Cambridge IAL Mathematics study plan

Derive and apply the laws of indices and surds to simplify algebraic expressions without a calculator

Complete the square on general quadratics and use the result to identify vertex, range and discriminant conditions

Sketch quadratic, cubic and reciprocal graphs stating intercepts, asymptotes and turning points from the equation alone

Solve simultaneous equations where one is linear and one is quadratic, interpreting the geometric meaning of the number of solutions

Apply the factor and remainder theorems to factorise cubics fully and find unknown coefficients

Perform polynomial long division and express improper algebraic fractions in quotient-plus-remainder form

Use the binomial expansion for positive integer and general rational powers, stating the range of validity

Manipulate logarithmic and exponential expressions, including solving equations that reduce to a quadratic in e^x or ln x

Differentiate from first principles and apply the chain, product and quotient rules fluently

Find and classify stationary points using the second derivative, including points of inflection

Differentiate implicit and parametric relations and construct tangent and normal equations

Set up and solve optimisation problems with a justified domain for the variable

Integrate standard functions and use substitution, including trigonometric and exponential integrands

Evaluate definite integrals and apply them to areas between curves and volumes of revolution about the x-axis

Resolve forces in two dimensions and solve equilibrium problems for a particle under concurrent forces

Model motion in a straight line using constant-acceleration formulae and interpret velocity-time graphs

Apply Newton's second law to particles on inclines, in lifts and connected by strings over pulleys

Analyse friction problems at the point of slipping using the limiting-equilibrium condition

Solve projectile problems by separating horizontal and vertical motion and choosing an appropriate origin

Use momentum conservation to solve direct collision and coalescence problems, including coefficient of restitution

Resolve and compose forces using vector notation and find resultant magnitudes and directions

Apply the work-energy principle and conservation of mechanical energy to motion on rough and smooth paths

Calculate moments and solve rigid-body equilibrium problems with distributed loads

Model motion under a variable force using calculus-based acceleration relationships

Summarise and compare data sets using measures of location and spread, including coding of variables

Calculate and interpret probabilities using Venn diagrams, tree diagrams and the addition and multiplication rules

Apply the binomial and geometric distributions, stating the conditions for each model

Standardise normally distributed variables and find probabilities and inverse normal values

Complete two full-length timed papers under exam conditions and mark them strictly

Rank your weakest six topics by marks lost per question attempted and re-teach each from first principles

Drill calculator technique and standard exam phrasing for show-that and hence questions

Build a final revision checklist mapping every syllabus topic to your confidence level