·        concepts of sequence and series for finite and infinite cases

·        sequence as function y = f(n) where n is a positive integer

·        relationship between un (the nth term) and Sn (the sum to n terms)

·        sequence given by a formula for the nth term

·        use of Σ notation

·        sum and difference of two series

·        summation of series by the method of differences

·        convergence of a series and the sum to infinity

·        formula for the nth term and the sum of a finite arithmetic series

·        formula for the nth term and the sum of a finite geometric series

·        condition for convergence of an infinite geometric series

·        formula for the sum to infinity of a convergent geometric series

·        addition and subtraction of vectors, multiplication of a vector by a scalar, and their geometrical interpretations

æ x ö æ x ö

·        use of notations such as ç ÷, ç y ÷   xi + y j ,

è y ø ç ÷,

ç z ÷

è   ø

xi + y j + zk , AB, a

·        position vectors, displacement vectors and direction vectors

·        magnitude of a vector

·        unit vectors

·        distance between two points

·        collinearity

·        use of the ratio theorem in geometrical applications

·        concepts of scalar product and vector product of vectors and their properties

·        angle between two vectors

·        geometrical meanings of | a ·  nˆ |  and | a ´ nˆ |,

where nˆ  is a unit vector

Exclude triple products a · b ´ c and a ´ b ´ c .

·        vector and cartesian equations of lines and planes

·        finding the foot of the perpendicular and distance from a point to a line or to a plane

·        finding the angle between two lines, between a line and a plane, or between two planes

·        relationships between

(i)           two lines (coplanar or skew)

(ii)          a line and a plane

(iii)        two planes

Exclude:

·        finding the shortest distance between two skew lines

·        finding an equation for the common perpendicular to two skew lines

·        extension of the number system from real numbers to complex numbers

·        complex roots of quadratic equations

·        conjugate of a complex number

·        four operations of complex numbers

·        equality of complex numbers

·        conjugate roots of a polynomial equation with real coefficients

·      representation of complex numbers in the Argand diagram

·      complex numbers expressed in the form

r(cos θ + i sin θ), or re^iθ where r > 0 and

– π < θ ⩽ π

·         calculation of modulus (r) and argument (θ) of a complex number

·      multiplication and division of two complex numbers expressed in polar form

·        graphical interpretation of

(i) f ′(x) > 0, f ′(x) = 0 and f ′(x) < 0

(ii) f ″(x) > 0 and f ″(x) < 0

·        relating the graph of y = f ′(x) to the graph of

y = f(x)

·        differentiation of simple functions defined implicitly or parametrically

·        determining the nature of the stationary points (local maximum and minimum points and points of inflexion) analytically, in simple cases, using the first derivative test or the second derivative test

·        locating maximum and minimum points using a graphing calculator

·        finding the approximate value of a derivative at a given point using a graphing calculator

·        finding equations of tangents and normals to curves, including cases where the curve is defined implicitly or parametrically

·        local maxima and minima problems

·        connected rates of change problems

Exclude finding non-stationary points of inflexion and finding second derivatives of functions defined parametrically.

·        standard series expansion of (1 + x)ⁿ for any rational n, e^x, sin x, cos x and In(1 + x)

·        derivation of the first few terms of the Maclaurin series by

–      repeated differentiation, e.g. sec x

–      repeated implicit differentiation, e.g.

y³ + y² + y = x² – 2x

–      using standard series, e.g. e^x cos 2x,

æ 1+ x ö

Inç         ÷

è 1- x ø

·        range of values of x for which a standard series converges

·        concept of “approximation”

·        small angle approximations: sin x » x,

cos x » 1 – 1 x ², tan x » x

2

Exclude derivation of the general term of the series.

