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## 16 Further Pure 1

### Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1 and Core 2.

Candidates will also be expected to know for section 16.6 that the roots of an equation $\mbox{f}(x) = 0$ can be located by considering changes of sign of $\mbox{f}(x)$ in an interval of $x$ in which $\mbox{f}(x)$ is continuous.

Candidates may use relevant formulae included in the formulae booklet without proof.

### 16.1 Algebra and Graphs

 Graphs of rational functions of the form. $\frac{ax + b}{cx + d}, \space \frac{ax+b}{cx^2 + dx + e}$ or $\frac{x^2 + ax + b}{x^2 + cx + d}$ Sketching the graphs. Finding the equations of the asymptotes which will always be parallel to the coordinate axes. Finding points of intersection with the coordinate axes or other straight lines. Solving associated inequalities. Using quadratic theory (not calculus) to find the possible values of the function and the coordinates of the maximum or minimum points on the graph. Eg for $y = \frac{x^2 + 2}{x^2-4x},\space y = k \Rightarrow x^2 + 2 = kx^2 - 4kx$ which has real roots if $16k^2 + 8k - 8 \geq -0$, ie if $k \leq -1 \space or \space k \geq \frac{1}{2}$; stationary points are $(1, -1)$ and $(-2, \frac{1}{2})$ Graphs of parabolas, ellipses and hyperbolas with equations $y^2=4ax, \space \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$ $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $xy = c^2$ Sketching the graphs. Finding points of intersection with the coordinate axes or other straight lines. Candidates will be expected to interpret the geometrical implication of equal roots, distinct real roots or no real roots. Knowledge of the effects on these equations of single transformations of these graphs involving translations, stretches parallel to the $x$-axis or $y$-axis, and reflections in the line $y = x$. Including the use of the equations of the asymptotes of the hyperbolas given in the formulae booklet.

### 16.2 Complex Numbers

 Non-real roots of quadratic equations. Complex conjugates – awareness that non-real roots of quadratic equations with real coefficients occur in conjugate pairs. Sum, difference and product of complex numbers in the form $x + \mbox{i}y$ Comparing real and imaginary parts. Including solving equations eg $2z + z^{*} = 1 + \mbox{i}$ where $z^{ * }$ is theconjugate of $z$ .

### 16.3 Roots and coefficients of a quadratic equation

 Manipulating expressions involving $\alpha + \beta$ and $\alpha \beta$ . Eg $\alpha^3 + \beta^3 = (\alpha+\beta)^3 = (\alpha + \beta)^3-3\alpha \beta(\alpha + \beta)$ Forming an equation with roots $\alpha^3, \beta^3,$ or $\frac{1}{\alpha},\frac{1}{\beta},\alpha+\frac{2}{\beta}, \beta+\frac{2}{\alpha}$ etc.

### 16.4 Series

 Use of formulae for the sum of the squares and the sum of the cubes of the natural numbers. Eg to find a polynomial expression for $\displaystyle\sum_{r=1}^{n} r^2(r+2)$ or $\displaystyle\sum_{r=1}^{n} (r^2-r+1)$

### 16.5 Calculus

 Finding the gradient of the tangent to a curve at a point, by taking the limit as $h$ tends to zero of the gradient of a chord joining two points whose $x$-coordinates differ by $h$. The equation will be given as $y=\mbox{f}(x)$, where $\mbox{f}(x)$ is a simple polynomial such as $x^2-2x \mbox{ or } x^4+3$. Evaluation of simple improper integrals. E.g. $\displaystyle\int_{1}^4 \frac{1}{\sqrt{x}} \mbox{d}x , \int_{4}^{\infty} x^{-\frac{3}{2}} \mbox{d}x$

### 16.6 Numerical Methods

 Finding roots of equations by interval bisection, linear interpolation and the Newton-Raphson method. Graphical illustration of these methods. Solving differential equations of the form $\displaystyle {\operatorname{d}\!y\over\operatorname{d}\!x} = \mbox{f}(x)$ Using a step-by-step method based on the linear approximations $y_{n+1} \approx y_n + h \mbox{f}(x_n); x_{n+1} = x_n + h,$ with given values for $x_0, y_0$ and $h$. Reducing a relation to a linear law. E.g. $\frac{1}{x} + \frac{1}{y} = k; \space y^2 = ax^3+b; \space y = a x^n; \space y = ab^x$ Use of logarithms to base 10 where appropriate. Given numerical values of $(x, y)$, drawing a linear graph and using it to estimate the values of the unknown constants.

### 16.7 Trigonometry

 General solutions of trigonometric equations including use of exact values for the sine, cosine and tangent of $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$ Eg $\sin 2x = \frac{\sqrt{3}}{2}, \mbox{ } \cos\Big(x + \frac{\pi}{6}\Big) = -\frac{1}{\sqrt{2}}, \mbox{ } \tan \Big( \frac{\pi}{3}-2x \Big) = 1$, $\sin 2x=0.3, \mbox{ } \cos(3x-1) = -0.2$

### 16.8 Matrices and Transformations

 $2 \times 2$ and $2 \times 1$ matrices; addition and subtraction, multiplication by a scalar. Multiplying a $2 \times 2$ matrix by a $2 \times 2$ matrix or by a $2 \times 1$ matrix. The identity matrix $\mathbf{I}$ for a $2 \times 2$ matrix. Transformations of points in the $x - y$ plane represented by $2 \times 2$ matrices. Transformations will be restricted to rotations about the origin, reflections in a line through the origin, stretches parallel to the $\mbox{x}$-axis and $\mbox{y}$-axis, and enlargements with centre the origin. Use of the standard transformation matrices given in the formulae booklet. Combinations of these transformations e.g. $\Bigg[ \begin{matrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix} \Bigg], \mbox{ } \Bigg[ \begin{matrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} \end{matrix} \Bigg], \mbox{ } \Big[ \begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix} \Big], \mbox{ } \Big[ \begin{matrix} 2 & 0 \\ 0 & 2 \end{matrix} \Big]$

## 17 Further Pure 2

### Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1, Core 2, Core 3, Core 4 and Further Pure 1.

Candidates may use relevant formulae included in the formulae booklet without proof except where proof is required in this module and requested in a question.

### 17.1 Roots of Polynomials

 The relations between the roots and the coefficients of a polynomial equation; the occurrence of the non-real roots in conjugate pairs when the coefficients of the polynomial are real.

### 17.2 Complex Numbers

 The Cartesian and polar coordinate forms of a complex number, its modulus, argument and conjugate. $x+\mbox{i}y$ and $r(\cos \theta + \mbox{i} \sin \theta)$. The sum, difference, product and quotient of two complex numbers. The parts of this topic also included in module Further Pure 1 will be examined only in the context of the content of this module. The representation of a complex number by a point on an Argand diagram; geometrical illustrations. Simple loci in the complex plane. For example, $|z - 2 - \mbox{i} | \leqslant 5, \space \space \mbox{arg}(z - 2)= \frac{ \pi }{3}$ Maximum level of difficulty $| z - a | = | z - b |$ where $a$ and $b$ are complex numbers.

### 17.3 De Moivre's Theorem

 De Moivre's theorem for integral $n$. Use of $z+\frac{1}{z} = 2 \cos \theta$ and $z-\frac{1}{z} = 2 \mbox{i} \sin \theta$, leading to, for example, expressing $\sin^5 \theta$ in terms of multiple angles and $\tan 5 \theta$ in term of powers of $\tan \theta$. Applications in evaluating integrals, for example,$\int \sin^5 \theta \mbox{d}\theta$. De Moivre's theorem; the $n \text{th}$ roots of unity, the exponential form of a complex number. The use, without justification, of the identity $e^{ix}= \cos x + \mbox{i} \sin x$ Solutions of equations of the form $z^n = a + \mbox{i}b$ To include geometric interpretation and use, for example, in expressing $\cos \frac{5 \pi}{12}$ in surd form.

