# Subject content

## 12 Pure Core 1

### Introduction

Candidates will be required to demonstrate:

a. construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language;

b. correct understanding and use of mathematical language and grammar in respect of terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, ‘implies’, ‘is implied by’, ‘necessary’, ‘sufficient’ and notation such as ∴ , ⇒, ⇐ and ⇔ .

Candidates are not allowed to use a calculator in the assessment unit for this module.

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.

Quadratic Equations $ax^2 + bx^2+c = 0 \space$ has roots $\displaystyle \space \frac{-b \pm \sqrt{(b^2 - 4ac)}}{2a}$
Circles A circle, centre (,) and radius , has equation
$(x-a)^2 + (y-b)^2 = r^2$
Differentiation
 function derivative $ax^n$ $anx^{n-1} \space \space \space n$ is a whole number $\mbox{f}(x) + \mbox{g}(x)$ $\mbox{f}'(x) + \mbox{g}'(x)$
Integration
 function integral $ax^n$ $\frac{a}{n+1} x^{n+1}+c \space \space \space n$ is a whole number $\mbox{f}'(x) + \mbox{g}'(x)$ $\mbox{f}(x) + \mbox{g}(x) + c$
Area Area under a curve = $\int_a^b y \, \mbox{d}x (y \geqslant 0)$

### 12.1 Algebra

 Use and manipulation of surds. To include simplification and rationalisation of the denominator of a fraction. Eg $\displaystyle \sqrt{12} + 2 \sqrt{27} = 8 \sqrt{3} \space ; \space \frac{1}{\sqrt{2}-1} = \sqrt{2}+1 \space ; \space \frac{2 \sqrt{3} + \sqrt{2}}{3 \sqrt{2}+\sqrt{3}} = \frac{\sqrt{6}}{3}$ Quadratic functions and their graphs. To include reference to the vertex and line of symmetry of the graph. The discriminant of a quadratic function. To include the conditions for equal roots, for distinct real roots and for no real roots Factorisation of quadratic polynomials. Eg factorisation of $2x^2+x-6$ Completing the square. Eg $x^2+6x-1 = (x+3)^2 - 10 \space \space ;\space \space \space 2x^2-6x+2 = 2(x-1.5)^2 - 2.5$ Solution of quadratic equations. Use of any of factorisation, $\displaystyle \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ or completing the sqaure will be accepted. Simultaneous equations, e.g. one linear and one quadratic, analytical solution by substitution. Solution of linear and quadratic inequalities. Eg $2x^2 + x \geqslant 6$ Algebraic manipulation of polynomials, including expanding brackets and collecting like terms. Simple algebraic division. Applied to a quadratic or a cubic polynomial divided by a linear term of the form $(x+a) \space \text{or} \space (x-a)$ where $a$ is a small whole number. Any method will be accepted, e.g. by inspection, by equating coefficients or by formal division e.g. $\displaystyle \frac{x^3-x^2-5x+2}{x+2}$ Use of the Remainder Theorem. Knowledge that when a quadratic or cubic polynomial $\mbox{f}(x)$ is dividedby $(x-a)$ the remainder is $\mbox{f}(a)$ and, that when $\mbox{f}(a) = 0$, then $(x-a)$ is a factor and vice versa. Use of the Factor Theorem. Greatest level of difficulty as indicated by $x^3-5x^2+7x-3$, i.e. acubic always with a factor $(x+a)\space \text{or} \space (x-a)$where $a$ is a small whole number but including the cases of three distinct linear factors, repeated linear factors or a quadratic factor which cannot be factorized in the real numbers. Graphs of functions; sketching curves defined by simple equations. Linear, quadratic and cubic functions. The $\mbox{f}(x)$ notation may be used but only a very general idea of the concept of a function is required. Domain and range are not included. Graphs of circles are included. Geometrical interpretation of algebraic solution of equations and use of intersection points of graphs of functions to solve equations. Interpreting the solutions of equations as the intersection points of graphs and vice versa. Knowledge of the effect of translations on graphs and their equations. Applied to quadratic graphs and circles, i.e. $y = (x-a)^2 + b$ as a translation of $y=x^2$ and $(x-a)^2 + (y-b)^2 = r^2$ as a translation of $x^2+y^2 = r^2$.

