Syllabus

Cambridge International A Level Further Mathematics Syllabus code 9231 For examination in June and November 2013

Contents

Cambridge A Level Further Mathematics Syllabus code 9231
1. Introduction ..................................................................................... 2
1.1 1.2 1.3 1.4 Why choose Cambridge? Why choose Cambridge International A Level Further Mathematics? Cambridge Advanced International Certificate of Education (AICE) How can I find out more?

2. Assessment at a glance .................................................................. 5 3. Syllabus aims and objectives ........................................................... 7 4. Curriculum content .......................................................................... 8
4.1 Paper 1 4.2 Paper 2

5. Mathematical notation................................................................... 17 6. Resource list .................................................................................. 22 7 Additional information.................................................................... 26 .
7 .1 7 .2 7 .3 7 .4 7 .5 7 .6 Guided learning hours Recommended prior learning Progression Component codes Grading and reporting Resources

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013. © UCLES 2010

1. Introduction

1.1 Why choose Cambridge?
University of Cambridge International Examinations (CIE) is the world’s largest provider of international qualifications. Around 1.5 million students from 150 countries enter Cambridge examinations every year. What makes educators around the world choose Cambridge?

Recognition
A Cambridge International A or AS Level is recognized around the world by schools, universities and employers. The qualifications are accepted as proof of academic ability for entry to universities worldwide, though some courses do require specific subjects. Cambridge International A Levels typically take two years to complete and offer a flexible course of study that gives students the freedom to select subjects that are right for them. Cambridge International AS Levels often represent the first half of an A Level course but may also be taken as a freestanding qualification. They are accepted in all UK universities and carry half the weighting of an A Level. University course credit and advanced standing is often available for Cambridge International A/AS Levels in countries such as the USA and Canada. Learn more at www.cie.org.uk/recognition.

Support
CIE provides a world-class support service for teachers and exams officers. We offer a wide range of teacher materials to Centres, plus teacher training (online and face-to-face) and student support materials. Exams officers can trust in reliable, efficient administration of exams entry and excellent, personal support from CIE Customer Services. Learn more at www.cie.org.uk/teachers.

Excellence in education
Cambridge qualifications develop successful students. They build not only understanding and knowledge required for progression, but also learning and thinking skills that help students become independent learners and equip them for life.

Not-for-profit, part of the University of Cambridge
CIE is part of Cambridge Assessment, a not-for-profit organisation and part of the University of Cambridge. The needs of teachers and learners are at the core of what we do. CIE invests constantly in improving its qualifications and services. We draw upon education research in developing our qualifications.

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

2

1. Introduction

1.2 Why choose Cambridge International A Level Further Mathematics?
Cambridge International A Level Further Mathematics is accepted by universities and employers as proof of mathematical knowledge and understanding. Successful candidates gain lifelong skills, including: • • • • • a deeper understanding of mathematical principles; the further development of mathematical skills including the use of applications of mathematics in the context of everyday situations and in other subjects that they may be studying; the ability to analyse problems logically, recognising when and how a situation may be represented mathematically; the use of mathematics as a means of communication; a solid foundation for further study.

1.3 Cambridge Advanced International Certificate of Education (AICE)
Cambridge AICE is the group award of Cambridge International Advanced Supplementary Level and Advanced Level (AS Level and A Level). Cambridge AICE involves the selection of subjects from three curriculum groups – Mathematics and Science; Languages; Arts and Humanities. An A Level counts as a double-credit qualification and an AS Level as a single-credit qualification within the Cambridge AICE award framework. To be considered for an AICE Diploma, a candidate must earn the equivalent of six credits by passing a combination of examinations at either double credit or single credit, with at least one course coming from each of the three curriculum areas. The examinations are administered in May/June and October/November sessions each year. Further Mathematics (9231) falls into Group A, Mathematics and Sciences. Learn more about AICE at http://www.cie.org.uk/qualifications/academic/uppersec/aice.

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

3

1. Introduction

1.4 How can I find out more?
If you are already a Cambridge Centre
You can make entries for this qualification through your usual channels, e.g. CIE Direct. If you have any queries, please contact us at international@cie.org.uk.

If you are not a Cambridge Centre
You can find out how your organisation can become a Cambridge Centre. Email us at international@cie.org.uk. Learn more about the benefits of becoming a Cambridge Centre at www.cie.org.uk.

