Additional Mathematics | 0606

Functions - Understand the terms function, domain, range (image set), one–one function, many–one function, inverse function and composition of functions Functions - Find the domain and range of functions, including inverse and composite functions Functions - Recognise and use function notation such as f(x), f⁻¹(x), fg(x) and f²(x) Functions - Understand the relationship between y = f(x) and y = |f(x)| for linear, quadratic, cubic or trigonometric functions Functions - Explain in words why a given function does not have an inverse Functions - Find the inverse of a one–one function using correct notation Functions - Form and use composite functions, understanding that the order of composition is important Functions - Use sketch graphs to show the relationship between a function and its inverse as reflections in the line y = x Quadratic functions - Find the maximum or minimum value of a quadratic function by completing the square or by differentiation Quadratic functions - Use the maximum or minimum value of a quadratic function to sketch its graph or to determine the range for a given domain Quadratic functions - Know the conditions for a quadratic equation to have two real roots, two equal roots or no real roots and the related conditions for a line to intersect, be tangent to or not intersect a curve Quadratic functions - Solve quadratic equations for real roots using factorisation, the quadratic formula or completing the square Quadratic functions - Find the solution set for quadratic inequalities either graphically or algebraically, expressing solutions using correct inequality notation Factors of polynomials - Know and use the remainder and factor theorems Factors of polynomials - Find factors of polynomials, including factorising a cubic into a linear factor and a quadratic factor Factors of polynomials - Solve cubic equations using factorisation or other appropriate methods Equations, inequalities and graphs - Solve equations involving moduli such as |ax + b| = c, |ax + b| = cx + d, |ax + b| = |cx + d| and |ax² + bx + c| = d using algebraic or graphical methods Equations, inequalities and graphs - Solve graphically or algebraically inequalities involving moduli, including forms such as k|ax + b| > c, k|ax + b| ≤ c, k|ax + b| ≤ |cx + d|, |ax + b| ≤ cx + d and |ax² + bx + c| > or ≤ d Equations, inequalities and graphs - Use substitution to form and solve a quadratic equation in order to solve a related equation Equations, inequalities and graphs - Sketch graphs of cubic polynomials and their moduli when given as a product of three linear factors, clearly showing intercepts Equations, inequalities and graphs - Solve graphically cubic inequalities of the form f(x) ≥ d, f(x) > d, f(x) ≤ d or f(x) < d where f(x) is a product of three linear factors Simultaneous equations - Solve simultaneous equations in two unknowns by elimination or substitution, including equations that are linear and non-linear Logarithmic and exponential functions - Know and use simple properties and graphs of logarithmic and exponential functions, including ln x and e^x, and understand their inverse relationship and asymptotes Logarithmic and exponential functions - Know and use the laws of logarithms, including change of base, and express combinations of logarithms as a single logarithm Logarithmic and exponential functions - Solve equations of the form a^x = b using logarithms or other appropriate methods Straight-line graphs - Use the equation of a straight line to solve problems involving gradients and intercepts Straight-line graphs - Know and use the condition for two lines to be parallel or perpendicular in terms of their gradients Straight-line graphs - Solve problems involving midpoint and length of a line segment and find and use the equation of a perpendicular bisector Straight-line graphs - Transform given relationships to and from straight-line form and determine unknown constants by calculating gradient or intercept of the transformed graph Coordinate geometry of the circle - Know and use the equation of a circle with radius r and centre (a, b) and identify the centre and radius from a circle equation in various forms Coordinate geometry of the circle - Solve problems involving the intersection of a circle and a straight line, including deciding whether the line is a tangent, a chord or does not meet the circle Coordinate geometry of the circle - Solve problems involving tangents to a circle, including finding equations of tangents Coordinate geometry of the circle - Solve problems involving the intersection of two circles, including finding points of intersection, the equation of a common chord and deciding whether two circles intersect, touch or do not meet Circular measure - Solve problems involving arc length and sector area of a circle using radian measure, including compound shapes Trigonometry - Know and use the six trigonometric functions of angles of any magnitude: sine, cosine, tangent, secant, cosecant and cotangent Trigonometry - Understand and use the amplitude and period of a trigonometric function and the relationships between graphs of related trigonometric functions Trigonometry - Draw and use graphs of y = a sin(bx) + c, y = a cos(bx) + c and y = a tan(bx) + c over a given domain, including identification of asymptotes for tangent graphs Trigonometry - Use the identities sin²A + cos²A = 1, sec²A = 1 + tan²A and cosec²A = 1 + cot²A Trigonometry - Solve, for a given domain, trigonometric equations involving any of the six trigonometric functions, possibly using standard identities Trigonometry - Prove trigonometric relationships involving the six trigonometric functions using standard identities Permutations and combinations - Recognise the difference between permutations and combinations and know when each should be used Permutations and combinations - Know and use the notation n! and the standard expressions for permutations and combinations of n items taken r at a time Permutations and combinations - Solve arrangement and selection problems in context using permutations or combinations Series - Use the binomial theorem for the expansion of (a + b)^n for positive integer n, including simplification of coefficients Series - Use the general term in the binomial expansion to find a specified term or to solve related problems Series - Recognise arithmetic and geometric progressions and understand the difference between them Series - Use the formulas for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions Series - Use the condition for convergence of a geometric progression and the formula for the sum to infinity of a convergent geometric progression Vectors in two dimensions - Understand and use vector notation in various forms, including column vectors and directed line segments Vectors in two dimensions - Know and use position vectors and unit vectors, including forming a unit vector parallel to a given vector Vectors in two dimensions - Find the magnitude of a vector and add, subtract and multiply vectors by scalars, including solving vector geometry problems Vectors in two dimensions - Compose and resolve velocities, determine resultant vectors and use velocity vectors to solve problems in context such as motion and collisions Calculus - Understand the idea of a derived function as the rate of change and the link with gradients of curves Calculus - Use standard differentiation notation including first and second derivatives Calculus - Know and use derivatives of standard functions x^n for rational n, sin x, cos x, tan x, e^x and ln x, including constant multiples, sums and simple composite functions using the chain rule Calculus - Differentiate products and quotients of functions using the product and quotient rules Calculus - Use differentiation to find gradients, tangents and normals to curves Calculus - Use differentiation to find stationary points of functions Calculus - Apply differentiation to connected rates of change, small increments and approximations Calculus - Apply differentiation to practical problems involving maxima and minima in context Calculus - Use first and second derivative tests to distinguish between maxima and minima and justify conclusions Calculus - Understand integration as the reverse process of differentiation and include an arbitrary constant in indefinite integrals Calculus - Integrate sums of terms in powers of x, including x⁻¹ and 1/(ax + b), including an arbitrary constant Calculus - Integrate functions of the form (ax + b)^n, sin(ax + b), cos(ax + b), sec²(ax + b) and e^(ax + b), including the case n = –1 Calculus - Evaluate definite integrals and apply integration to find plane areas between curves and lines Calculus - Apply differentiation and integration to kinematics of a particle moving in a straight line, relating displacement, velocity and acceleration for constant or variable acceleration Calculus - Use displacement–time, distance–time, velocity–time, speed–time and acceleration–time graphs to interpret and solve kinematics problems