### 12.2 Coordinate Geometry

 Equation of a straight line, including the forms $y-y_1 = m(x-x_1)$ and $ax+by+c=0$. To include problems using gradients, mid-points and the distancebetween two points. The form $y = mx + c$ is also included. Conditions for two straight lines to be parallel or perpendicular to each other. Knowledge that the product of the gradients of two perpendicular lines is $-1$. Coordinate geometry of the circle. Candidates will be expected to complete the square to find the centre and radius of a circle where the equation of the circle is for example given as $x^2+4x+y^2-6y-12 = 0$. The equation of a circle in the form $(x-a)^2 + (y-b)^2 = r^2 \space.$ The use of the following circle properties is required: (i) the angle in a semicircle is a right angle; (ii) the perpendicular from the centre to a chord bisects the chord; (iii) the tangent to a circle is perpendicular to the radius at its point of contact. The equation of the tangent and normal at a given point to a circle. Implicit differentiation is not required. Candidates will be expected to use the coordinates of the centre and a point on the circle or of other appropriate points to find relevant gradients. The intersection of a straight line and a curve. Using algebraic methods. Candidates will be expected to interpret the geometrical implication of equal roots, distinct real roots or no real roots. Applications will be to either circles or graphs of quadratic functions.

### 12.3 Differentiation

 The derivative of $\mbox{f}(x)$ as the gradient of the tangent to the graph of $y = \mbox{f}(x)$ at a point; the gradient of the tangent as a limit; interpretation as a rate of change. The notations $\mbox{f}'(x)$ or $\displaystyle {\operatorname{d}\!y\over\operatorname{d}\!x}$ will be used. A general appreciation only of the derivative when interpreting it is required. Differentiation from first principles will not be tested. Differentiation of polynomials. Applications of differentiation to gradients, tangents and normals, maxima and minima and stationary points, increasing and decreasing functions. Questions will not be set requiring the determination of or knowledge of points of inflection. Questions may be set in the form of a practical problem where a function of a single variable has to be optimised. Second order derivatives. Application to determining maxima and minima.

### 12.4 Integration

 Indefinite integration as the reverse of differentiation Integration of polynomials. Evaluation of definite integrals. Interpretation of the definite integral as the area under a curve. Integration to determine the area of a region between a curve and the $x$-axis. To include regions wholly below the $x$-axis, i.e. knowledge that the integral will give a negative value. Questions involving regions partially above and below the $x$-axis will not be set. Questions may involve finding the area of a region bounded by a straight line and a curve, or by two curves.

## 13 Pure Core 2

### Introduction

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

a. Construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language;

b. correct understanding and use of mathematical language and grammar in respect of terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, ‘implies’, ‘is implied by’, ‘necessary’, ‘sufficient’ and notation such as ∴ , ⇒, ⇐ and ⇔ .

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.

Trigonometry In the triangle $ABC$

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

$\text{area} = \frac{1}{2} ab \sin C$

$\text{arc length of a circle}, l = r \theta$

$\text{area of a sector of a circle}, A = \frac{1}{2} r^2 \theta$

$\tan \theta = \dfrac{\sin \theta}{\cos \theta}$

$\sin^2 \theta + \cos^2 \theta = 1$

Laws of Logarithms

$\log_a x + \log_a y = \log_a (xy)$

$\log_a x - \log_a y = \log_a \Big( \dfrac{x}{y} \Big)$

$k \log_a x = \log_a (x^k)$

Differentiation
 Function derivative $ax^n$ $nax^{n-1}$,      is a rational number
Integration
 Function integral $ax^n$ $\dfrac{a}{n+1} x^{n+1}$,      is a rational number, $n \not= -1$

