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AS Core Content |
A2 Core Content |
Further Pure Content |
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Indices and Surds |
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Rational indices (positive, negative and zero) Laws of indices Equivalence of Surd and Index notation Properties of Surds; rationalising denominators.
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Polynomials |
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Addition, subtraction, multiplication of polynomials; collecting like terms, expansion of brackets, simplifying. Completing the square; using this to find the vertex. The discriminant of a quadratic polynomial; using the discriminant to determine the number of real roots. Solution of quadratic equations, and linear and quadratic inequalities in one unknown. Solution of simultaneous equations, one linear and one quadratic. Solutions of equations in x which are quadratic in some function of x. |
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Using relationships between the roots of a quadratic/cubic and the coefficients. Using substitution to get equations with roots simply related to the roots of an original equation. |
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Coordinate Geometry and Graphs |
Polar Coordinates |
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Finding length, gradient and midpoint of a line segment given its endpoints Equations of straight lines (y=mx+c, y-y1=m(x-x1), ax+by+c=0 Gradients of parallel or perpendicular lines Equation of a circle with centre (a,b) and radius r: (x-a)2+(y-b)2=r2 Circle geometry: equation of a circle in expanded form x2+y2+2gx+2fy+c=0, angle in a semicircle is a right angle, perpendicular from centre to chord bisects the chord, radius is perpendicular to tangent. Solving equations using intersections of graphs, interpreting geometrically the algebraic solution of equations. Curve sketching: y=kxn, where n is an integer and k is a constant y=k?x where k is a constant y=ax2+bx+c where a, b and c are constants y=f(x), where f(x) is the product of at most 3 linear factors, not necessarily distinct
Transformations of graphs: Relationship between y=f(x) and y=af(x), y=f(x) + a, y=f(x+a), y=f(ax) where a is constant. |
Composition of transformations of graphs ? relationship between y=f(x) and y=af(x+b) The modulus function, the relationship between the graphs y=f(x) and y=|f(x)|
Parametric equations of curves; converting between parametric and cartesian forms |
Converting equations between Cartesian and polar form. Sketching simple polar curves. Finding the area of a sector using integration. |
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Differentiation and Integration |
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Gradient of a curve as the limit of gradients of a sequence of chords. Derivative and second derivative; notation f'(x) and f''(x), dy/dx, d2y/dx2
The derivative of xn where n is rational, together with constant multiples, sums, differences.
Gradients, tangents, normals, rates of change, increasing/decreasing functions, stationary points, classifying stationary points. Indefinite integration as the reverse process of differentiation. Integrating xn for rational n (n?-1) together with constant multiples, sums and differences. Definite integrals, constants of integration. Using integration to find the area of a region bounded by curves and lines. Estimating areas under curves using the Trapezium Rule.
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Derivative of ex and ln x, together with constant multiples, sums and differences. Chain rule, product rule, quotient rule. dx/dy as 1 ÷ dy/dx Integral of ex and 1/x together with constant multiples, sums and differences Integrating expressions involving a linear substitution. Volumes of revolution Brimful A Derivative of sin x, cos x and tan x together with constant multiples, sums and differences. Derivatives of functions defined parametrically. Integration of trigonometric functions (through the notion of "reverse differentiation) Integration of rational functions Integration of functions of the form y=kf'(x)/f(x) Integration by parts |
Derivatives of inverse trig functions, hyperbolic functions, inverse hyperbolic functions. Derivation of first few terms of Maclaurin series of simple functions. Integrals such as 1/?(a2-x2), 1/?(x2-a2), 1/( a2+x2), 1/?(x2+a2), using appropriate trigonometric or hyperbolic substitutions.
Reduction formulae to evaluate definite integrals
Using areas of rectangles to estimate or bound the area under a curve or to derive inequalities concerning sums. |
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Trigonometry |
Hyperbolic Functions |
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Sine and Cosine rules. Area formula for triangles A=½ab sinC Relationship between degrees and radians Arc length s=r?, Area of a sector A = ½r2? Graphs, periodicity and symmetry for sine, cosine and tangent functions Trigger W Identities tan ? = sin ?/cos ?, cos2? + sin2?=1 Exact values of sine, cosine and tangent of 30° , 45° , 60° Finding solutions of sin(kx)=c, cos(kx)=c, tan(kx)=c and equations which can be reduced to these forms within a specified interval.
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Inverse trigonomic relations sin-1, cos -1, tan-1, and their graphs on an appropriate domain. Properties of sec, cosec and cot.