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

4

2. Assessment at a glance

Cambridge A Level Further Mathematics Syllabus code 9231
All candidates take two papers. Paper 1 3 hours

There are about 11 questions of different marks and lengths on Pure Mathematics. Candidates should answer all questions except for the final question (worth 12–14 marks) which will offer two alternatives, only one of which must be answered. 100 marks weighted at 50% of total Paper 2 3 hours

There are 4 or 5 questions of different marks and lengths on Mechanics (worth a total of 43 or 44 marks) followed by 4 or 5 questions of different marks and lengths on Statistics (worth a total of 43 or 44 marks) and one final question worth 12 or 14 marks. The final question consists of two alternatives, one on Mechanics and one on Statistics. Candidates should answer all questions except for the last question where only one of the alternatives must be answered. 100 marks weighted at 50% of total Electronic Calculators Candidates should have a calculator with standard ‘scientific’ functions for use in the examination. Graphic calculators will be permitted but candidates obtaining results solely from graphic calculators without supporting working or reasoning will not receive credit. Computers, and calculators capable of algebraic manipulation, are not permitted. All the regulations in the Handbook for Centres apply with the exception that, for examinations on this syllabus only, graphic calculators are permitted. Mathematical Instruments Apart from the usual mathematical instruments, candidates may use flexicurves in this examination. Mathematical Notation Attention is drawn to the list of mathematical notation at the end of this booklet. Examiners’ Reports (SR(I) booklets) Reports on the June examinations are distributed to Caribbean Centres in November/December and reports on the November examinations are distributed to other International Centres in April/May.

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

5

2. Assessment at a glance

Availability
This syllabus is examined in the May/June examination session and the October/November examination session. This syllabus is available to private candidates. Centres in the UK that receive government funding are advised to consult the CIE website www.cie.org.uk for the latest information before beginning to teach this syllabus.

Combining this with other syllabuses
Candidates can combine this syllabus in an examination session with any other CIE syllabus, except: • syllabuses with the same title at the same level

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

6

3. Syllabus aims and objectives

3.1 Aims
The aims for Advanced Level Mathematics 9709 apply, with appropriate emphasis. The aims are to enable candidates to: • • • • develop their mathematical knowledge and skills in a way which encourages confidence and provides satisfaction and enjoyment; develop an understanding of mathematical principles and an appreciation of mathematics as a logical and coherent subject; acquire a range of mathematical skills, particularly those which will enable them to use applications of mathematics in the context of everyday situations and of other subjects they may be studying; develop the ability to analyse problems logically, recognise when and how a situation may be represented mathematically, identify and interpret relevant factors and, where necessary, select an appropriate mathematical method to solve the problem; use mathematics as a means of communication with emphasis on the use of clear expression; acquire the mathematical background necessary for further study in this or related subjects.

• •

3.2 Assessment objectives
The assessment objectives for Advanced Level Mathematics 9709 apply, with appropriate emphasis. The abilities assessed in the examinations cover a single area: technique with application. The examination will test the ability of candidates to: • • • • • understand relevant mathematical concepts, terminology and notation; recall accurately and use successfully appropriate manipulative techniques; recognise the appropriate mathematical procedure for a given situation; apply combinations of mathematical skills and techniques in solving problems; present mathematical work, and communicate conclusions, in a clear and logical way.

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

7

4. Curriculum content

4.1 Paper 1
Knowledge of the syllabus for Pure Mathematics (units P1 and P3) in Mathematics 9709 is assumed, and candidates may need to apply such knowledge in answering questions. Theme or topic 1. Polynomials and rational functions Curriculum objectives Candidates should be able to: • recall and use the relations between the roots and coefficients of polynomial equations, for equations of degree 2, 3, 4 only; use a given simple substitution to obtain an equation whose roots are related in a simple way to those of the original equation; sketch graphs of simple rational functions, including the determination of oblique asymptotes, in cases where the degree of the numerator and the denominator are at most 2 (detailed plotting of curves will not be required, but sketches will generally be expected to show significant features, such as turning points, asymptotes and intersections with the axes). understand the relations between cartesian and polar coordinates (using the convention r ğ 0), and convert equations of curves from cartesian to polar form and vice versa; sketch simple polar curves, for 0 Ğ θ < 2π or −π < θ Ğ π or a subset of either of these intervals (detailed plotting of curves will not be required, but sketches will generally be expected to show significant features, such as symmetry, the form of the curve at the pole and least/greatest values of r ); β recall the formula 1 ∫ r 2 dθ for the area of a sector, and α 2 use this formula in simple cases.