### 13.1 Algebra and Functions

 Laws of indices for all rational exponents. Knowledge of the effect of simple transformations on the graph of $y = \mbox{f} (x)$as represented by $y = a \mbox{f} (x), y = \mbox{f} (x) + a,$$y = \mbox{f} (x + a), y = \mbox{f} (ax)$ . Candidates are expected to use the terms reflection, translation and stretch in the $x \mbox{ or } y$ direction in their descriptions of these transformations. Eg graphs of $y = \sin 2x$ ; $y = \cos (x + 30^\circ )$ ; $y = 2^{x+3}$ ; $y = 2^{-x}$ Descriptions involving combinations of more than one transformation will not be tested.

### 13.2 Sequences and Series

 Sequences, including those given by a formula for the th term. To include $\Sigma$ notation for sums of series. Sequences generated by a simple relation of the form $\space x_{n+1} = \mbox{f}(x_n)$ To include their use in finding of a limit as $\space n \rightarrow \infty \space$ by putting $L = \mbox{f} (L)$ . Arithmetic series, including the formula for the sum of the first natural numbers. The sum of a finite geometric series. The sum to infinity of a convergent $\space (-1 < r < 1) \space$ geometric series. Candidates should be familiar with the notation $\space|r|<1 \space$ in this context. The binomial expansion of$\space (1 + x)^n \space$ for positive integer . To include the notations $n!$ and $\dbinom{n}{r}$. Use of Pascal’s triangle or formulae to expand $(a + b)^n$ will be accepted.

### 13.3 Trigonometry

 The sine and cosine rules. The area of a triangle in theform $\frac{1}{2} ab\sin C$. Degree and radian measure. Arc length, area of a sector of a circle. Knowledge of the formulae $l = r \theta, \space A = \frac{1}{2}r^2 \theta$. Sine, cosine and tangent functions. Their graphs,symmetries and periodicity. The concepts of odd and even functions are not required. Knowledge and use of $\tan \theta = \dfrac{\sin \theta}{\cos \theta^ {\prime}}$ and $\sin^2 \theta + \cos^2 \theta = 1$. Solution of simple trigonometric equations in a given interval of degrees or radians. Maximum level of difficulty as indicated by $\sin 2 \theta = -0.4 ,$ $\sin (\theta - 20^{\circ} ) = 0.2, \space 2 \sin \theta - \cos \theta = 0$ and

### 13.4 Exponentials and logarithms

 $y = a^x$ and its graph. Using the laws of indices where appropriate. Logarithms and the laws oflogarithms. $\log_a x + \log_a y = log_a (xy) \space ; \space \space \log_a x - \log_a y = log_a \Big(\dfrac{x}{y}\Big) \space ;$ $k \log_a x = \log_a (x^k).$ The equivalence of $y= a^x$ and $x = \log_a y.$ The solution of equations ofthe form $a^x = b$ . Use of a calculator logarithm function to solve for example $3^{2x} = 2$.

### 13.5 Differentiation

 Differentiation of $x^n$ , where is a rational number, andrelated sums and differences. i.e. expressions such as $x^\frac{3}{2} + \dfrac{3}{x^2}$ , including terms which can be expressed as a single power such as $x \sqrt{x}$. Applications to techniques included in module Core 1.

### 13.6 Integration

 Integration of $x^n , \space n \ne -1$, and related sums and differences. i.e. expressions such as $x^\frac{3}{2} + 2x^{- \frac{1}{2}}$ or $\frac{x+2}{\sqrt{x}} = x^\frac{1}{2} + 2x^{- \frac{1}{2}}$ Applications to techniques included in module Core 1. Approximation of the area under a curve using the trapezium rule. The term 'ordinate' will be used. To include a graphical determination of whether the rule over- or under- estimates the area and improvement of an estimate by increasing the number of steps.