### 17.4 Proof by Induction

 Applications to sequences and series, and other problems. Eg proving that $7^n + 4^n+1$is divisible by 6, or $(\cos \theta + \mbox{i} \sin \theta)^n = \cos n \theta + \mbox{i} \sin n \theta$ where n is a positive integer.

### 17.5 Finite Series

 Summation of a finite series by any method such as induction, partial fractions or differencing. Eg $\displaystyle\sum_{r=1}^n r.r! = \displaystyle\sum_{r=1}^n \big[(r+1)! - r! \big]$

### 17.6 The calculus of inverse trigonometrical functions

 Use of the derivatives of $\sin^{-1}x, \space \cos^{-1}x, \space \tan^{-1}x$ as given in the formulae booklet. To include the use of the standard integrals $\int \frac{1}{a^2 + x^2} \mbox{d}x; \space \int \frac{1}{\sqrt{a^2-x^2}}\mbox{d}x$ given in the formulae booklet.

### 17.7 Hyperbolic Functions

 Hyperbolic and inverse hyperbolic functions and their derivatives; applications to integration. The proofs mentioned below require expressing hyperbolic functions in terms of exponential functions. To include solution of equations of the form $a \sinh x + b \cosh x = c$. Use of basic definitions in proving simple identities. Maximum level of difficulty: $\sinh(x+y) \equiv \sinh x \cosh y + \cosh x \sinh y$. The logarithmic forms of the inverse functions, given in the formulae booklet, may be required. Proofs of these results may also be required. Proofs of the results of differentiation of the hyperbolic functions, given in the formula booklet, are included. Knowledge, proof and use of: $\cosh^2 x - \sinh^2 x = 1$ $1 - \tanh^2 x = \mbox{sech}^2 x$ $\coth^2 x - 1 = \mbox{cosech}^2 x$ Familiarity with the graphs of $\sinh x, \space \cosh x, \space \tanh x, \space \sinh^{-1} x, \space \cosh^{-1} x, \space \tanh^{-1} x$.

### 17.8 Arc length and Area of surface of revolution about the x-axis

 Calculation of the arc length of a curve and the area of a surface of revolution using Cartesian or parametric coordinates. Use of the following formulae will be expected: $s = \displaystyle\int_{x_1}^{x_2} \Bigg[ 1 + \bigg({\operatorname{d}\!y\over\operatorname{d}\!x} \bigg)^2 \Bigg]^{\frac{1}{2}} \mbox{d}x = \int_{t_1}^{t_2} \Bigg[ \bigg( {\operatorname{d}\!x\over\operatorname{d}\!t} \bigg)^2 + \bigg( {\operatorname{d}\!y\over\operatorname{d}\!t} \bigg)^2 \Bigg] ^ {\frac{1}{2}} \mbox{d}t$ $S = 2 \pi \displaystyle\int_{x_1}^{x_2} y \Bigg[ 1 + \bigg( {\operatorname{d}\!y\over\operatorname{d}\!x} \bigg)^2 \Bigg]^{\frac{1}{2}} \mbox{d}x = 2 \pi \int_{t_1}^{t_2} y \Bigg[ \bigg( {\operatorname{d}\!x\over\operatorname{d}\!t} \bigg)^2 + \bigg( {\operatorname{d}\!y\over\operatorname{d}\!t} \bigg)^2 \Bigg]^{\frac{1}{2}} \mbox{d}t$

## 18 Further Pure 3

### Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1, Core 2, Core 3, Core 4 and Further Pure 1.

Candidates may use relevant formulae included in the formulae booklet without proof.

### 18.1 Series and Limits

 Maclaurin series Expansions of $e^x, \space \ln(1+x)$,$\cos x$ and $\sin x$, and $(1+x)^n$ for rational values of $n.$ Use of the range of values of $\space x \space$ for which these expansions are valid, as given in the formulae booklet, is expected to determine the range of values for which expansions of related functions are valid; eg. $\ln\Big(\frac{1+x}{1-x}\Big);\space (1-2x)^{\frac{1}{2}}e^x$. Knowledge and use, for $k > 0$, of $\mbox{lim}x^ke^{-x}$ as $x$ tends to infinity and $\mbox{lim} x^k \ln x$ as $\space x \space$ tends to zero. Improper integrals. E.g.$\int_{0}^{e}x\mbox{ln}x \space \mbox{d}x \space, \int^{\infty}_{0}xe^{-x} \space\mbox{d}x$. Candidates will be expected to show the limiting processes used. Use of series expansion to find limits. E.g. $\displaystyle \lim_{x \to 0} \frac{e^x-1}{x} \space ; \space \lim_{x \to 0} \frac{\sin3x}{x} \space ; \space \lim_{x \to 0} \frac{x^2 e^x}{\cos2x-1} \space ; \space \lim_{x \to 0}\frac{\sqrt{2+x}-\sqrt{2}}{x}$

### 18.2 Polar Coordinates

 Relationship between polar and Cartesian coordinates. The convention  $r > 0$  wil be used. The sketching of curves given by equations of the form  $r = \mbox{f}(\theta)$   may be required. Knowledge of the formula $\tan\phi=r{\operatorname{d}\!\theta\over\operatorname{d}\!r}$   is not required. Use of the formula$\mbox{area} = \int_{\alpha}^{\beta}\frac{1}{2}r^2 \space \mbox{d}\theta$.

### 18.3 Differential Equations

 The concept of a differential equation and its order. The relationship of order to the number of arbitrary constants in the general solution will be expected. Boundary values and initial conditions, general solutions and particular solutions.

### 18.4 Differential Equations - First Order

 Analytical solution of first order linear differential equations of the form ${\operatorname{d}\!y\over\operatorname{d}\!x} + \mbox{P}y = \mbox{Q}$ where $P$ and $Q$ are functions of $x$. To include use of an integrating factor and solution by complementary function and particular integral. Numerical methods for the solution of differential eqations of the form ${\operatorname{d}\!y\over\operatorname{d}\!x} = \mbox{f}(x,y)$. Euler's formula and extensions to second order methods for this first order differential equation. Formulae to be used will be stated explicitly in questions, but candidates should be familiar with standard notation such as used in Euler's formula $y_{r+1}=y_{r} + h \mbox{f}(x_r, y_r),$ the formula $y_{r+1} = y_{r-1} + 2h\mbox{f}(x_r, y_r)$, and the formula $y_{r+1}=y_{r}+\frac{1}{2}(k_{1} + k_{2})$ where $k_{1} = h \mbox{f}(x_{r}, y_{r}) \space$ and $\space k_{2} = h \mbox{f}(x_r + h, y_r + k_1)$.

### 18.5 Differential Equations - Second Order

 Solution of differential equations of the form $\displaystyle a{\operatorname{d}^2\!y\over\operatorname{d}\!x^2} + b {\operatorname{d}\!y\over\operatorname{d}\!x}+cy = 0$, where $a$, $b$ and $c$ are integers, by using an auxiliary equation whose roots may be real or complex. Including repeated roots. Solution of equations of the form $\displaystyle a{\operatorname{d}^2\!y\over\operatorname{d}\!x^2} + b {\operatorname{d}\!y\over\operatorname{d}\!x}+cy = \mbox{f}(x)$ where $a$, $b$ and $c$ are integers by finding the complementary function and a particular integral Finding particular integrals will be restricted to cases where $\mbox{f}(x)$ is of the form $e^{kx} \space , \cos kx, \space \sin kx$ or a polynomial of degree at most 4, or a linear combination of any of the above. Solutions of differential equations of the form: $\displaystyle {\operatorname{d}^2\!y\over\operatorname{d}\!x^2} + P{\operatorname{d}\!y\over\operatorname{d}\!x}+Qy = R$ where $P,Q$ and $R$ are functions of $x$. A substitution will always be given which reduces the differential equation to a form which can be directly solved using the other analytical methods in 18.4 and 18.5 of this specification or by separating variables. Level or difficulty as indicated by: (a) Given $\displaystyle x^2{\operatorname{d}^2\!y\over\operatorname{d}\!x^2}-2y = x$ use the substitution $x = e^t$ to show that $\displaystyle {\operatorname{d}^2\!y\over\operatorname{d}\!t^2}-{\operatorname{d}\!y\over\operatorname{d}\!t}-2y = e^t$. Hence find $y$ in terms of $t$ Hence find $y$ in terms of $x$ (b) $\displaystyle (1-x^2){\operatorname{d}^2\!y\over\operatorname{d}\!x^2}-2x{\operatorname{d}\!y\over\operatorname{d}\!x} = 0$  use the subsitution $\displaystyle u={\operatorname{d}\!y\over\operatorname{d}\!x}$ to show that $\displaystyle {\operatorname{d}\!u\over\operatorname{d}\!x} = \frac{2xu}{1-x^2}$ and hence that $u = \frac{A}{1-x^2}$, where $A$ is an arbitrary constant. Hence find $y$ in terms of $x$.