Solving equations using: sec2 ? = 1+ tan2 ? cosec2 ? = 1 + cot2 ? expansions of sin(A+B), cos(A+B), tan(A+B) formulae for sin 2A, cos 2A, tan 2A
expression of a sin ? + b cos ? in the form Rsin(?+?) and Rcos(?+?)
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Definition of sinh, cosh, tanh, sech, cosech and coth in terms of ex. Graphs of simple hyperbolic functions.
cosh 2x ? sinh 2x = 1, sinh 2x = 2 sinh x cosh x, etc.
Expressing in terms of logarithms the inverse hyperbolic relations sinh-1x, cosh-1x, tanh-1x. |
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Sequences and Series |
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Definitions such as un=n2 or un+1=2un, and deducing simple properties from such definitions. ? notation Arithmetic and geometric progressions, finding the sum of an AP or GP, including the formula ½n(n+1) for the sum of the first n natural numbers. AP Train W Sum to infinity of a GP with |r|<1. Expansion of (a+b)n where n is a positive integer.
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Expansion of (1+x)n where n is a rational number and |x|<1 |
?r, ?r2, ?r3 and related sums. Summing finite series using the method of differences. Recognising when a series is convergent, finding the sum to infinity.
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Algebra and functions |
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Factor Theorem and Remainder Theorem. Algebraic division of polynomials by a linear polynomial. Sketching y=ax where a>0 Relationship between logarithms and indices. Laws of logarithms. Power Match C
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Simplifying rational functions. Algebraic division of polynomials by a linear or quadratic polynomial. Expressing rational functions using partial fractions. Inverting Rational Functions A Identifying domain and range. Composition of functions. One-one functions, finding inverses. Graphical illustration of the relation between a one-one function and its inverse. Exponential and logarithmic functions ex and ln x, and their graphs. Exponential growth and decay.
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Partial fractions with (x2+a2) in the denominator, and where the numerator is of higher degree than the denominator.
Determining asymptotic behaviour for rational functions.
Relationship between graphs of y=f(x) and y2=f(x)
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Numerical methods |
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Locating roots by graphical considerations or sign-change Simple iterative methods, xn+1=F(xn), relating such an iterative formula to the equation being solved. Numerical integration: Simpson's rule.
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Staircase and cobweb diagrams. Properties of successive errors in a converging iteration. Newton-Raphson method for finding roots. |
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Differential Equations |
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Forming differential equations from situations involving rate of change First order differential equations with separable variables: general form, and particular solutions from initial conditions. Interpreting solutions to differential equations within the context of a problem being modelled.
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Integrating factors for first order differential equations Reducing a first order differential equation to linear form or variable-separable using substitution. Complementary functions, particular integrals and general solutions of differential equations. Finding particular solutions using initial conditions, interpreting solutions in the context of a problem modelled by a differential equation. |
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Vectors |
Vectors and Matrices |
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Addition and subtraction of vectors, multiplication of a vector by a scalar, geometrical interpretation of these. Unit vectors, position vectors, displacement vectors Magnitude of a vector Scalar product of two vectors; determining the angle between two vectors Equation of a straight line in the form r = a + tb Angle between straight lines, point of intersection of straight lines, parallel or skew lines.
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Matrix addition, subtraction and multiplication. Singular and non-singular matrices, finding determinants and inverses. 2x2 matrices as transformations in the x-y plane. Solving linear simultaneous equations using matrices.
Equation of a line in the form (x-a)/p = (y-b)/q = (z-c)/r
Equation of a plane in the form ax + by + cz = d or (r ? a).n=0 or r = a + ?b + ?c
Vector product of two vectors Cross with the Scalar Product B
Determining whether a line is in a plane, parallel to a plane or intersects a plane, finding point of intersection. Line of intersection of two planes Perpendicular distance from point to plane or line Angle between two planes or a line and a plane Shortest distance between skew lines |
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Complex Numbers |
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Real and imaginary parts, modulus and argument, complex conjugate. Addition, subtraction, multiplication, division, square roots of complex numbers x + iy Conjugate pairs of roots of a polynomial Complex conjugates and addition/subtraction of complex numbers on an Argand diagram, loci of simple equations and inequalities.
Multiplication and division of complex numbers in polar form. de Moivre's theorem sin ? and cos ? in terms of ei? nth roots of unity.
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Proof by Induction |
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Establishing a given result using induction. Making conjectures based on some trial cases, then proving the conjectures using induction. |
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Groups |
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Definition of a group Establishing whether a structure is or is not a group Order of group, order of elements in a group. Subgroups Lagrange's theorem Cyclic groups Isomorphic groups Rose B
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