## 14 Pure Core 3

### 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 be required to demonstrate:

a. construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language;

b. correct understanding and use of mathematical language and grammar in respect of terms such as ‘equals’, ‘identically equals’, ‘therefore’, ‘because’, ‘implies’, ‘is implied by’, ‘necessary’, ‘sufficient’ and notation such as ∴ , ⇒, ⇐ and ⇔ ;

c. methods of proof, including proof by contradiction and disproof by counter-example.

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.

Trigonometry $\sec^2 A = 1 + \tan^2 A$

$\mbox{cosec}^2 A = 1 + \cot^2 A$

Differentiation
 function derivative $\mathrm {e}^{kx}$ $k\mathrm{e}^{kx}$ $\ln x$ $\frac{1}{x}$ $\sin kx$ $k \cos kx$ $\cos kx$ $-k \sin kx$ $\mbox{f}(x)\mbox{g}(x)$ $\mbox{f}'(x)\mbox{g} (x) + \mbox{f} (x)\mbox{g}'(x)$ $\mbox{f}\big(\mbox{g}(x)\big)$ $\mbox{f}'\big(\mbox{g}(x)\big)\mbox{g}'(x)$
Volumes Volume of solid of revolution:

About the $x$-axis: $V = \int_a^b \pi y^2 \space \mbox{d}x$

About the $y$-axis: $V = \int_c^d \pi x^2 \space \mbox{d}y$

Integration
 Function integral $\cos kx$ $\dfrac{1}{k}\sin kx + c$ $\sin kx$ $-\dfrac{1}{k}\cos kx + c$ $\mathrm {e}^{kx}$ $\frac{1}{k} \mathrm{e}^{kx}+ c$ $\frac{1}{x}$ $\ln|x| + c \space \space (x \not= 0)$ $\mbox{f}'\big(\mbox{g}(x)\big)\mbox{g}'(x)$ $\mbox{f}\big(\mbox{g}(x)\big)+c$

### 14.1 Algebra and Functions

 Definition of a function. Domain and range of a function. Notation such as $\space \mbox{f}(x) = x^2-4$ may be used. Domain may be expressed as$\space x > 1 \space$for example and range may be expressed as $\mbox{f}(x)> -3$ for example. Composition of functions. $\mbox{f} \mbox{g}(x) = \mbox{f} \big( \mbox{g}(x) \big)$ Inverse functions and their graphs. The notation $\mbox{f}^{-1}$ will be used for the inverse of $\mbox{f}$.To include reflection in $y = x$. The modulus function. To include related graphs and the solution from them of inequalities such as $|x+2| < 3|x|$ using solutions of $|x+2| = 3|x|$. Combinations of the transformations on the graph of $y = \mbox{f} (x)$as represented by $y = a\mbox{f} (x)$, $y = \mbox{f} (x) + a$, $y = \mbox{f} (x + a)$, $y = \mbox{f} (ax)$. For example the transformations of: $\mathrm{e}^x$ leading to $\mathrm{e}^{2x}-1$ ; $\ln x$ leading to $2 \ln (x-1)$; $\sec x$ leading to $3 \sec 2x$ Transformations on the graphs of functions included in modules Core 1 and Core 2.

### 14.2 Trigonometry

 Knowledge of $\sin^{-1}, \cos^{-1}$ and $\tan^{-1}$ functions. Understanding of their domains and graphs. Knowledge that $-\dfrac{\pi}{2} \leqslant \sin^{-1} x \leqslant \dfrac{\pi}{2} \space ; 0 \leqslant \cos^{-1} x \leqslant \pi \space ; -\dfrac{\pi}{2} < \tan^{-1} x < \dfrac{\pi}{2}$ The graphs of these functions as reflections of the relevant parts of trigonometric graphs in $\space y = x$ are included. The addition formulae for inverse functions are not required. Knowledge of secant, cosecant and cotangent. Their relationships to cosine, sine and tangent functions. Understanding of their domains and graphs. Knowledge and use of $1+ \tan^2 x = \sec^2 x$, $1 + \cot^2 x = \mbox{cosec}^2 x$. Use in simple identities. Solution of trigonometric equations in a given interval, using these identities.