•

•

2. Polar coordinates

•

•

•

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

8

4. Curriculum content

3. Summation of series

• •

use the standard results for sums;

∑ r, ∑ r , ∑ r
2

3

to find related

use the method of differences to obtain the sum of a finite series, e.g. by expressing the general term in partial fractions; recognise, by direct consideration of a sum to n terms, when a series is convergent, and find the sum to infinity in such cases. use the method of mathematical induction to establish a given result (questions set may involve divisibility tests and inequalities, for example); recognise situations where conjecture based on a limited trial followed by inductive proof is a useful strategy, and carry this out in simple cases e.g. find the nth derivative of xex. d2y in cases where the relation dx 2 between y and x is defined implicitly or parametrically; obtain an expression for derive and use reduction formulae for the evaluation of definite integrals in simple cases; use integration to find mean values and centroids of two- and three-dimensional figures (where equations are expressed in cartesian coordinates, including the use of a parameter), using strips, discs or shells as appropriate, arc lengths (for curves with equations in cartesian coordinates, including the use of a parameter, or in polar coordinates), surface areas of revolution about one of the axes (for curves with equations in cartesian coordinates, including the use of a parameter, but not for curves with equations in polar coordinates).

•

4. Mathematical induction

•

•

5. Differentiation and integration

•

• •

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

9

4. Curriculum content

6. Differential equations

•

recall the meaning of the terms ‘complementary function’ and ‘particular integral’ in the context of linear differential equations, and recall that the general solution is the sum of the complementary function and a particular integral; find the complementary function for a second order linear differential equation with constant coefficients; recall the form of, and find, a particular integral for a second order linear differential equation in the cases where a polynomial or ebx or a cos px + b sin px is a suitable form, and in other simple cases find the appropriate coefficient(s) given a suitable form of particular integral; use a substitution to reduce a given differential equation to a second order linear equation with constant coefficients; use initial conditions to find a particular solution to a differential equation, and interpret a solution in terms of a problem modelled by a differential equation. understand de Moivre’s theorem, for a positive integral exponent, in terms of the geometrical effect of multiplication of complex numbers; prove de Moivre’s theorem for a positive integral exponent; use de Moivre’s theorem for positive integral exponent to express trigonometrical ratios of multiple angles in terms of powers of trigonometrical ratios of the fundamental angle; use de Moivre’s theorem, for a positive or negative rational exponent in expressing powers of sin θ and cos θ in terms of multiple angles, in the summation of series, in finding and using the nth roots of unity.

• •

• •

7.

Complex numbers

•

• •

•

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

10

4. Curriculum content

8. Vectors

•

use the equation of a plane in any of the forms ax + by + cz = d or r.n = p or r = a + λb + µc, and convert equations of planes from one form to another as necessary in solving problems; recall that the vector product a × b of two vectors can be expressed either as |a| |b| sin θ n , where n is a unit vector, ˆ ˆ or in component form as (a2b3 – a3b2) i + (a3b1 – a1b3) j + (a1b2 – a2b1) k; use equations of lines and planes, together with scalar and vector products where appropriate, to solve problems concerning distances, angles and intersections, including determining whether a line lies in a plane, is parallel to a plane or intersects a plane, and finding the point of intersection of a line and a plane when it exists, finding the perpendicular distance from a point to a plane, finding the angle between a line and a plane, and the angle between two planes, finding an equation for the line of intersection of two planes, calculating the shortest distance between two skew lines, finding an equation for the common perpendicular to two skew lines. recall and use the axioms of a linear (vector) space (restricted to spaces of finite dimension over the field of real numbers only); understand the idea of linear independence, and determine whether a given set of vectors is dependent or independent; understand the idea of the subspace spanned by a given set of vectors; recall that a basis for a space is a linearly independent set of vectors that spans the space, and determine a basis in simple cases; recall that the dimension of a space is the number of vectors in a basis; understand the use of matrices to represent linear transformations from on → om;

•

•

9. Matrices and linear spaces

•

•

• •

• •

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

11

4. Curriculum content

•

understand the terms ‘column space’, ‘row space’, ‘range space’ and ‘null space’, and determine the dimensions of, and bases for, these spaces in simple cases; determine the rank of a square matrix, and use (without proof) the relation between the rank, the dimension of the null space and the order of the matrix; use methods associated with matrices and linear spaces in the context of the solution of a set of linear equations; evaluate the determinant of a square matrix and find the inverse of a non-singular matrix (2 × 2 and 3 × 3 matrices only), and recall that the columns (or rows) of a square matrix are independent if and only if the determinant is nonzero; understand the terms ‘eigenvalue’ and ‘eigenvector’, as applied to square matrices; find eigenvalues and eigenvectors of 2 × 2 and 3 × 3 matrices (restricted to cases where the eigenvalues are real and distinct); express a matrix in the form QDQ−1, where D is a diagonal matrix of eigenvalues and Q is a matrix whose columns are eigenvectors, and use this expression, e.g. in calculating powers of matrices.