## 19 Further Pure 4

### Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1, Core 2, Core 3, Core 4 and Further Pure 1.

Candidates may use relevant formulae included in the formulae booklet without proof.

### 19.1 Vectors and Three-Dimensional Coordinate Geometry

 Definition and properties of the vector product. Calculation of vector products. Including the use of vector products in the calculation of the area of a triangle or parallelogram. Calculation of scalar triple products. Including the use of the scalar triple product in the calculation of the volume of a parallelepiped and in identifying coplanar vectors. Proof of the distributive law and knowledge of particular formulae is not required. Applications of vectors to two- and three-dimensional geometry, involving points, lines and planes. Including the equation of a line in the form $({\bf r - a}) \times {\bf b} = 0$. Vector equation of a plane in the form ${\bf r.n} = d$ or $\bf{\mbox{r}} = \mbox{a}+ { \lambda } \bf{ \mbox{b} } + \mu \bf{ \mbox{c}}$ . Intersection of a line and a plane. Angle between a line and a plane and between two planes. Cartesian coordinate geometry of lines and planes. Direction ratios and direction cosines. To include finding the equation of the line of intersection of two non-parallel planes. Including the use of $l^2 + m^2 + n^2 = 1$ where $l, m, n$ are the direction cosines. Knowledge of formulae other than those in the formulae booklet will not be expected.

### 19.2 Matrix Algebra

 Matrix algebra of up to 3 x 3 matrices, including the inverse of a 2 x 2 or 3 x 3 matrix. Including non-square matrices and use of the results $({\bf AB})^{-1} = {\bf B}^{-1}{\bf A}^{-1}$ and $({\bf AB}{^T}) = {\bf B}{^T}{\bf A}{^T}$ Singular and non-singular matrices. The identity matrix $\bf{I}$ for 2 x 2 and 3 x 3 matrices. Matrix transformations in two dimensions: shears. Candidates will be expected to recognise the matrix for a shear parallel to the $x$ or $y$ axis. Where the line of invariant points is not the $x$ or $y$ axis candidates will be informed that the matrix represents a shear. The combination of a shear with a matrix transformation from MFP1 is included. Rotations, reflections and enlargements in three dimensions, and combinations of these. Rotations about the coordinate axes only. Reflections in the planes $x = 0, y=0, z=0, x=y, x=z, y=z$ only. Invariant points and invariant lines. Eigenvalues and eigenvectors of 2 x 2 and 3 x 3 matrices. Characteristic equations. Real eigenvalues only. Repeated eigenvalues may be included. Diagonalisation of 2 x 2 and 3 x 3 matrices. $\bf{M} = \bf{UDU}^{-1}$ where $\bf{D}$ is diagonal matrix featuring the eigenvalues and $\bf{U}$ is a matrix whose columns are the eigenvectors. Use of the result $\bf{\mbox{M}}^n = \bf{\mbox{UD}}^n \bf{\mbox{U}}^{-1}$

### 19.3 Solution of Linear Equations

 Consideration of up to three linear equations in up to three unknowns. Their geometrical interpretation and solution. Any method of solution is acceptable.

### 19.4 Determinants

 Second order and third order determinants, and their manipulation. Including the use of the result ${det} ({\bf AB}) = {det} {\bf A} {det} {\bf B}$, but a general treatment of products is not required. Factorisation of determinants. Using row and/or column operations or other suitable methods. Calculation of area and volume scale factors for transformation representing enlargements in two and three dimensions.

### 19.5 Linear Independence

 Linear independence and dependence vectors.

## 20 Statistics 1

### Introduction

Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formula, which is not included in the formulae booklet, but which may be required to answer questions.

$(\textbf{residual})_i = y_i - a - bx_i$

### 20.1 Numerical Measures

 Standard deviation and variance calculated on ungrouped and grouped data. Where raw data are given, candidates will be expected to be able to obtain standard deviation and mean values directly from calculators. Where summarised data are given, candidates may be required to use the formula from the booklet provided for the examination. It is advisable for candidates to know whether to divide by   $n \space \text{or} \space (n-1)$ when calculating the variance; either divisor will be accepted unless a question specifically requests an unbiased estimate of a population variance. Linear scaling. Artificial questions requiring linear scaling will not be set, but candidates should be aware of the effect of linear scaling on numerical measures. Choice of numerical measures. Candidates will be expected to be able to choose numerical measures, including mean, median, mode, range and interquartile range, appropriate to given contexts. Linear interpolation will not be required.

### 20.2 Probability

 Elementary probability; the concept of a random event and its probability. Assigning probabilities to events using relative frequencies or equally likely outcomes. Candidates will be expected to understand set notation but its use will not be essential. Addition law of probability. Mutually exclusive events. $\mbox{P}(A \cup B)=\mbox{P}(A) + \mbox{P}(B) - \mbox{P}(A \cap B)$; two events only. $\mbox{P}(A \cup B)=\mbox{P}(A) + \mbox{P}(B)$; two or more events. $\mbox{P}(A')= 1-\mbox{P}(A)$. Multiplication law of probability and conditional probability. Independent events. $\mbox{P}(A \cap B) = \mbox{P}(A) \times \mbox{P}(B|A) = \mbox{P}(B) \times \mbox{P}(A|B)$; two or more events. $\mbox{P}(A \cap B) = \mbox{P}(A) \times \mbox{P}(B)$; two or more events. Application of probability laws. Only simple problems will be set that can be solved by direct application of the probability laws, by counting equally likely outcomes and/or the construction and the use of frequency tables or relative frequency (probability) tables. Questions requiring the use of tree diagrams or Venn diagrams will not be set, but their use will be permitted.

### 20.3 Binomial Distribution

 Discrete random variables. Only an understanding of the concepts; not examined beyond binomial distributions. Conditions for application of a binomial distribution. Calculation of probabilities using formula. Use of $\space \displaystyle\binom{n}{x}$ notation. Use of tables. Mean, variance and standard deviation of a binomial distribution. Knowledge, but not derivations, will be required.

### 20.4 Normal Distribution

 Continuous random variables. Only an understanding of the concepts; not examined beyond normal distributions. Properties of normal distributions. Shape, symmetry and area properties. Knowledge that approximately   $\frac{2}{3}$ of observations lie within and equivalent results. Calculation of probabilities. Transformation to the standardised normal distribution and use of the supplied tables. Interpolation will not be essential; rounding − values to two decimal places will be accepted. Mean, variance and standard deviation of a normal distribution. To include finding unknown mean and/or standard deviation by making use of the table of percentage points. (Candidates may be required to solve two simultaneous equations.)