### 14.3 Exponentials and Logarithms

 The function $\mathrm{e}^x$ and its graph. The function $\ln x$ and its graph; $\space \ln x \space$as the inverse function of $\mathrm{e}^x$ .

### 14.4 Differentiation

 Differentiation of $\mathrm{e}^x ,$ $\ln x, \sin x, \cos x, \tan x ,$ andlinear combinations of these functions. Differentiation using the product rule, the quotient rule, the chain rule and by the use of . E.g $\displaystyle x^2 \ln x \space ; \space \space \mathrm{e}^{3x} \sin x \space ; \space \space \frac{\mathrm{e}^{2x} - 1}{\mathrm{e}^{2x} + 1} \space ; \space \space \frac{2x+1}{3x-2}$ Eg A curve has equation $x = y^2 -4y +1.$ Find $\displaystyle {\operatorname{d}\!y\over\operatorname{d}\!x}$ when $y = 1.$

### 14.5 Integration

 Integration of $\mathrm{e}^x, \dfrac{1}{x}$,$\sin x, \cos x.$ Simple cases of integration:by inspection or substitution; Eg $\int \mathrm{e}^{-3x} \,\mbox{d}x \space ; \space \int \sin 4x \, \mbox{d}x \space ; \space \int x\sqrt{1+x^2} \, \mbox{d}x$ by substitution; Eg $\int x(2+x)^6 \, \mbox{d}x \space ; \space \int x\sqrt{2x-3} \, \mbox{d}x$ and integration by parts. Eg $\int x \mathrm{e}^{2x} \, \mbox{d}x \space ; \space \int x \sin 3x \, \mbox{d}x \space ; \space \int x \ln x \, \mbox{d}x$ These methods as the reverse processes of the chain and product rules respectively. Including the use of $\int \dfrac{\mbox{f}'(x)}{\mbox{f}(x)} \mbox{d}x = \ln | \mbox{f}(x) | + c$ by inspection or substitution. Evaluation of a volume ofrevolution. The axes of revolution will be restricted to the $x$-axis and $y$-axis.

### 14.6 Numerical Methods

 Location of roots of $\mbox{f}(x) = 0$ by considering changes of sign of $\mbox{f}(x)$ in an interval of $x$ in which $\mbox{f}(x)$ is continuous. Approximate solutions of equations using simple iterative methods, including recurrence relations of the form $x_{n+1}= \mbox{f}(x_n).$ Rearrangement of equations to the form $x=\mbox{g}(x).$ Staircase and cobweb diagrams to illustrate the iteration and their use in considerations of convergence. Numerical integration of functions using the mid-ordinate rule and Simpson's rule. To include improvement of an estimate by increasing the number of steps.

## 15 Pure Core 4

### Introduction

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

Candidates will be required to demonstrate:

a: construction and presentation of mathematical arguments through appropriate use of logical deduction and precise statements involving correct use of symbols and appropriate connecting language;

b: correct understanding and use of mathematical language and grammar in respect of terms such as 'equals', 'identically equals', 'therefore', 'because', 'implies', 'is implied by', 'necessary', 'sufficient' and notation such as ∴ , ⇒ , ⇐ and ⇔ ;

c: methods of proof, including proof by contradiction and disproof by counter-example.

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.

