Maths Extension 2 syllabus, dot point by dot point

Sketch curves and regions in the complex plane defined by conditions on modulus and argument, such as circles, perpendicular bisectors, rays and half-planes

How do equations and inequalities in the complex variable describe curves and regions in the Argand plane?

Represent complex numbers in Cartesian and polar form, perform arithmetic, and interpret modulus, argument and conjugate geometrically on the Argand plane

How are complex numbers represented, added, multiplied and divided, and how does the Argand plane give them geometric meaning?

Use de Moivre's theorem to find powers and nth roots of complex numbers and to derive the roots of unity and their geometric arrangement

How does de Moivre's theorem let us compute powers and roots of complex numbers, and what is the geometry of the nth roots of unity?

Express complex numbers in modulus-argument form and use it to multiply and divide, interpreting these operations geometrically as scaling and rotation

How does the modulus-argument form of a complex number turn multiplication and division into operations on lengths and angles?

Solve quadratic equations with complex coefficients and factorise polynomials over the complex field, using the conjugate root theorem for real polynomials

How do quadratic and higher polynomial equations behave over the complex numbers, and why do real polynomials have complex roots in conjugate pairs?

Further Integration (MEX-C1)

Apply integration by parts to evaluate integrals of products, including repeated application and the recovery of the original integral

How does integration by parts reverse the product rule, and how do we choose the parts to evaluate products such as $x e^x$ or $x \ln x$ ?

Decompose rational functions into partial fractions and use the decomposition to integrate, including linear, repeated and irreducible quadratic factors

How does decomposing a rational function into partial fractions turn an awkward integral into a sum of standard logarithm and arctangent integrals?

Evaluate integrals using trigonometric substitution and the t = tan(x/2) substitution, including completing the square to reach standard inverse-trigonometric and logarithmic forms

How do trigonometric and rationalising substitutions transform integrals containing square roots or awkward rational functions into standard forms?

Integrate powers and products of trigonometric functions: powers of sin and cos (odd and even), powers of tan and sec, the integral of sec x, and products of sines and cosines via product-to-sum identities

How do we integrate powers and products of trigonometric functions by choosing between the Pythagorean-identity substitution, the double-angle reduction, and the product-to-sum identities?

Applications of Calculus to Mechanics (MEX-M1)

Model rectilinear motion under gravity with a resistive force proportional to velocity or to the square of velocity, and determine terminal velocity

How does a resistance proportional to speed change rectilinear motion, and what is the terminal velocity of a body falling under gravity with resistance?

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How do we use Newton's laws to turn the forces acting on a body, including weight, tension, normal and resistive forces, into its equation of motion?

Use Newton's laws to resolve forces and form the equation of motion: weight, tension, normal and resistive forces, equilibrium of concurrent forces, and the conical pendulum

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How does the defining equation of simple harmonic motion lead to its displacement, velocity and period, and how do we model an oscillating particle?

Derive and apply the equations of simple harmonic motion, relating acceleration, velocity, displacement, amplitude and period for an oscillating particle

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How do we analyse rectilinear motion when acceleration is given as a function of displacement or of velocity rather than of time?

Apply calculus to rectilinear motion where acceleration is expressed as a function of displacement or velocity, using the forms

a = v\,dv/dx

and

a = d(v^2/2)/dx

Analyse projectile motion using calculus, resolving into horizontal and vertical components, and extend to projectiles experiencing a resistance proportional to velocity

How is two-dimensional projectile motion analysed with calculus, and how does air resistance proportional to velocity change the horizontal and vertical motion?

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Proof (MEX-P1)

Prove inequalities using algebraic manipulation, the fact that squares are non-negative, and standard results such as the arithmetic mean-geometric mean inequality

How do we prove inequalities rigorously, and how do foundational results such as the arithmetic-geometric mean inequality follow from elementary algebra?

Use the language of proof, prove results by contradiction and contrapositive, and disprove statements by counterexample

How do rigorous logical arguments such as proof by contradiction and counterexample establish or refute mathematical statements?

Prove results involving sums, divisibility and inequalities for all integers using the principle of mathematical induction

How does mathematical induction prove statements about all positive integers, including inequalities, divisibility, and results needing more than one base case?

Further Work with Vectors (MEX-V1)

Represent three-dimensional vectors in component form, compute the scalar product and magnitude, and find vector equations of lines and the equation of a sphere

How are vectors in three dimensions described, how do we find lines and spheres, and how does the scalar product extend to space?

Prove geometric results using vectors, including properties of triangles, parallelograms and the diagonals of quadrilaterals, by expressing points as position vectors

How can vector methods prove classical geometric results such as concurrency, midpoint theorems and properties of parallelograms more cleanly than coordinate geometry?

Compute the projection of one vector onto another, distinguishing the scalar projection from the vector projection, and resolve a vector into parallel and perpendicular components

How do we resolve one vector along the direction of another, and what do the scalar and vector projections measure geometrically?