### 20.5 Estimation

 Population and sample. To include the terms ‘parameter’ and ‘statistic’. Candidates will be expected to understand the concept of a simple random sample. Methods for obtaining simple random samples will not be tested directly in the written examination. Unbiased estimators of a population mean and variance. $\bar{X} \space \mbox{and} \space S^2 \space$ respectively. The sampling distribution of the mean of a random sample from a normal distribution. To include the standard error of the sample mean, , and its estimator, $\space \displaystyle{\frac{S}{\sqrt{n}}}$. A normal distribution as an approximation to the sampling distribution of the mean of a large sample from any distribution. Knowledge and application of the Central Limit Theorem. Confidence intervals for the mean of a normal distribution with known variance. Only confidence intervals symmetrical about the mean will be required. Confidence intervals for the mean of a distribution using a normal approximation. Large samples only. Known and unknown variance. Inferences from confidence intervals. Based on whether a calculated confidence interval includes or does not include a ’hypothesised’ mean value.

### 20.6 Correlation and Regression

 Calculation and interpretation of the product moment correlation coefficient. Where raw data are given, candidates should be encouraged to obtain correlation coefficient values directly from calculators. Where summarised data are given, candidates may be required to use a formula from the booklet provided for the examination. Calculations from grouped data are excluded. Importance of checking for approximate linear relationship but no hypothesis tests. Understanding that association does not necessarily imply cause and effect. Identification of response (dependent) and explanatory (independent) variables in regression. Calculation of least squares regression lines with one explanatory variable. Scatter diagrams and drawing a regression line theorem. Where raw data are given, candidates should be encouraged to obtain gradient and intercept values directly from calculators. Where summarised data are given, candidates may be required to use formulae from the booklet provided for the examination. Practical interpretation of values for the gradient and intercept. Use of line for prediction within range of observed values of explanatory variable. Appreciation of the dangers of extrapolation. Calculation of residuals. Use of $\space (\text{residual})_i = y_i-a-bx_i$. Examination of residuals to check plausibility of model and to identify outliers. Appreciation of the possible large influence of outliers on the fitted line. Linear scaling. Artificial questions requiring linear scaling will not be set, but candidates should be aware of the effect of linear scaling in correlation and regression.

## 21 Statistics 2

### Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the module Statistics 1 and Core 1.

Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

$\mbox{E}(aX + b) = a\mbox{E}(X) + b \space \space$ and $\space \space \text{Var}(aX) = a^2\text{Var}(X)$

$\mbox{P}(a < X < b) = \int_a^b \mbox{f}(x) \mbox{d}$

$\mbox{P}(\text{Type I error}) = \mbox{P}(\text{reject}\space\mbox{H}_0\space|\space\mbox{H}_0\space\text{true})\space$ and

$\mbox{P}(\text{Type II error}) = \mbox{P}(\text{accept}\space\mbox{H}_0\space|\space\mbox{H}_0\space\text{false})$

$\displaystyle E_{ij} = \frac{R_i \times C_j}{T}$ and   $\nu = (\text{rows} - 1)(\text{columns} - 1)$

Yates’ correction (for table) is $\space \displaystyle \chi^2 = \sum\frac{(|O_i - E_i| - 0.5)^2}{E_i}$

### 21.1 Discrete Random Variables

 Discrete random variables and their associated probability distributions. The number of possible outcomes will be finite. Distributions will be given or easily determined in the form of a table or simple function. Mean, variance and standard deviation. Knowledge of the formulae $\mbox{E}(X) = \displaystyle\sum x_i p_i\space,\space\mbox{E}\big(\mbox{g}(X)\big) = \sum \mbox{g}(x_i)p_i\space,\space\text{Var}(X) = \mbox{E}(X^2) - (\mbox{E}(X))^2\space,\space$ $\mbox{E}(aX+b) = a\mbox{E}(X) + b \space$ and $\space \text{Var}(aX+b) = a^2 \text{Var}(X)$ will be expected. Mean, variance and standard deviation of a simple function of a discrete random variable. Eg   $\mbox{E}(2X + 3) \space,\space \mbox{E}(5X^2) \space, \space \mbox{E}(10X^{-1})\space,\space \mbox{E}(100X^{-2})$ Eg   .

### 21.2 Poisson Distribution

 Conditions for application of a Poisson distribution. Calculation of probabilities using formula. To include calculation of values of $\space \mathrm{e}^{- \lambda}$ from a calculator. Use of Tables. Mean, variance and standard deviation of a Poisson distribution. Knowledge, but not derivations, will be required. Distribution of sum of independent Poisson distributions. Result, not proof.

### 21.3 Continuous Random Variables

 Differences from discrete random variables. Probability density functions, cumulative distribution functions and their relationship. $\displaystyle \mbox{F}(x) = \int_{- \infty}^x \mathrm{f}(t)\mbox{d}t \space\space$ and $\space \space \displaystyle \space\mathrm{f}(x) = \frac{\mbox{d}}{\mbox{d}\!x}\mbox{F}(x)$. Polynomial integration only. The probability of an observation lying in a specified interval. $\displaystyle \mbox{P}(a < X < b) = \int_a^b \mbox{f}(x) \mbox{d}x \space \space$ and $\space \space \mbox{P}(X = a) = 0$. Median, quartiles and percentiles. Mean, variance and standard deviation. Knowledge of the formulae $\text{Var}(X) = \mbox{E}(X^2)-\big(\mbox{E}(X)\big)^2\space, \space \mbox{E}(aX + b) = a \mbox{E}(X) + b\space$ and $\text{Var}(aX + b) = a^2\text{Var}(X)$ will be expected. Mean, variance and standard deviation of a simple function of a continuous random variable. Eg $\space \mbox{E}(2X+3)\space ,\space \mbox{E}(5X^2) \space , \space \mbox{E}(10X^{-1}) \space, \space \mbox{E}(100X^{-2})$. Eg $\space \mathrm{Var}(3X)\space,\space \mathrm{Var}(4X-5)\space,\space \mathrm{Var}(6X^{-1})$. Rectangular distribution. Calculation of probabilities, proofs of mean, variance and standard deviation.

### 21.4 Estimation

 Confidence intervals for the mean of a normal distribution with unknown variance. Using a   $t$ distribution. Only confidence intervals symmetrical about the mean will be required. Questions may involve a knowledge of confidence intervals from the module Statistics 1.

### 21.5 Hypothesis Testing

 Null and alternative hypotheses. The null hypothesis to be of the form that a parameter takes a specified value. One tailed and two tailed tests, significance level, critical value, critical region, acceptance region, test statistic, $\space \text{Type I}$ and $\text{Type II}\space$errors. The concepts of $\space \text{Type I errors}\space(\text{reject}\space\mbox{H}_0\space|\space\mbox{H}_0\space\text{true})$ and $\text{Type II errors}\space(\text{accept}\space\mbox{H}_0\space|\space\mbox{H}_0\space\text{false})$ should be understood but questions which require the calculation of the risk of a $\space \text{Type II error}$ will not be set. The significance level to be used in a hypothesis test will usually be given. Tests for the mean of a normal distribution with known variance. Using a   $z$-statistic. Tests for the mean of a normal distribution with unknown variance. Using a   $t$-statistic. Tests for the mean of a distribution using a normal approximation. Large samples only. Known and unknown variance.

### 21.6 Chi-Squared Contingency Table Tests

 Introduction to   $\chi^2$ distribution. To include use of the supplied tables. Use of  $\displaystyle \sum \frac{(O_i - E_i)^2}{E_i}$ as an approximate  $\chi^2$-statistic. Conditions for approximation to be valid. The convention that all  $E_i$ should be greater than 5 will be expected. Test for independence in contingency tables. Use of Yates' correction for table will be required.

## 22 Statistics 3

### Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Statistics 1 and 2 and Core 1 and 2.

Candidate may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

For $\space X_i$ independently distributed $\space (\mu_i, \space \sigma_i^2)$, then

$\Sigma a_i X_i \space$ is distributed$\space \big(\Sigma a_i \mu_i, \space \Sigma a_i^2 \sigma_i^2 \big)$

Power = 1 – P(Type $\space \mathrm{II}$ error)

### 22.1 Further Probability

 Bayes’ Theorem. Knowledge and application to at most three events. The construction and use of tree diagrams

### 22.2 Linear Combinations of Random Variables

 Mean, variance and standard deviation of a linear combination of two (discrete or continuous) random variables. To include covariance and correlation. Implications of independence. Applications, rather than proofs, will be required. Mean, variance and standard deviation of a linear combination of independent (discrete or continuous) random variables. Use of these, rather than proofs, will be required. Linear combinations of independent normal random variables. Use of these only.