•

• •

• •

•

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

12

4. Curriculum content

4.2 Paper 2
Knowledge of the syllabuses for Mechanics (units M1 and M2) and Probability and Statistics (units S1 and S2) in Mathematics 9709 is assumed. Candidates may need to apply such knowledge in answering questions; harder questions on those units may also be set. Theme or topic Curriculum objectives Candidates should be able to: MECHANICS (Sections 1 to 5) 1. Momentum and impulse • • recall and use the definition of linear momentum, and show understanding of its vector nature (in one dimension only); recall Newton’s experimental law and the definition of the coefficient of restitution, the property 0 Ğ e Ğ 1, and the meaning of the terms ‘perfectly elastic’ (e = 1) and ‘inelastic’ (e = 0); use conservation of linear momentum and/or Newton’s experimental law to solve problems that may be modelled as the direct impact of two smooth spheres or the direct or oblique impact of a smooth sphere with a fixed surface; recall and use the definition of the impulse of a constant force, and relate the impulse acting on a particle to the change of momentum of the particle (in one dimension only). recall and use the radial and transverse components of acceleration for a particle moving in a circle with variable speed; solve problems which can be modelled by the motion of a particle in a vertical circle without loss of energy (including finding the tension in a string or a normal contact force, locating points at which these are zero, and conditions for complete circular motion).

•

•

2. Circular motion

•

•

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

13

4. Curriculum content

3. Equilibrium of a rigid body under coplanar forces

•

understand and use the result that the effect of gravity on a rigid body is equivalent to a single force acting at the centre of mass of the body, and identify the centre of mass by considerations of symmetry in suitable cases; calculate the moment of a force about a point in 2 dimensional situations only (understanding of the vector nature of moments is not required); recall that if a rigid body is in equilibrium under the action of coplanar forces then the vector sum of the forces is zero and the sum of the moments of the forces about any point is zero, and the converse of this; use Newton’s third law in situations involving the contact of rigid bodies in equilibrium; solve problems involving the equilibrium of rigid bodies under the action of coplanar forces (problems set will not involve complicated trigonometry). understand and use the definition of the moment of inertia mr 2 and of a system of particles about a fixed axis as the additive property of moment of inertia for a rigid body composed of several parts (the use of integration to find moments of inertia will not be required);

•

•

• •

4. Rotation of a rigid body

•

∑

• •

use the parallel and perpendicular axes theorems (proofs of these theorems will not be required); ¨ recall and use the equation of angular motion C = Iθ for the motion of a rigid body about a fixed axis (simple cases only, where the moment C arises from constant forces such as weights or the tension in a string wrapped around the circumference of a flywheel; knowledge of couples is not included and problems will not involve consideration or calculation of forces acting at the axis of rotation); recall and use the formula 1 I ω2 for the kinetic energy of a 2 rigid body rotating about a fixed axis; use conservation of energy in solving problems concerning mechanical systems where rotation of a rigid body about a fixed axis is involved.

•

•

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

14

4. Curriculum content

5. Simple harmonic motion

• • •

recall a definition of SHM and understand the concepts of period and amplitude; use standard SHM formulae in the course of solving problems; set up the differential equation of motion in problems leading to SHM, recall and use appropriate forms of solution, and identify the period and amplitude of the motion; recognise situations where an exact equation of motion may be approximated by an SHM equation, carry out necessary approximations (e.g. small angle approximations or binomial approximations) and appreciate the conditions necessary for such approximations to be useful.

•

STATISTICS (Sections 6 to 9) 6. Further work on distributions • use the definition of the distribution function as a probability to deduce the form of a distribution function in simple cases, e.g. to find the distribution function for Y, where Y = X 3 and X has a given distribution; understand conditions under which a geometric distribution or negative exponential distribution may be a suitable probability model; recall and use the formula for the calculation of geometric or negative exponential probabilities; recall and use the means and variances of a geometric distribution and negative exponential distribution. formulate hypotheses and apply a hypothesis test concerning the population mean using a small sample drawn from a normal population of unknown variance, using a t-test; calculate a pooled estimate of a population variance from two samples (calculations based on either raw or summarised data may be required); formulate hypotheses concerning the difference of population means, and apply, as appropriate, a 2-sample t-test, a paired sample t-test, a test using a normal distribution (the ability to select the test appropriate to the circumstances of a problem is expected);
Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

