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Pure Mathematics
Pure Mathematics develops the algebraic, graphical, trigonometric, logarithmic, calculus and numerical methods that underpin AQA A-level Mathematics. Students must select methods carefully, use exact notation, show connected working and explain restrictions such as domains, intervals, convergence conditions and modelling assumptions. The topic links proof, functions, coordinate geometry, sequences, calculus and numerical methods so that exam answers show both technique and reasoning.
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Objectives
10
Flashcards
10
Questions
90 min
Study time
AqaA LevelMathematicsPaper 1
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Proof1 objectives
- A1 Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof including proof by deduction and proof by exhaustion; use disproof by counter example; use proof by contradiction including proof of the irrationality of sqrt(2), the infinity of primes and application to unfamiliar proofs.
Laws of indices1 objectives
- B1 Understand and use the laws of indices for all rational exponents.
Surds1 objectives
- B2 Use and manipulate surds, including rationalising the denominator.
Quadratic functions1 objectives
- B3 Work with quadratic functions and their graphs; use the discriminant including the conditions for real and repeated roots; complete the square; solve quadratic equations including solving quadratic equations in a function of the unknown.
Simultaneous equations1 objectives
- B4 Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation.
Inequalities1 objectives
- B5 Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions; express solutions through correct use of and and or, or through set notation; represent linear and quadratic inequalities graphically.
Polynomial and rational expression manipulation1 objectives
- B6 Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use the factor theorem; simplify rational expressions including by factorising and cancelling, and algebraic division by linear expressions only.
Graphs of functions1 objectives
- B7 Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, the modulus of a linear function, y = a/x and y = a/x^2 including their vertical and horizontal asymptotes; interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations; understand and use proportional relationships and their graphs.
Composite and inverse functions1 objectives
- B8 Understand and use composite functions, inverse functions and their graphs.
Transformations of function graphs1 objectives
- B9 Understand the effect of simple transformations on the graph of y = f(x), including sketching associated graphs y = af(x), y = f(x) + a, y = f(x + a), y = f(ax), and combinations of these transformations.
Partial fractions1 objectives
- B10 Decompose rational functions into partial fractions, with denominators not more complicated than squared linear terms and with no more than three terms, and numerators constant or linear.
Functions in modelling1 objectives
- B11 Use functions in modelling, including consideration of limitations and refinements of the models.
Straight line coordinate geometry1 objectives
- C1 Understand and use the equation of a straight line, including the forms y - y1 = m(x - x1) and ax + by + c = 0; use gradient conditions for two straight lines to be parallel or perpendicular; use straight line models in a variety of contexts.
Circle coordinate geometry1 objectives
- C2 Understand and use the coordinate geometry of the circle including the equation of a circle in the form (x - a)^2 + (y - b)^2 = r^2; complete the square to find the centre and radius; use circle properties including the angle in a semicircle, the perpendicular from the centre to a chord, and the radius perpendicular to a tangent.
Parametric equations1 objectives
- C3 Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms.
Parametric modelling1 objectives
- C4 Use parametric equations in modelling in a variety of contexts.
Binomial expansion1 objectives
- D1 Understand and use the binomial expansion of (a + bx)^n for positive integer n; use the notations n!, nCr and binomial coefficients; link to binomial probabilities; extend to any rational n including use for approximation; be aware that the expansion is valid for |bx/a| < 1, with proof not required.
Sequences1 objectives
- D2 Work with sequences including those given by a formula for the nth term and those generated by a simple relation of the form x_(n+1) = f(x_n); work with increasing, decreasing and periodic sequences.
Sigma notation1 objectives
- D3 Understand and use sigma notation for sums of series.
Arithmetic sequences and series1 objectives
- D4 Understand and work with arithmetic sequences and series, including the formulae for nth term and the sum to n terms.
Geometric sequences and series1 objectives
- D5 Understand and work with geometric sequences and series, including formulae for the nth term and the sum of a finite geometric series; use the sum to infinity of a convergent geometric series, including the use of |r| < 1 and modulus notation.
Sequences and series in modelling1 objectives
- D6 Use sequences and series in modelling.
Trigonometric definitions and radians1 objectives
- E1 Understand and use the definitions of sine, cosine and tangent for all arguments; use the sine and cosine rules; use the area of a triangle in the form 1/2 ab sin C; work with radian measure, including use for arc length and area of sector.
Small angle approximations1 objectives
- E2 Understand and use the standard small angle approximations of sine, cosine and tangent, including sin theta approximately theta, cos theta approximately 1 - theta^2/2, and tan theta approximately theta where theta is in radians.
Trigonometric functions and exact values1 objectives
- E3 Understand and use the sine, cosine and tangent functions, their graphs, symmetries and periodicity; know and use exact values of sin and cos for 0, pi/6, pi/4, pi/3, pi/2, pi and multiples thereof, and exact values of tan for 0, pi/6, pi/4, pi/3, pi and multiples thereof.
Reciprocal and inverse trigonometric functions1 objectives
- E4 Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; understand their relationships to sine, cosine and tangent, their graphs, ranges and domains.
Trigonometric identities1 objectives
- E5 Understand and use tan theta = sin theta / cos theta; understand and use sin^2 theta + cos^2 theta = 1, sec^2 theta = 1 + tan^2 theta, and cosec^2 theta = 1 + cot^2 theta.