### 22.3 Distributional Approximations

 Mean, variance and standard deviation of binomial and Poisson distributions. Proofs using  $\mbox{E}(X)$  and  $\mathrm{E}(X(X-1))$ together with  $\sum p_i = 1 \space$. A Poisson distribution as an approximation to a binomial distribution. Conditions for use. A normal distribution as an approximation to a binomial distribution. Conditions for use. Knowledge and use of continuity corrections. A normal distribution as an approximation to a Poisson distribution. Conditions for use. Knowledge and use of continuity corrections.

### 22.4 Estimation

 Estimation of sample sizes necessary to achieve confidence intervals of a required width with a given level of confidence. Questions may be set based on a knowledge of confidence intervals from the module Statistics 1. Confidence intervals for the difference between the means of two independent normal distributions with known variances. Symmetric intervals only. Using a normal distribution. Confidence intervals for the difference between the means of two independent distributions using normal approximations. Large samples only. Known and unknown variances. The mean, variance and standard deviation of a sample proportion. Unbiased estimator of a population proportion. $\hat{P}$ A normal distribution as an approximation to the sampling distribution of a sample proportion based on a large sample. $\displaystyle\mbox{N}\Bigg(p, \frac{p(1-p)}{n} \Bigg)$ Approximate confidence intervals for a population proportion and for the mean of a Poisson distribution. Using normal approximations. The use of a continuity correction will not be required in these cases. Approximate confidence intervals for the difference between two population proportions and for the difference between the means of two Poisson distributions. Using normal approximations. The use of continuity corrections will not be required in these cases.

### 22.5 Hypothesis Testing

 The notion of the power of a test. Candidates may be asked to calculate the probability of a Type $\space\mathrm{II}$ error or the power for a simple alternative hypothesis of a specific test, but they will not be asked to derive a power function. Questions may be set which require the calculation of a $\space z$-statistic using knowledge from the module Statistics 1. The significance level to be used in a hypothesis test will usually be given. Tests for the difference between the means of two independent normal distributions with known variances. Using a  $z$-statistic. Tests for the difference between the means of two independent distributions using normal approximations. Large samples only. Known and unknown variances. Tests for a population proportion and for the mean of a Poisson distribution. Using exact probabilities or, when appropriate, normal approximations where a continuity correction will not be required. Tests for the difference between two population proportions and for the difference between the means of two Poisson distributions. Using normal approximations where continuity corrections will not be required. In cases where the null hypothesis is testing an equality, a pooling of variances will be expected. Use of the supplied tables to test $\mbox{H}_0 :\rho = 0 \space$ for a bivariate normal population. Where $\space \rho$ denotes the population product moment correlation coefficient.

## 23 Statistics 4

### Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Statistics 1 and Statistics 2 and Core 1, Core 2 and Core 3.

Those candidates who have not studied the module Statistics 3 will also require knowledge of the mean, variance and standard deviation of a difference between two independent normal random variables.

Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

For an exponential distribution, $\mbox{F}(x) = 1-\mathrm{e}^{- \lambda x}$

Efficiency of $\space Estimator A\space$ relative to $\space \displaystyle Estimator B = \frac{1/\text{Var}(Estimator A)}{1/\text{Var}(Estimator B)}$

### 23.1 Geometric and Exponential Distributions

 Conditions for application of a geometric distribution. Calculation of probabilities for a geometric distribution using formula. Mean, variance and standard deviation of a geometric distribution. Knowledge and derivations will be expected. Conditions for application of an exponential distribution. Knowledge that lengths of intervals between Poisson events have an exponential distribution. Calculation of probabilities for an exponential distribution. Using cumulative distribution function or integration of probability density function. Mean, variance and standard deviation of an exponential distribution. Knowledge and derivations will be expected.

### 23.2 Estimators

 Review of the concepts of a sample statistic and its sampling distribution, and of a population parameter. Estimators and estimates. Properties of estimators. Unbiasedness, consistency, relative efficiency. Mean and variance of pooled estimators of means and proportions. Proof that $\space \mbox{E}(S^2) = \sigma^2$.

### 23.3 Estimation

 Confidence intervals for the difference between the means of two normal distributions with unknown variances. Independent and paired samples. For independent samples, only when the population variances may be assumed equal so that a pooled estimate of variance may be calculated. Small samples only. Using a $\space t$ distribution. Confidence intervals for a normal population variance (or standard deviation) based on a random sample. Using a $\space \chi^2$ distribution. Confidence intervals for theratio of two normal population variances (or standard deviations) based on independent random samples. Introduction to  $F$ distribution. To include use of the supplied tables. Using an  $F$ distribution.

### 23.4 Hypothesis Testing

 The significance level to be used in a hypothesis test will usually be given. Tests for the difference between the means of two normal distributions with unknown variances. Independent and paired samples. For independent samples, only when the population variances may be assumed equal so that a pooled estimate of variance may be calculated. Small samples only. Using a  $t$-statistic. Tests for a normal population variance (or standard deviation) based on a random sample. Using a  $\chi^2$-statistic. Tests for the ratio of two normal population variances (or standard deviations) based on independent random samples. Using a  $F$-statistic.

### 23.5 Chi-Squared Goodness of Fit Tests

 Use of  $\sum \frac{(O_i - E_i)^2}{E_i}$ as an approximate  $\chi^2$-statistic. Conditions for approximation to be valid. The convention that all  $E_i$ should be greater than 5 will be expected. Goodness of fit tests. Discrete probabilities based on either a discrete or a continuous distribution. Questions may be set based on a knowledge of discrete or continuous random variables from the module Statistics 2. Integration may be required for continuous random variables.

## 24 Mechanics 1

### Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the modules Core 1 and Core 2. Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

Constant Acceleration Formulae
 $v^2 = u^2 + 2as$
Weight $W = mg$
Momentum $\text{Momentum} = mv$
Newton's Second Law $F = ma \space \space$  or   Force = rate of change of momentum
Friction, dynamic $F = \mu R$
Friction, static $F \leqslant \mu R$

### 24.1 Mathematical Modelling

 Use of assumptions in simplifying reality. Candidates are expected to use mathematical models to solve problems. Mathematical analysis of models. Modelling will include the appreciation that: it is appropriate at times to treat relatively large moving bodies as point masses; the friction law  $F \leqslant \mu R \space$ is experimental. Interpretation and validity of models. Candidates should be able to comment on the modelling assumptions made when using terms such as particle, light, inextensible string, smooth surface and motion under gravity. Refinement and extension of models.

### 24.2 Kinematics in One and Two Dimensions

Displacement, speed, velocity, acceleration. Understanding the difference between displacement and distance.
Sketching and interpreting kinematics graphs. Use of gradients and area under graphs to solve problems.
Use of constant acceleration equations
 $v^2 = u^2 + 2as$
Vertical motion under gravity.
Average speed and average velocity.
Application of vectors in two dimensions to represent position, velocity or acceleration. Resolving quantities into two perpendicular components.
Use of unit vectors $\space \bf{\mbox{i}} \space$ and $\space \bf{\mbox{j}}$. Candidates may work with column vectors.
Magnitude and direction of quantities represented by a vector.
Finding position, velocity, speed and acceleration of a particle moving in two dimensions with constant acceleration. The solution of problems such as when a particle is at a specified position or velocity, or finding position, velocity or acceleration at a specified time. Use of constant acceleration equations in vector form, for example,

$\bf{\mbox{v}} = \bf{\mbox{u}} + \bf{\mbox{a}}\mathit{t}\space$.

Problems involving resultant velocities. To include solutions using either vectors or vector triangles.