•

• • 7. Inference using normal and t-distributions •

•

•

15

4. Curriculum content

•

determine a confidence interval for a population mean, based on a small sample from a normal population with unknown variance, using a t-distribution; determine a confidence interval for a difference of population means, using a t-distribution, or a normal distribution, as appropriate. fit a theoretical distribution, as prescribed by a given hypothesis, to given data (questions will not involve lengthy calculations); use a χ 2-test, with the appropriate number of degrees of freedom, to carry out the corresponding goodness of fit analysis (classes should be combined so that each expected frequency is at least 5); use a χ 2-test, with the appropriate number of degrees of freedom, for independence in a contingency table (Yates’ correction is not required, but classes should be combined so that the expected frequency in each cell is at least 5). understand the concept of least squares, regression lines and correlation in the context of a scatter diagram; calculate, both from simple raw data and from summarised data, the equations of regression lines and the product moment correlation coefficient, and appreciate the distinction between the regression line of y on x and that of x on y ; recall and use the facts that both regression lines pass – – through the mean centre (x, y ) and that the product moment correlation coefficient r and the regression coefficients b1, b2 are related by r 2 = b1b2; select and use, in the context of a problem, the appropriate regression line to estimate a value, and understand the uncertainties associated with such estimations; relate, in simple terms, the value of the product moment correlation coefficient to the appearance of the scatter diagram, with particular reference to the interpretation of cases where the value of the product moment correlation coefficient is close to +1, −1 or 0; carry out a hypothesis test based on the product moment correlation coefficient.

•

8. χ 2 –tests

•

•

•

9. Bivariate data

• •

•

•

•

•

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

16

5. Mathematical notation

The list which follows summarises the notation used in the CIE’s Mathematics examinations. Although primarily directed towards Advanced/HSC (Principal) level, the list also applies, where relevant, to examinations at O Level/S.C.

1

Set Notation

∈ ∉

{x1, x2,…} {x : …} n(A)
∅ A′ »=

is an element of is not an element of the set with elements x1, x2… the set of all x such that … the number of elements in set A the empty set the universal set the complement of the set A the set of natural numbers, {1, 2, 3, …} the set of integers, {0, ± 1, ± 2, ± 3,…} the set of positive integers, {1, 2, 3,…} the set of integers modulo n, {0, 1, 2,…, n − 1} the set of rational numbers,  : p∈ », q∈ » + } the set of positive rational numbers, {x ∈ » : x > 0} set of positive rational numbers and zero, {x ∈ » : x [ 0} the set of real numbers the set of positive real numbers, {x ∈ » : x > 0} the set of positive real numbers and zero, {x ∈ » : x [ 0} the set of complex numbers the ordered pair x, y the cartesian product of sets A, and B, i.e. A × B = {(a, b) : a ∈ A, b ∈ B} is a subset of is a proper subset of union intersection the closed interval {x ∈ » : a Y x Y b} the interval {x ∈ » : a Y x < b} the interval {x ∈ » : a < x Y b} the open interval {x ∈ » : a < x < b}
p q

»= »
+

»n »= »+ »+ 0 »= »
+

»+ 0 »=

(x, y) A×B
⊆ ⊂ ∪ ∩

[a, b] [a, b) (a, b] (a, b) yRx y~x

y is related to x by the relation R y is equivalent to x, in the context of some equivalence relation

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

17

5. Mathematical notation

2

Miscellaneous Symbols
= ≠

≡

≈ ≅ ∝ < Y > [ ∞ p∧q p∨q ~p

p⇒q p⇐q p⇔q
∃ ∀

is equal to is not equal to is identical to or is congruent to is approximately equal to is isomorphic to is proportional to is less than is less than or equal to, is not greater than is greater than is greater than or equal to, is not less than infinity p and q p or q (or both) not p p implies q (if p then q) p is implied by q (if q then p) p implies and is implied by q (p is equivalent to q) there exists for all

3

Operations

a+b a–b a × b, ab, a.b a a ÷ b, , a/b b

a plus b a minus b a multiplied by b a divided by b a1 + a2 + ... + an

∑a i =1

n

i

∏a i =1

n

i

a1 × a2 × ... × an the positive square root of a.

a
|a| n!  n   r  

n factorial

the modulus of a.

the binomial coefficient or

n! for n ∈ »+ r!(n − r )!

n(n − 1)...(n − r + 1) for n ∈ » r!

