involving exponential functions (e.g. radioactive decay of a substance).
- Simplify and evaluate, without using calculator, using laws of logarithm.
- Sketch logarithmic and exponential curves.

4. Modulus functions
- Sketch graphs of modulus functions.
- Solve equations and/or inequalities (by graphical approach) involving modulus functions.

5. Circles
- Find equations of circles and show appreciation of radius and centre.
- Find new equations given some transformation (e.g. reflection about x-axis).

6. Quadratic equations
- Appreciate the nature of roots (real, equal, imaginary) for graphs which are always above/below the x-axis, for graphs that do not intersect, for graphs that are tangent to each other and the like.
- Form new quadratic equations via sum and product of roots.
- Solve quadratic inequalities.

7. Binomial expansion
- Expand.
- Find terms independent of x or terms with certain powers.
- Approximate values by suitable substitutions.
- Know how to expand nCr where n is an unknown.

8. Linearisation of non-linear expressions
- Express a non-linear equation by manipulation to become Y = mX + c.
- Find slopes and intercepts in order to find unknowns in the original non-linear form.
- Find values from graphs.
- Interpretations of linear graphs.
- Draw another graph to find the number of roots of an equation.

9. Differentiation and its applications
- Differentiation techniques, especially the use of product rule, quotient rule and differentiation of composite functions like cos (3/x).

- Tangents and normals
(i) find gradients of tangents and normals;
(ii) find equations of tangents and normals;
(iii) find axial intercepts of tangents and normals.

- Rates of change via the chain rule.

- Maxima and minima
(i) form an expression based on the given information;
(ii) carry out differentiation, set the derivative to zero, find the stationary value(s);
(iii) determine the nature of stationary value via first or second derivative test.

- Curve sketching
(i) find axial intercepts.
(ii) carry out differentiation, set the derivative to zero, find the stationary values;
(iii) determine the nature of stationary points via first or second derivative test.

10. Integration and its applications
- Integration of standard functions.
- Integration via partial fractions.
- Find areas of regions via integration.
- Given known definite integrals, evaluate some other integrals.

11. Applications involving both differentiation and integration
- Kinematics (involving displacement, velocity, acceleration).
- Applying the derivative of a given expression to find the integral of a related one, e.g. differentiate x ln x, then find the definite integral of ln x.

12. Coordinate geometry
- Find gradients or equations of lines (parallel to or perpendicular to given lines).
- Find points of intersections.
- Find areas.
- Apply known geometrical properties of quadrilaterals.
- Find mid-points.
- Prove collinearity of points.

13. Remainder and factor theorems
- Factorise completely some expression and find its solutions.
- Form simultaneous equations based on given information and solve for unknowns.
- Use the fact that polynomial = quotient * divisor + remainder to solve problems.
- Carry out long division of polynomials.
- Three types of partial fractions (linear, repeated linear, non-factorisable quadratic factors).

14. Geometrical proofs
- Prove congruency and similarity.
- Prove results by applying mid-point and intercept theorems for triangles.
- Prove results by applying tangent-chord, intersecting-chord and tangent-secant theorems for circles.
- Know and apply angle properties and symmetric properties of circles.
- Know and apply the alternate segment theorem.

15. Simultaneous equations in two unknowns
- Solve both linear equations.
- Solve one linear, the other non-linear equations.
- Solve via substitution or inverse matrix method.