### 24.3 Statics and Forces

 Drawing force diagrams, identifying forces present and clearly labelling diagrams. When drawing diagrams, candidates should distinguish clearly between forces and other quantities such as velocity. Force of gravity (Newton's Universal Law not required). The acceleration due to gravity,$\space g$, will be taken as $\space 9.8 \space\mbox{ms}^{-2}$. Friction, limiting friction, coefficient of friction and the relationship of $\space F \leqslant \mu R$. Candidates should be able to derive and work with inequalities from the relationship $\space F \leqslant \mu R$. Normal reaction forces. Tensions in strings and rods, thrusts in rods. Modelling forces as vectors. Candidates will be required to resolve forces only in two dimensions. Finding the resultant of a number of forces acting at a point Candidates will be expected to express the resultant using components of a vector and to find the magnitude and direction of the resultant. Finding the resultant force acting on a particle. Knowledge that the resultant force is zero if a body is in equilibrium Find unknown forces on bodies that are at rest.

### 24.4 Momentum

 Concept of momentum Momentum as a vector in one or two dimensions. (Resolving velocities is not required.) Momentum $= mv \space$ The principle of conservation of momentum applied to two particles. Knowledge of Newton’s law of restitution is not required.

### 24.5 Newton's Laws of Motion.

 Newton’s three laws of motion. Problems may be set in one or two dimensions Simple applications of the above to the linear motion of a particle of constant mass. Including a particle moving up or down an inclined plane. Use of $\space F = \mu R \space$ as a model for dynamic friction.

### 24.6 Connected Particles

 Connected particle problems. To include the motion of two particles connected by a light inextensible string passing over a smooth fixed peg or a smooth light pulley, when the forces on each particle are constant. Also includes other connected particle problems, such as a car and trailer.

### 24.7 Projectiles

 Motion of a particle under gravity in two dimensions. Candidates will be expected to state and use equations of the form $x = V \space \space \cos \alpha t \space$ and $\space \space y = V \sin \alpha t - \frac{1}{2} gt^2$. Candidates should be aware of any assumptions they are making. Calculate range, time of flight and maximum height. Formulae for the range, time of flight and maximum height should not be quoted in examinations. Inclined plane and problems involving resistance will not be set. The use of the identity$\space \sin 2 \theta = 2 \sin \theta \cos \theta \space$ will not be required. Candidates may be expected to find initial speeds or angles of projection. Modification of equations to take account of the height of release.

## 25 Mechanics 2

### Introduction

A knowledge of the topics and associated formulae from Modules Core 1 – Core 4, and Mechanics 1 is required. Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

 Centres of Mass $\displaystyle\bar{X} \Sigma m_i = \Sigma m_i x_i \space$ and $\displaystyle \space \bar{Y} \Sigma m_i = \Sigma m_i y_i$ Circular Motion $\displaystyle v = r \omega, \space a = r \omega^2 \space$ and $\displaystyle \space a = \frac{v^2}{r}$ Work and Energy Work done, constant force:   $\text{Work}= Fd \cos \theta$ Work done, variable force in direction of motion in a straight line: $\text{Work}= \int \mbox{F}\space \mbox{d}x$ Gravitational Potential Energy $= mgh$ Kinetic Energy $= \frac{1}{2}mv^2$ Elastic potential energy $\displaystyle = \frac{\lambda}{2l}e^2$ Hooke’s Law $\displaystyle T = \frac{\lambda}{l}e$

### 25.1 Mathematical Modelling

 The application of mathematical modelling to situations that relate to the topics covered in this module.

### 25.2 Moments and Centres of Mass

 Finding the moment of a force about a given point. Knowledge that when a rigid body is in equilibrium, the resultant force and the resultant moment are both zero. Determining the forces acting on a rigid body when in equilibrium. This will include situations where all the forces are parallel, as on a horizontal beam or where the forces act in two dimensions, as on a ladder leaning against a wall. Centres of Mass. Integration methods are not required. Finding centres of mass by symmetry (eg for circle, rectangle). Finding the centre of mass of a system of particles. Centre of mass of a system of particles is given by   $(\bar{X}, \bar{Y})$   where  $\bar{X} \Sigma m_i = \Sigma m_i x_i \space$ and $\space \bar{Y} \Sigma m_i = \Sigma m_i y_i$ Finding the centre of mass of a composite body. Finding the position of a body when suspended from a given point and in equilibrium.

### 25.3 Kinematics

 Relationship between position, velocity and acceleration in one, two or three dimensions, involving variable acceleration. Application of calculus techniques will be required to solve problems. Finding position, velocity andacceleration vectors, by thedifferentiation or integrationof   $\mbox{f}(t){\bf \mbox{i}} + \mbox{g}(t){\bf \mbox{j}} + \mbox{h}(t){\bf\mbox{k}}$, with respect to $\space t$. If       $\space \space \space \space \space{\bf \mbox{r}} = \mbox{f}(t){\bf \mbox{i}} + \mbox{g}(t){\bf \mbox{j}} + \mbox{h}(t){\bf \mbox{k}}$ then       ${\bf \mbox{v}} = \mbox{f}'(t){\bf \mbox{i}} + \mbox{g}'(t){\bf \mbox{j}} + \mbox{h}'(t){\bf \mbox{k}}$ then       ${\bf \mbox{a}} = \mbox{f}''(t){\bf \mbox{i}} + \mbox{g}''(t){\bf \mbox{j}} + \mbox{h}''(t){\bf \mbox{k}}$ Vectors may be expressed in the form $\space \mbox{a}{\bf\mbox{i}}+\mbox{b}{\bf \mbox{j}}+\mbox{c}{\bf \mbox{k}}$ as columnvectors. Candidates may use either notation.

### 25.4 Newton's Laws of Motion

 Application of Newton’s laws to situations, with variable acceleration. Problems will be posed in one, two or three dimensions and may require the use of integration or differentiation.

### 25.5 Application of differential equations

 One-dimensional problemswhere simple differential equations are formed as a result of the application of Newton’s second law. Use of  $\displaystyle \frac{\mbox{d}v}{\mbox{d}t}$  for acceleration, to form simple differential equations, for example, $\displaystyle \space m\frac{\mbox{d}v}{\mbox{d}t} = -\frac{k}{\sqrt{v}} \space \text{or} \space m\frac{\mbox{d}v}{\mbox{d}t} = k(v-2)$. Use of $\space \displaystyle a=\frac{\mbox{d}^2 x}{\mbox{d}t^2} = \frac{\mbox{d}v}{\mbox{d}t};\space v = \frac{\mbox{d}x}{\mbox{d}t}$. The use of  $\displaystyle \frac{\mbox{d}v}{\mbox{d}t} = v\frac{\mbox{d}v}{\mbox{d}x}$ is not required. Problems will require the use of the method of separation of variables.

### 25.6 Uniform Circular Motion

 Motion of a particle in a circle with constant speed. Problems will involve either horizontal circles or situations, such as a satellite describing a circular orbit, where the gravitational force is towards the centre of the circle. Knowledge and use of the relationships $\displaystyle {v} = r \omega, \space {a} = r \omega^2 \space$ and $\displaystyle \space a=\frac{v^2}{r}$. Angular speed in radians $\space \mbox{s}^{-1}$   converted from other unitssuch as revolutions per minute or time for one revolution. Use of the term angular speed. Position, velocity and acceleration vectors in relation to circular motion in terms of $\space{\bf \mbox{i}} \space$ and $\space {\bf \mbox{j}}$. Candidates may be required to show that motion is circular by showing that the body is at a constant distance from a given point. Conical pendulum.