Compound and double angle formulae1 objectives
- E6 Understand and use double angle formulae; use formulae for sin(A +/- B), cos(A +/- B) and tan(A +/- B); understand geometrical proofs of these formulae; understand and use expressions for a cos theta + b sin theta in equivalent forms r cos(theta +/- alpha) or r sin(theta +/- alpha).
Trigonometric equations1 objectives
- E7 Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle.
Trigonometric proofs1 objectives
- E8 Construct proofs involving trigonometric functions and identities.
Trigonometry in context1 objectives
- E9 Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.
Exponential functions1 objectives
- F1 Know and use the function a^x and its graph, where a is positive; know and use the function e^x and its graph.
Exponential gradient and modelling1 objectives
- F2 Know that the gradient of e^(kx) is equal to ke^(kx) and hence understand why the exponential model is suitable in many applications.
Logarithmic functions1 objectives
- F3 Know and use the definition of log_a x as the inverse of a^x, where a is positive and x >= 0; know and use the function ln x and its graph; know and use ln x as the inverse function of e^x.
Laws of logarithms1 objectives
- F4 Understand and use the laws of logarithms, including log_a x + log_a y = log_a(xy), log_a x - log_a y = log_a(x/y), and k log_a x = log_a(x^k), including cases such as k = -1 and k = -1/2.
Exponential equations1 objectives
- F5 Solve equations of the form a^x = b.
Logarithmic graphs1 objectives
- F6 Use logarithmic graphs to estimate parameters in relationships of the form y = ax^n and y = kb^x, given data for x and y.
Exponential growth and decay1 objectives
- F7 Understand and use exponential growth and decay; use exponential models in contexts such as continuous compound interest, radioactive decay, drug concentration decay and population growth; consider limitations and refinements of exponential models.
Derivative concept1 objectives
- G1 Understand and use the derivative of f(x) as the gradient of the tangent to y = f(x) at a general point; understand the gradient of the tangent as a limit and as a rate of change; sketch the gradient function for a given curve; understand second derivatives; differentiate from first principles for small positive integer powers of x and for sin x and cos x; understand and use the second derivative as the rate of change of gradient, including links to convex and concave sections and points of inflection.
Differentiation techniques1 objectives
- G2 Differentiate x^n for rational values of n and related constant multiples, sums and differences; differentiate e^(kx), a^(kx), sin kx, cos kx, tan kx and related sums, differences and constant multiples; understand and use the derivative of ln x.
Applications of differentiation1 objectives
- G3 Apply differentiation to find gradients, tangents and normals, maxima and minima, stationary points and points of inflection; identify where functions are increasing or decreasing.
Product, quotient and chain rules1 objectives
- G4 Differentiate using the product rule, quotient rule and chain rule, including problems involving connected rates of change and inverse functions.
Implicit and parametric differentiation1 objectives
- G5 Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only.
Differential equation construction1 objectives
- G6 Construct simple differential equations in pure mathematics and in context, including contexts such as kinematics, population growth and modelling the relationship between price and demand.
Fundamental Theorem of Calculus1 objectives
- H1 Know and use the Fundamental Theorem of Calculus.
Integration techniques1 objectives
- H2 Integrate x^n excluding n = -1 and related sums, differences and constant multiples; integrate e^(kx), 1/x, sin kx, cos kx and related sums, differences and constant multiples.
Definite integration and areas1 objectives
- H3 Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves.
Integration as limit of a sum1 objectives
- H4 Understand and use integration as the limit of a sum.
Integration by substitution and parts1 objectives
- H5 Carry out simple cases of integration by substitution and integration by parts; understand these methods as inverse processes of the chain and product rules respectively, with substitution limited to cases where one substitution leads to an integrable function and integration by parts excluding reduction formulae.
Integration using partial fractions1 objectives
- H6 Integrate using partial fractions that are linear in the denominator.
First order differential equations1 objectives
- H7 Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions, where separation of variables may require factorisation involving a common factor.
Differential equation interpretation1 objectives
- H8 Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution and links to kinematics.
Root location1 objectives
- I1 Locate roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x on which f(x) is sufficiently well behaved; understand how change of sign methods can fail.
Iterative methods and Newton-Raphson1 objectives
- I2 Solve equations approximately using simple iterative methods; draw associated cobweb and staircase diagrams; solve equations using the Newton-Raphson method and other recurrence relations of the form x_(n+1) = g(x_n); understand how such methods can fail.
Numerical integration1 objectives
- I3 Understand and use numerical integration of functions, including the trapezium rule and estimating the approximate area under a curve and limits that it must lie between.
Numerical methods in context1 objectives
- I4 Use numerical methods to solve problems in context.
Key terms
deductionindicessurdsquadraticsimultaneous equationsinequalitiespolynomialsfunctionscomposite functionsunderstandrational functionsstraight line
Exam tips
- Proof exam tip 1: Write the method before the answer so the examiner can follow each step. Apply this to a1 Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof including proof by deduction and proof by exhaustion; use disproof by counter example; use proof by contradiction including proof of the irrationality of sqrt(2), the infinity of primes and application to unfamiliar proofs..
- Laws of indices exam tip 1: Write the method before the answer so the examiner can follow each step. Apply this to b1 Understand and use the laws of indices for all rational exponents..
Common mistakes
- Proof common mistake 1: Show the method first, then give the final answer in the required form. Apply this directly to Proof.
- Laws of indices common mistake 1: Show the method first, then give the final answer in the required form. Apply this directly to Laws of indices.
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