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

18

5. Mathematical notation

4

Functions

f(x) f :A→ B f :xa y f gf
−1

the value of the function f at x f is a function under which each element of set A has an image in set B the function f maps the element x to the element y the inverse function of the function f the composite function of f and g which is defined by gf(x) = g(f(x)) the limit of f(x) as x tends to a an increment of x the derivative of y with respect to x the n th derivative of y with respect to x
(n)

x→ a

lim f( x)

∆x, δx dy dx dn y dx n

f ′(x), f ″(x), … , f

∫ y dx

(x) the first, second, ..., n th derivatives of f(x) with respect to x the indefinite integral of y with respect to x the definite integral of y with respect to x between the limits x = a and x = b the partial derivative of V with respect to x the first, second, ... derivatives of x with respect to t

∫

b

a

y dx

∂V ∂x

& x x, &&,...

5

Exponential and Logarithmic Functions

e ex, exp x loga x ln x, loge x lg x, log10 x
6

base of natural logarithms exponential function of x logarithm to the base a of x natural logarithm of x logarithm of x to base 10

Circular and Hyperbolic Functions sin, cos, tan, cosec, sec, cot

}

the circular functions the inverse circular functions the hyperbolic functions the inverse hyperbolic functions

sin −1, cos−1, tan −1,  cosec−1, sec−1, cot −1  sinh, cosh, tanh, cosech, sech, coth

}

sinh −1, cosh −1, tanh −1,  cosech −1, sech −1, coth −1 

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

19

5. Mathematical notation

7

Complex Numbers

i z Re z Im z |z| arg z z*
8 Matrices

square root of –1 a complex number, z = x + iy = r(cos θ + i sin θ) the real part of z, Re z = x the imaginary part of z, Im z = y the modulus of z, |z| =

x2 + y2

the argument of z, arg z = θ, – π < θ Y π the complex conjugate of z, x – i y

M M–1 MT det M or |M|
9 Vectors

a matrix M the inverse of the matrix M the transpose of the matrix M the determinant of the square matrix M

a AB
ˆ a i, j, k a,a

the vector a the vector represented in magnitude and direction by the directed line segment a unit vector in the direction of a unit vectors in the directions of the cartesian coordinate axes the magnitude of a the magnitude of AB the scalar product of a and b the vector product of a and b

AB

AB , AB
a.b a×b

10 Probability and Statistics

A, B, C, etc. A∪B A∩B P(A) A′ P(A | B) X, Y, R, etc. x, y, r, etc. x1, x2, ... f1, f2, ... p(x)

events union of the events A and B intersection of the events A and B probability of the event A complement of the event A probability of the event A conditional on the event B random variables values of the random variables X, Y, R etc observations frequencies with which the observations x1, x2 occur probability function P(X = x) of the discrete random variable X

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

20

5. Mathematical notation

p1, p2,… f(x), g(x),... F(x), G(x),.... E(X) E(g(X)) Var(X) G(t) B(n, p) Po(µ) N(µ, σ2) µ σ2 σ x, m
ˆ s 2, σ 2

probabilities of the values x1, x2 of the discrete random variable X the value of the probability density function of a continuous random variable X the value of the (cumulative) distribution function P(X Y x) of a continuous random variable X expectation of the random variable X expectation of g(X) variance of the random variable X probability generating function for a random variable which takes the values 0,

1, 2 … binomial distribution with parameters n and p Poisson distribution, mean µ normal distribution with mean µ and variance σ2 population mean population variance population standard deviation sample mean unbiased estimate of population variance from a sample, s 2 =

φ
Φ ρ

1 n −1

∑ ( x − x) i 2

probability density function of the standardised normal variable with distribution

N(0, 1) corresponding cumulative distribution function product moment correlation coefficient for a population product moment correlation coefficient for a sample covariance of X and Y

r Cov(X, Y)

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

21

6. Resource list

These titles represent some of the texts available in the UK at the time of printing this booklet. Teachers are encouraged to choose texts for class use which they feel will be of interest to their students and will support their own teaching style. ISBN numbers are provided wherever possible. A LEVEL MATHEMATICS (9709) AND A LEVEL FURTHER MATHEMATICS (9231)

Endorsed Textbooks for A Level Mathematics (9709)
The following textbooks are endorsed by CIE for use with the 9709 syllabus. Please contact Cambridge University Press for further information. Author Neill & Quadling Neill & Quadling Quadling Quadling Dobbs & Miller Dobbs & Miller Title Pure Mathematics 1 Pure Mathematics 2 & 3 Mechanics 1 Mechanics 2 Statistics 1 Statistics 2 Publisher Cambridge University Press Cambridge University Press Cambridge University Press Cambridge University Press Cambridge University Press Cambridge University Press ISBN 0 521 53011 3 0 521 53012 1 0 521 53015 6 0 521 53016 4 0 521 53013 X 0 521 53014 8