### 25.7 Work and Energy

 Work done by a constant force. Forces may or may not act in the direction of motion. Work done$\space=Fd \cos \theta$ Gravitational potential energy. Universal law of gravitation will not be required. Gravitational Potential Energy $= mgh$ Kinetic energy. Kinetic Energy $=\frac{1}{2}mv^2$ The work-energy principle. Use of Work Done = Change in Kinetic Energy. Conservation of mechanical energy. Solution of problems using conservation of energy. One-dimensional problems only for variable forces. Work done by a variable force. Use of $\int F \space \mbox {d}x$ will only be used for elastic strings and springs. Hooke’s law. $\displaystyle T = \frac{\lambda}{l}e$. Elastic potential energy for strings and springs. Candidates will be expected to quote the formula for elastic potential energy unless explicitly asked to derive it. Power, as the rate at which a force does work, and the relationship $\space P = Fv$.

### 25.8 Vertical Circular Motion

 Circular motion in a vertical plane. Includes conditions to complete vertical circles.

## 26 Mechanics 3

### Introduction

A knowledge of the topics and associated formulae from Modules Mechanics 1, Core 1 and Core 2 is required. A knowledge of the trigonometric identity $\space \sec^2x = 1 + \tan^2 \space x$ is also required. Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

Momentum and Collision
 $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ $m_1{\bf \mbox{u}}_1 + m_2{\bf \mbox{u}}_2 = m_1{\bf \mbox{v}}_1 + m_2{\bf \mbox{v}}_2$ $v = eu$ $v_1 - v_2 = -e(u_1 - u_2)$

### 26.1 Relative Motion

 Relative velocity. Use of relative velocity and initial conditions to find relative displacement. Velocities may be expressed in the form $\space \mbox{a}\space {\bf \mbox{i}} + \mbox{b}\space {\bf \mbox{j}} + \mbox{c}\space {\bf \mbox{k}}\space$or as column vectors. Interception and closest approach. Use of calculus or completing the square. Geometric approaches may be required.

### 26.2 Dimensional Analysis

 Finding dimensions of quantities. Prediction of formulae. Checks on working, using dimensional consistency. Finding the dimensions of quantities in terms of M, L and T. Using this method to predict the indices in proposed formulae, for example, for the period of a simple pendulum. Use dimensional analysis to find units, and as a check on working.

### 26.3 Collisions in one dimension

 Momentum. Impulse as change of momentum. Knowledge and use of the equation  $I = mv - mu$. Impulse as Force $\times$ Time. Impulse as$\int F\space \mbox{d}t$ $I = Ft$  Applied to explosions as well as collisions. Conservation of momentum. Newton’s Experimental Law. Coefficient of restitution. $m_1u_1 + m_2u_2 = m_1v_1+m_2v_2$ $v=eu$ $v_1 - v_2 = -e(u_1-u_2)$

### 26.4 Collisions in two dimensions

 Momentum as a vector. Impulse as a vector. ${\bf \mbox{I}} = m {\bf \mbox{v}} - m {\bf \mbox{u}} \space$ and $\space {\bf \mbox{ I }} = {\bf \mbox{F}}t \space$ will be required. Conservation of momentum in two dimensions. $m_1 {\bf \mbox{u}_1} + m_2 {\bf \mbox{u}_2} = m_1{\bf \mbox{v}_1} + m_2{\bf \mbox{v}_2}$ Coefficient of restitution and Newton’s experimental law. Impacts with a fixed surface. The impact may be at any angle to the surface. Candidates may be asked to find the impulse on the body. Questions that require the use of trigonometric identities will not be set. Oblique Collisions Collisions between two smooth spheres. Candidates will be expected to consider components of velocities parallel and perpendicular to the line of centres.

### 26.5 Further Projectiles

 Elimination of time from equations to derive the equation of the trajectory of a projectile Candidates will not be required to know the formula $\displaystyle y=x \tan \alpha-\frac{gx^2}{2v^2}(1+\tan^2 \alpha)$, but should be able to derive it when needed. The identity  $1+ \tan^2 \theta = \sec^2 \theta \space$ will be required.

### 26.6 Projectiles on Inclined Planes

 Projectiles launched onto inclined planes. Problems will be set on projectiles that are launched and land on an inclined plane. Candidates may approach these problems by resolving the acceleration parallel and perpendicular to the plane. Questions may be set which require the use of trigonometric identities, but any identities which are needed, apart from $\space \tan^2 \space x + 1 = \sec^2 \space x$ and those in Core 2, will be given in the examination paper. Candidates will be expected to find the maximum range for a given slope and speed of projection. Candidates may be expected to determine whether a projectile lands at a higher or lower point on the plane after a bounce.

## 27 Mechanics 4

### Introduction

A knowledge of the topics and associated formulae from Modules Core 1 – Core 4, Mechanics 1 and Mechanics 2 is required. Candidates may use relevant formulae included in the formulae booklet without proof.

Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

 Rotations Moment of a Force $=\bf \mbox{r} \times \mbox{F}$ Moment of Inertia $\displaystyle = \sum_{i=1}^n m_ix_i^2$ Rational Kinetic Energy $= \frac{1}{2}I \omega^2 \space \text{or} \space \frac{1}{2}I \dot{\theta}^2$ Resultant Moment $= I \ddot{\theta}$ Centre of Mass For a Uniform Lamina $\displaystyle \bar{x} =\frac {\int_a^b xy \space\mbox{d}x} {\int_a^b y \space\mbox{d}x} \space \space \space$ $\displaystyle \bar{y} = \frac{\int_a^b \frac{1}{2} y^2 \space\mbox{d}x}{\int_a^b y \space\mbox{d}x}$ For a Solid of Revolution (about the $x$-axis) $\displaystyle \bar{x} = \frac{\int_a^b \pi xy^2 \space\mbox{d}x}{\int_a^b \pi y^2 \space\mbox{d}x}$

### 27.1 Moments

 Couples. Understanding of the concept of a couple. Reduction of systems of coplanar forces. Reduction to a single force, a single couple or to a couple and a force acting at a point. The line of action of a resultant force may be required. Conditions for sliding and toppling. Determining how equilibrium will be broken in situations, such as a force applied to a solid on a horizontal surface or on an inclined plane with an increasing slope. Derivation of inequalities that must be satisfied for equilibrium.

### 27.2 Frameworks

 Finding unknown forces acting on a framework. Finding the forces in the members of a light, smoothly jointed framework. Determining whether rods are in tension or compression. Awareness of assumptions made when solving framework problems.

### 27.3 Vector Product and Moments

 The vector product $\bf\mbox{i} \times \mbox{i} = \mbox{0}\mathrm{,}\space \mbox{i} \times \mbox{j}=\mbox{k}\mathrm{,}\space \mbox{j} \times \mbox{i} = -\mbox{k}$ etc Candidates may use determinants to find vector products. The result $|\bf\mbox{a} \times \mbox{b}| = |\mbox{a}|| \mbox{b}|\sin \theta$ The moment of a force as $\bf \mbox{r} \times \mbox{F}$. Vector methods for resultant force and moment. Application to simple problems. Finding condition for equilibrium, unknown forces or points of application.

### 27.4 Centres of Mass by Integration for Uniform Bodies

 Centre of mass of a uniform lamina by integration. Centre of mass of a uniform solid formed by rotating a region about the $x$-axis. Finding $x$ and $y$ coordinates of the centre of mass.

### 27.5 Moments of Inertia

 Moments of inertia for a system of particles. About any axis $\displaystyle I = \sum_{i=1}^n m_i x_i^2$ Moments of inertia for uniform bodies by integration. Candidates should be able to derive standard results, ie rod, rectangular lamina, hollow or solid sphere and cylinder. Moments of inertia of composite bodies. Bodies formed from simple shapes. Parallel and perpendicular axis theorems. Application to finding moments of inertia about different axes.