·     integration of  f ¢(x )[f (x )]n   (including n = –1), f ′(x)e^f(x)

sin² x, cos² x, tan² x,

sin mx cos nx, cos mx cos nx and sin mx sin nx

1     ,       1      ,     1      and      1

a² + x ²     a2 – x 2     a² – x ²        x ² – a²

·     integration by a given substitution

·     integration by parts

·        concept of definite integral as a limit of sum

·        definite integral as the area under a curve

·        evaluation of definite integrals

·        finding the area of a region bounded by a curve and lines parallel to the coordinate axes, between a curve and a line, or between two curves

·        area below the x-axis

·        finding the area under a curve defined parametrically

·        finding the volume of revolution about the x– or

y-axis

·        finding the approximate value of a definite integral using a graphing calculator

Exclude finding the volume of revolution about the x-axis or y-axis where the curve is defined parametrically.

·        solving for the general solutions and particular solutions of differential equations of the forms

(i)                         dy = f (x )

dx

(ii)                    dy = f (y )

dx

d2 y = ( )

(iii)                                           f x

dx ²

including those that can be reduced to (i) and (ii) by means of a given substitution

·        formulating a differential equation from a problem situation

·        interpreting a differential equation and its solution in terms of a problem situation

·        addition and multiplication principles for counting

·       concepts of permutation (ⁿPr) and combination (ⁿCr)

·        arrangements of objects in a line or in a circle, including cases involving repetition and restriction

·        addition and multiplication of probabilities

·        mutually exclusive events and independent events

·        use of tables of outcomes, Venn diagrams, tree diagrams, and permutations and combinations techniques to calculate probabilities

·        calculation of conditional probabilities in simple cases

·        use of:

P ( A‘ ) = 1 – P ( A)

P ( A È B) = P ( A) + P (B) – P ( A Ç B)

P ( A | B) = P ( A Ç B)

P (B)

·        concept of discrete random variables, probability distributions, expectations and variances

·        concept of binomial distribution B(n, p) as an example of a discrete probability distribution and use of B(n, p) as a probability model, including conditions under which the binomial distribution is a suitable model

·        use of mean and variance of binomial distribution (without proof)

Exclude finding cumulative distribution function of a discrete random variable.

·        concept of a normal distribution as an example of a continuous probability model and its mean and variance; use of N(μ, σ ²) as a probability model

·        standard normal distribution

·        finding the value of P(X < x1) or a related probability, given the values of x1, μ, σ

·        symmetry of the normal curve and its properties

·        finding a relationship between x1, μ, σ given the value of P(X < x1), or a related probability

·        solving problems involving the use of E(aX + b) and Var (aX + b)

·        solving problems involving the use of E(aX + bY) and Var (aX + bY), where X and Y are independent

Exclude normal approximation to binomial distribution.

·        concepts of population and simple random sample

·        concept of the sample mean X as a random

variable with E(X ) = μ and Var ( ) = σ

2

X

n

·        distribution of sample means from a normal population

·        use of the Central Limit Theorem to treat sample mean as having normal distribution when the sample size is sufficiently large (e.g. n ⩾ 30)

·        calculation and use of unbiased estimates of the population mean and variance from a sample, including cases where the data are given in summarised form Σx and Σx², or Σ(x – a) and Σ(x – a)²

·        concepts of null hypothesis (H0) and alternative hypotheses (H1), test statistic, critical region, critical value, level of significance and p-value

·        formulation of hypotheses and testing for a population mean based on:

–     a sample from a normal population of known variance

–     a large sample from any population

·        1-tail and 2-tail tests

·        interpretation of the results of a hypothesis test in the context of the problem

Exclude the use of the term ‘Type I’ error, concept of Type II error and testing the difference between two population means.

·      use of scatter diagram to determine if there is a plausible linear relationship between the two variables

·     correlation coefficient as a measure of the fit of a linear model to the scatter diagram

·     finding and interpreting the product moment correlation coefficient (in particular, values close to

-1, 0 and 1)

·     concepts of linear regression and method of least squares to find the equation of the regression line

·     concepts of interpolation and extrapolation

·     use of the appropriate regression line to make prediction or estimate a value in practical situations, including explaining how well the situation is modelled by the linear regression model

·     use of a square, reciprocal or logarithmic transformation to achieve linearity

Exclude:

·        derivation of formulae

·        relationship r ² = b1b2, where b1 and b2 are regression coefficients

·        hypothesis tests