Suggested Books
Pure Mathematics Author Backhouse, Houldsworth & Horrill Backhouse, Houldsworth & Horrill Backhouse, Houldsworth, Horrill & Wood Bostock & Chandler Butcher & Megeny Emanuel, Wood & Crawshaw Title Pure Mathematics 1 Pure Mathematics 2 Essential Pure Mathematics Publisher Longman, 1985 Longman, 1985 Longman, 1991 ISBN 0 582 35386 6 0 582 35387 4 0582 066581

Core Maths for Advanced Level Access to Advanced Level Maths (short introductory course) Pure Mathematics 1

Nelson Thornes, 2000 Nelson Thornes, 1997 Longman, 2001

0 7487 5509 8 0 7487 2999 2 0 582 40550 5

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

22

6. Resource list

Emanuel, Wood & Crawshaw Hunt

Pure Mathematics 2 Graded Exercises in Pure Mathematics (Practice questions) Complete Advanced Level Mathematics : Pure Mathematics: Core Text Practice for Advanced Mathematics – Pure Mathematics (Practice questions) Understanding Pure Mathematics Introducing Pure Mathematics Mathematics for AS and A Level – Pure Mathematics Advanced Level Mathematics : Pure Mathematics

Longman, 2001 Cambridge University Press, 2001 Nelson Thornes, 2000

0 582 40549 1 0 521 63753 8

Martin, Brown, Rigby & Riley Morley

0 7487 3558 5

Hodder & Stoughton Educational, 1999 Oxford University Press, 1987 Oxford University Press, 2001 Cambridge University Press, 1997 John Murray, 1995

0 340 701676

Sadler & Thorning Smedley & Wiseman SMP Solomon

019 914243 2 0 19 914803 1 0 521 56617 7 0 7195 5344 X

Further Pure Mathematics Author Gaulter & Gaulter Title Further Pure Mathematics Publisher Oxford University Press, 2001 ISBN 0 19 914735 3

Integrated Courses Author Berry, Fentern, Francis & Graham Berry, Fentern, Francis & Graham Title Discovering Advanced Mathematics – AS Mathematics Discovering Advanced Mathematics – A2 Mathematics Publisher Collins Educational, 2000 Collins Educational, 2001 ISBN 0 00 322502 X 0 00 322503 8

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

23

6. Resource list

Mechanics Author Adams, Haighton, Trim Bostock & Chandler Jefferson & Beadsworth Kitchen & Wake Nunn & Simmons Sadler & Thorning SMP Solomon Young Title Complete Advanced Level Mathematics : Mechanics : Core Text Mechanics for A Level Introducing Mechanics Graded Exercises in Mechanics (Practice questions) Practice for Advanced Mathematics (Practice questions) Understanding Mechanics Mathematics for A and AS Level – Mechanics Advanced Level Mathematics : Mechanics Maths in Perspective 2: Mechanics Publisher Nelson Thornes, 2000 ISBN 0 7487 3559 3

Nelson Thornes, 1996 Oxford University Press, 2000 Cambridge University Press, 2001 Hodder & Stoughton Educational, 1998 Oxford University Press, 1996 Cambridge University Press, 1997 John Murray, 1995 Hodder & Stoughton Educational, 1989

07487 2596 2 0 19 914710 8 0 521 64686 3 0 340 70166 8 019 914675 6 0 521 56615 0 07195 7082 4 07131 78221

Further Mechanics Author Jefferson & Beadworth Title Further Mechanics Advanced Modular Mathematics – Mechanics 3 & 4 Statistics Author Clarke & Cooke Crawshaw & Chambers Title A Basic Course in Statistics A Concise Course in Advanced Level Statistics Publisher Hodder & Stoughton Educational, 1998 Nelson Thornes, 2001 ISBN 0 340 71995 8 0 7487 5475X Publisher Oxford University Press, 2001 Collins Educational, 1995 ISBN 0 19 914738 8 0 00 322401 5

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

24

6. Resource list

Crawshaw & Chambers McGill, McLennan, Migliorini Norris Rees Smith

A-Level Statistics Study Guide Complete Advanced Level Mathematics : Statistics : Core Text Graded Exercises in Statistics (Practice questions) Foundations of Statistics Practice for Advanced Mathematics: Statistics (Practice questions) Mathematics for AS and A Level – Statistics Advanced Level Mathematics: Statistics Introducing Statistics Understanding Statistics