### 27.6 Motion of a rigid body about a smooth fixed axis.

 Angular velocity and acceleration of a rigid body. To exclude small oscillations of a compound pendulum. Motion of a rigid body about a fixed horizontal or vertical axis. $I \ddot{\theta} =$ Resultant Moment Including motion under the action of a couple. Rotational kinetic energy and the principle of conservation of energy. Rotational Kinetic Energy $= \frac{1}{2} I \omega^2 \space$ and $\space \frac{1}{2} I \dot{\theta}^2$ To include problems such as the motion of a mass falling under gravity while fixed to the end of a light inextensible string wound round a pulley of given moment of inertia. Moment of momentum (angular momentum). The principle of conservation of angular momentum. To include simple collision problems, eg a particle colliding with a rod rotating about a fixed axis. Forces acting on the axis of rotation

## 28 Mechanics 5

### Introduction

A knowledge of the topics and associated formulae from Modules Core 1 – Core 4, Mechanics 1 and Mechanics 2 is required. Candidates may use relevant formulae included in the formulae booklet without proof.
Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

 Energy Formulae $\mbox{PE} = mgh \space\space\space$Gravitational Potential Energy $\mbox{EPE} = \frac{1}{2} k \space e^2 \space\space\space$ or $\space \displaystyle \frac{\lambda}{2l}e^2\space \space \space$ Elastic Potential Energy Simple Harmonic Motion $\displaystyle v^2 = \omega^2(a^2-x^2)$ $\displaystyle \frac{\mbox{d}^2x}{\mbox{d}t^2} = - \omega^2 x$

### 28.1 Simple Harmonic Motion

 Knowledge of the definition of simple harmonic motion. Finding frequency, period and amplitude Knowledge and use of the formula $\space v^2 = \omega^2 (a^2-x^2)$. Formation of simple second order differential equations to show that simple harmonic motion takes place. Problems will be set involving elastic strings and springs. Candidates will be required to be familiar with both modulus of elasticity and stiffness. They should be aware of and understand the relationship $\displaystyle \space k = \frac{\lambda}{l}$. Solution of second order differential equations of the form $\displaystyle \frac{\mbox{d}^2x}{\mbox{d}t^2}= -\omega^2x$ State solutions in the form $\displaystyle \space x = A\cos(\omega t + \alpha)$ or $\displaystyle \space x = A \cos(\omega t) + B \sin( \omega t) \space$and use these in problems. Simple Pendulum. Formation and solution of the differential equation, including the use of a small angle approximation. Finding the period.

### 28.2 Forced and Damped Harmonic Motion

 Understanding the terms forcing and damping and solution of problems involving them. Candidates should be able to set up and solve differential equations in situations involving damping and forcing. Damping will be proportional only to velocity. Forcing forces will be simple polynomials or of the form $\displaystyle a \sin(\omega t + \alpha),\space \omega a \sin t + b \cos \omega t \space$ or $\displaystyle \space ae^{bt}$. Light, critical and heavy damping. Candidates should be able to determine which of these will take place. Resonance. Solutions may be required for the case where the forcing frequency is equal to the natural frequency. Application to spring/mass systems.

### 28.3 Stability

 Finding and determining whether positions of equilibrium are stable or unstable. Use of potential energy methods. Problems will involve gravitational and elastic potential energy.

### 28.4 Variable Mass Problems

 Equation of motion for variable mass. Derive equations of motion for variable mass problems, for example, a rocket burning fuel, or a falling raindrop. Rocket problems will be set in zero or constant gravitational fields.

### 28.5 Motion in a Plane using Polar Coordinates

 Polar coordinates Transverse and radial components of velocity in polar form. These results may be stated. No proof will be required. Transverse and radial components of acceleration in polar form. Application of polar form of velocity and acceleration. Application to simple central forces. No specific knowledge of planetary motion will be required.

## 29 Decision 1

### 29.1 Simple Ideas of Algorithms

 Correctness, finiteness and generality. Stopping conditions. Candidates should appreciate that for a given input an algorithm produces a unique output. Candidates will not be required to write algorithms in examinations, but may be required to trace, correct, complete or amend a given algorithm, and compare or comment on the number of iterations required. The algorithm may be presented as a flow diagram. Bubble, shuttle, shell, quicksort algorithms. Candidates should appreciate the relative advantages of the different algorithms in terms of the number of iterations required. When using the quicksort algorithm, the first number in each list will be taken as the pivot.

### 29.2 Graphs and Networks

 Vertices, edges, edge weights, paths, cycles, simple graphs. Adjacency/distance matrices. For storage of graphs. Connectedness. Directed and undirected graphs. Degree of a vertex, odd and even vertices, Eulerian trails and Hamiltonian cycles. Trees. Bipartite and complete graph. Use of the notations $\space K_n$ and $\space K_{m,n}$

### 29.3 Spanning Tree Problems

 Prim's and Kruskal's algorithms to find minimum spanning trees. Relative advantage of the two algorithms. Minimum length spanning trees are also called minimum connectors. Candidates will be expected to apply these algorithms in graphical, and for Prim's algorithm also in tabular, form. Candidates may be required to comment on the appropriateness of their solution in its context. Greediness.

### 29.4 Matchings

 Use of bipartite graphs. Improvement of matching using an algorithm. Use of an alternating path.

### 29.5 Shortest Paths in Networks

 Dijkstra's algorithm. Problems involving shortest and quickest routes and paths of minimum cost. Including a labelling technique to identify the shortest path. Candidates may be required to comment on the appropriateness of their solution in its context.

### 29.6 Route Inspection Problem

 Chinese Postman problem. Candidates should appreciate the significance of the odd vertices. Although problems with more than four odd vertices will not be set, candidates must be able to calculate the number of possible pairings for $n$ odd vertices. Candidates may be required to comment on the appropriateness of their solution in its context.

### 29.7 Travelling Salesperson Problem

 Conversion of a practical problem into the classical problem of finding a Hamiltonian cycle. Determination of upper bounds by nearest neighbour algorithm. Determination of lower bounds on route lengths using minimum spanning trees. By deleting a node, then adding the two shortest distances to the node and the length of the minimum spanning tree for the remaining graph. Candidates may be required to comment on the appropriateness of their solution in its context.

### 29.8 Linear Programming

 Candidates will be expected to formulate a variety of problems as linear programmes. They may be required to use up to a maximum of 3 variables, which may reduce to two variable requiring a graphical solution. Graphical solution of two variable problems. In the case of two decision variables candidates may be expected to plot a feasible region and objective line. Candidates may be required to comment on the appropriateness of their solution in its context.

### 29.9 Mathematical modelling

 The application of mathematical modelling to situations that relate to the topics covered in this module. Including the interpretation of results in context.

## 30 Decision 2

### Introduction

Candidates will be expected to be familiar with the knowledge, skills and understanding implicit in the linear programming section of Decision 1.

### 30.1 Critical Path Analysis

 Representation of compound projects by activity networks, algorithm to find the critical path(s); cascade (or Gantt) diagrams; resource histograms and resource levelling. Activity-on-node representation will be used for project networks. Heuristic procedures only are required for resource levelling. Candidates may be required to comment on the appropriateness of their solution in its context

### 30.2 Allocation

 The Hungarian algorithm. Including the use of a dummy row or column for unbalanced problems. The use of an algorithm to establish a maximal matching may be required.

### 30.3 Dynamic Programming

 The ability to cope with negative edge lengths. A stage and state approach may be required in dynamic programming problems. Application to production planning. Finding minimum or maximum path through a network. Solving maximin and minimax problems.

### 30.4 Network Flows

 Maximum flow/minimum cut theorem. Problems may require super-sources and sinks, may have upper and lower capacities and may have vertex restrictions. Labelling procedure. For flow augmentation.

### 30.5 Linear Programming

 The Simplex method and the Simplex tableau. Candidates will be expected to introduce slack variables, iterate using a tableau and interpret the outcome at each stage.

### 30.6 Game Theory for Zero Sum Games

 Pay-off matrix, play-safe strategies and saddle points.Optimal mixed strategies for the graphical method. Reduction of pay-off matrix using dominance arguments. Candidates may be required to comment on the appropriateness of their solution in its context.

### 30.7 Mathematical modelling

 The application of mathematical modelling to situations that relate to the topics covered in this module. Including the interpretation of results in context.