Understand the meaning of |x|, sketch the graph of y = |ax + b| and use relations such as |a| = |b| ⇔ a2 = b2 and |x - a| < b ⇔ a - b < x < a + b when solving equations and inequalities [graphs of y = |f(x)| and y = f(|x|) for non-linear functions f are not included].
Divide a polynomial, of degree not exceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero).
Use the factor theorem and the remainder theorem.
Express rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than:
(ax + b)(cx + d)(ex + f)
(ax + b)(cx + d)2
(ax + b)(cx2 + d)
Excluding cases where the degree of the numerator exceeds that of the denominator.
Use the expansion of (1 + x)n, where n is a rational number and |x| < 1 [finding the general term in an expansion is not included].
Logarithmic and Exponential Functions
Understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base).
Use logarithms to solve equations and inequalities in which the unknown appears in indices.
Use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept.
Understand the relationship between ex and ln x, and their graphs [include graph of y = ekx for both positive and negative values of k].
Trigonometry
Understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude.
Use trigonometrical identities to simplify expressions and solve equations:
expansions of sin(A ± B), cos(A ± B) and tan(A ± B)
formulae for sin2A, cos2A and tan2A
expression of asinθ + bcosθ in the forms Rsin(θ ± \(\alpha\)) and Rcos(θ ± \(\alpha\))
Differentiation
Use the derivatives of ex, ln x, sin x, cos x, tan x, together with constant multiples, sums, differences and composites.
Differentiate products and quotients.
Use first derivative to solve problems involving tangents and normals.
Integration
Integration of the form eax + b, \(\frac{1}{ax + b}\), sin(ax + b), cos(ax + b), sec2(ax + b) and \(\frac{1}{x^2 + a^2}\).
Integrate rational functions by means of decomposition into partial fractions.
Integrate functions of the form \(\frac{kf'(x)}{f(x)}\).
Use integration by parts when an integrand can be regarded as a product.
Use a given substitution to simplify and evaluate an integral.
Numerical Solution of Equations
Find root of an equation, by means of graphs and/or searching for a sign change [e.g. finding a pair of consecutive integers between which a root lies].
Use the notation for a sequence of approximations which converges to a root of an equation.
Use a given iteration form xn + 1 = F(xn), or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy.
Vectors
Carry out addition and subtraction of vectors and multiplication of a vector by a scalar.
Calculate the magnitude of a vector, and use unit vectors, displacement vectors and position vectors [in 2 or 3 dimensions].
Equation of a straight line is expressed in the form r = a + tb [finding the equation of a line given the position vector of a point on the line and a direction vector, or the position vectors of two points on the line].
Determine whether two lines are parallel, intersect or are skew, and find the point of intersection of two lines when it exists.
Use formulae to calculate the scalar product of two vectors, and use scalar products in problems involving lines and points [e.g. finding the angle between two lines, and finding the foot of the perpendicular from a point to a line; questions may involve 3D objects such as cuboids, tetrahedra (pyramids), etc.].
Differential Equations
Formulate a simple statement involving a rate of change as a differential equation [use constant of proportionality, where necessary].
Find by integration a general form of solution for a first order differential equation in which the variables are separable.
Interpret the solution of a differential equation in the context of a problem being modelled by the equation.
Complex Numbers
Understand the terms real part, imaginary part, modulus, argument, conjugate, and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal [Notations Re z, Im z, |z|, arg z, z* should be known. The argument of a complex number will usually refer to an angle θ such that −π < θ ≤ π, but in some cases the interval 0 ≤ θ ≤ 2π may be more convenient].
Carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in Cartesian form x + iy.
Use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs.
Represent complex numbers geometrically by means of an Argand diagram.
Carry out operations of multiplication and division of two complex numbers expressed in polar form r(cosθ + i sinθ) or reiθ [including the results |z1z2| = |z1||z2| and arg(z1z2) = arg(z1) + arg(z2), and corresponding results for division].
Find the two square roots of a complex number.
Understand the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers.
Illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram [e.g. |z - a| < k, |z - a| = |z - b|, arg(z - a) = \(\alpha\)].
