Understand the terms: function, domain, range (image set), one-one function, inverse function and composition of functions
Use the notation f(x) = sin x, f:x ↦ lg x, (x > 0), f-1(x) and f2(x) [=f(f(x))]
Understand the relationship between y = f(x) and y = |f(x)|, where f(x) may be linear, quadratic or trigonometric
Explain in words why a given function is a function or why it does not have an inverse
Find the inverse of a one-one function and form composite functions
Use sketch graphs to show the relationship between a function and its inverse
[2] Quadratic Functions
Find the maximum or minimum value of the quadratic function f:x ↦ ax2 + bx + c by any method
Use the maximum or minimum value of f(x) to sketch the graph or determine the range for a given domain
Know the conditions for f(x) = 0 to have:
(i) two real roots, (ii) two equal roots, (iii) no real roots
and the related conditions for a given line to
(i) intersect a given curve, (ii) be a tangent to a given curve, (iii) not intersect a given curve
Solve quadratic equations for real roots and find the solution set for quadratic inequalities
[3] Equations, Inequalities and Graphs
Solve graphically or algebraically equations of the type |ax + b| = c (c \(\geq\) 0) and |ax + b| = |cx + d|
Solve graphically or algebraically inequalities of the type
|ax + b| > c (c \(\geq\) 0), |ax + b| \(\leq\) c (c > 0) and |ax + b| \(\leq\) |cx + d|
Use substitution to form and solve a quadratic equation in order to solve a related equation
Sketch the graphs of cubic polynomials and their moduli, when given in factorised form
y = k(x - a)(x - b)(x - c)
Solve cubic inequalities in the form k(x - a)(x - b)(x - c) ⩽ d graphically
[4] Indices and Surds
Perform simple operations with indices and with surds, including rationalising the denominator
[5] Factors of Polynomials
Know and use the remainder and factor theorems
Find factors of polynomials
Solve cubic equations
[6] Simultaneous Equations
Solve simple simultaneous equations in two unknowns by elimination or substitution
[7] Logarithmic and Exponential Functions
Know simple properties and graphs of the logarithmic and exponential functions including lnx and ex (series expansions are not required) and graphs of kenx + a and kln(ax + b) where n, k, a and b are integers
Know and use the laws of logarithms (including change of base of logarithms)
Solve equations of the form ax = b
[8] Straight Line Graphs
Interpret the equation of a straight line graph in the form y = mx + c
Transform given relationships, including y = axn and y = Abx, to straight line form and hence determine unknown constants by calculating the gradient or intercept of the transformed graph
Solve questions involving mid-point and length of a line
Know and use the condition for two lines to be parallel or perpendicular, including finding the equation of perpendicular bisectors
[9] Circular Measure
Solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure
[10] Trigonometry
Know the six trigonometric functions of angles of any magnitude (sine, cosine, tangent, secant, cosecant, cotangent)
Understand amplitude and periodicity and the relationship between graphs of related trigonometric functions, e.g. sin x and sin 2x
Draw and use the graphs of
y = a sin bx + c
y = a cos bx + c
y = a tan bx + c
where a is a positive integer, b is a simple fraction or integer (fractions will have a denominator of 2, 3, 4, 6 or 8 only), and c is an integer
Solve simple trigonometric equations involving the six trigonometric functions and the above relationships (not including general solution of trigonometric equations)
Prove simple trigonometric identities
[11] Permutations and Combinations
Recognise and distinguish between a permutation case and a combination case
Know and use the notation n! (with 0! = 1), and the expressions for permutations and combinations of n items taken r at a time
Answer simple problems on arrangement and selection (cases with repetition of objects, or with objects arranged in a circle, or involving both permutations and combinations, are excluded)
[12] Series
Use the Binomial theorem for expansion of (a + b)n for positive integer n
Use the general term nCr an-rbr, 0 \(\leq\) r \(\leq\) n (knowledge of the greatest term and properties of the coefficients is not required)
Recognise arithmetic and geometric progressions
Use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions
Use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression
[13] Vectors in Two Dimensions
Use vectors in any form, e.g. \(\begin{pmatrix}
a \\
b
\end{pmatrix}\), \(\overrightarrow{AB}\) and ai - bj
Know and use position vectors and unit vectors
Find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars
Compose and resolve velocities
[14] Differentiation and Integration
Use the notations f'(x), f"(x), \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\)
Use the derivatives of the standard functions xn (for any rational n), sin x, cos x, tan x, ex, ln x, together with constant multiples, sums and composite functions of these
Differentiate products and quotients of functions
Apply differentiation to gradients, tangents and normals, stationary points, connected rates of change, small increments and approximations and practical maxima and minima problems
Use the first and second derivative tests to discriminate between maxima and minima
Understand integration as the reverse process of differentiation
Integrate sums of terms in powers of x including \(\frac{1}{x}\) and \(\frac{1}{ax + b}\)
Integrate functions of the form (ax + b)n for any rational n, sin(ax + b), cos(ax + b), eax + b
Evaluate definite integrals and apply integration to the evaluation of plane areas
Apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration, and the use of x-t and v-t graphs