Nelson Thornes, 1997 Nelson Thornes, 2000

0 7487 2997 6 07487 3560 7

Cambridge University Press, 2000 Chapman & Hall, 1987 Hodder & Stoughton Educational, 1998 Cambridge University Press, 1997 John Murray, 1996 Oxford University Press, 2001 Oxford University Press, 1997

0 521 65399 1 0 412 28560 6 0 340 70165X

SMP Solomon Upton & Cook Upton & Cook

0 521 56616 9 0 7195 7088 3 0 19 914801 5 0 19 914391 9

Further Statistics Author Title Advanced Modular Mathematics – Statistics 3 & 4 Publisher Collins Educational, 1997 ISBN 0 00 322416 3

Resources are also listed on CIE’s public website at www.cie.org.uk. Please visit this site on a regular basis as the Resource lists are updated through the year. Access to teachers’ email discussion groups and regularly updated resource lists may be found on the CIE Teacher Support website at http://teachers.cie.org.uk. This website is available to teachers at registered CIE Centres.

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

25

7 Introduction 1. Additional information .

7 Guided learning hours .1
Advanced Level (‘A Level’) syllabuses are designed on the assumption that candidates have about 360 guided learning hours per subject over the duration of the course. (‘Guided learning hours’ include direct teaching and any other supervised or directed study time. They do not include private study by the candidate.) However, this figure is for guidance only, and the number of hours required may vary according to local curricular practice and the candidates’ prior experience of the subject.

7 Recommended prior learning .2
Knowledge of the syllabus for Pure Mathematics (units P1 and P3) in Mathematics 9709 is assumed for Paper 1, and candidates may need to apply such knowledge in answering questions. Knowledge of the syllabus for Mechanics units (M1 and M2) and Probability and Statistics units (S1 and S2) in Mathematics 9709 is assumed for Paper 2. Candidates may need to apply such knowledge in answering questions; harder questions on those units may also be set.

7 Progression .3
Cambridge International A Level Further Mathematics provides a suitable foundation for the study of Mathematics or related courses in higher education.

7 Component codes .4
Because of local variations, in some cases component codes will be different in instructions about making entries for examinations and timetables from those printed in this syllabus, but the component names will be unchanged to make identification straightforward.

7 Grading and reporting .5
A Level results are shown by one of the grades A*, A, B, C, D or E indicating the standard achieved, Grade A* being the highest and Grade E the lowest. ‘Ungraded’ indicates that the candidate has failed to reach the standard required for a pass at A Level. ‘Ungraded’ will be reported on the statement of results but not on the certificate.

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

26

7 Introduction 1. Additional information .

Percentage uniform marks are also provided on each candidate’s statement of results to supplement their grade for a syllabus. They are determined in this way: • A candidate who obtains… … the minimum mark necessary for a Grade A* obtains a percentage uniform mark of 90%. … the minimum mark necessary for a Grade A obtains a percentage uniform mark of 80%. … the minimum mark necessary for a Grade B obtains a percentage uniform mark of 70%. … the minimum mark necessary for a Grade C obtains a percentage uniform mark of 60%. … the minimum mark necessary for a Grade D obtains a percentage uniform mark of 50%. … the minimum mark necessary for a Grade E obtains a percentage uniform mark of 40%. … no marks receives a percentage uniform mark of 0%. Candidates whose mark is none of the above receive a percentage mark in between those stated according to the position of their mark in relation to the grade ‘thresholds’ (i.e. the minimum mark for obtaining a grade). For example, a candidate whose mark is halfway between the minimum for a Grade C and the minimum for a Grade D (and whose grade is therefore D) receives a percentage uniform mark of 55%. The uniform percentage mark is stated at syllabus level only. It is not the same as the ‘raw’ mark obtained by the candidate, since it depends on the position of the grade thresholds (which may vary from one session to another and from one subject to another) and it has been turned into a percentage.

7 Resources .6
Copies of syllabuses, the most recent question papers and Principal Examiners’ reports for teachers are available on the Syllabus and Support Materials CD-ROM, which is sent to all CIE Centres. Resources are also listed on CIE’s public website at www.cie.org.uk. Please visit this site on a regular basis as the Resource lists are updated through the year. Access to teachers’ email discussion groups and regularly updated resource lists may be found on the CIE Teacher Support website at http://teachers.cie.org.uk. This website is available to teachers at registered CIE Centres.

Cambridge A Level Further Mathematics 9231. Examination in June and November 2013.

27

University of Cambridge International Examinations 1 Hills Road, Cambridge, CB1 2EU, United Kingdom Tel: +44 (0)1223 553554 Fax: +44 (0)1223 553558 Email: international@cie.org.uk Website: www.cie.org.uk © University of Cambridge International Examinations 2010