 Trigonometry $\sin 2A = 2 \sin A \space \cos A$ $\cos 2A = \left\{ \begin{array}{ll} \mbox{cos}^2 A - \mbox{sin}^2 A & \\ 2 \mbox{cos}^2 A-1 & \\ 1-2 \mbox{sin}^2 A & \end{array} \right.$ $\tan 2A = \dfrac{2 \tan A}{1 - \tan{^2} A}$ $a \cos \theta + b \sin \theta = R \sin(\theta + \alpha)$ , where $\space R = \sqrt{a^2 + b^2}$ and $\tan \alpha = \dfrac{a}{b}$ $a \cos \theta - b \sin \theta = R \cos (\theta + \alpha)$ , where $\space R = \sqrt{a^2 + b^2}$ and $\tan \alpha = \dfrac{b}{a}$ Vectors $\left[ \begin{array}{c} x \\ y \\ z \end{array} \right] \centerdot \left[ \begin{array}{c} a \\ b \\ c \end{array} \right] = xa + yb + zc = \big(\sqrt{x^2 + y^2 + z^2}\big)\big(\sqrt{a^2 + b^2 + c^2}\big)\cos\theta$

### 15.1 Algebra and functions

 Rational functions. Including use of the Factor and Remainder Theorem for divisors ofthe form $(ax+b)$. Simplification of rationalexpressions includingfactorising and cancelling. Expressions of the type $\displaystyle \frac{x^2 - 4x}{x^2 - 5x + 4} = \frac{x(x-4)}{(x-4)(x-1)} = \frac{x}{x-1}$ Algebraic division. Any method will be accepted, e.g. by inspection, by equating coefficients or by formal division. $\displaystyle \frac{3x+4}{x-1} = 3 + \frac{7}{x-1}; \frac{2x^3 - 3x^2 - 2x + 2}{x-2}= 2x^2 + x + \frac{2}{x-2}$; $\displaystyle \frac{2x^2}{(x + 5)(x - 3)} = 2 - \frac{4x - 30}{(x + 5)(x - 3)}$ by using the given identity $\displaystyle \frac{2x^2}{(x + 5)(x - 3)} = \mbox{A} + \frac{\mbox{B}x + \mbox{C}}{(x + 5)(x - 3)}$ Partial fractions(denominators not more complicated than repeated linear terms). Greatest level of difficulty $\displaystyle \frac{3 + 2x^2}{(2x + 1)(x - 3)^2}$ Irreducible quadratic factors will not be tested.

### 15.2 Coordinate geometry in the (x, y) plane

 Cartesian and parametric equations of curves and conversion between the two forms. Eg $x = t^2$,  $y = 2t$;  $x = a\cos \theta$ , $y=b\sin\theta$; $x=\dfrac{1}{t}, \space y=3t \space ; \space x=t+\dfrac{1}{t}, \space y=t-\dfrac{1}{t} \Rightarrow (x+y)(x-y) = 4$

### 15.3 Sequences and series

 Binomial series for anyrational $n$. Expansion of $(1+x)^n, \space \left|x\right|<1 .$ Greatest level of difficulty $(2+3x)^{-2} = \dfrac{1}{4}\Bigg(1+\dfrac{3x}{2}\Bigg)^{-2}$, expansionvalid for $\left|x\right| < \dfrac{2}{3}$ Series expansion of rationalfunctions including the useof partial fractions Greatest level of difficulty $\displaystyle \frac{3+2x^2}{(2x+1)(x-3)^2}$.

### 15.4 Trigonometry

 Use of formulae for$\sin(A \pm B), \space \cos (A \pm B)$ and$\tan (A \pm B)$ and of expressions for $a \cos \theta + b \sin \theta$ in theequivalent forms of $r \cos(\theta \pm \alpha)$ or $r \sin(\theta \pm \alpha)$ . Use in simple identities. Solution of trigonometric equations in a given interval Eg $2 \sin x + 3 \cos x = 1.5, \space -180^{\circ} < x \leq 180^{\circ}$ Knowledge and use of doubleangle formulae. Knowledge that$\sin 2x = 2 \sin x \cos x$ $\cos 2x = \cos^2{x} - \sin^2{x}$ $\space \space \space \space \space = 2 \cos^2 x - 1$ $\space \space \space \space \space = 1-2\sin^2{x}$ $\tan 2x = \dfrac{2 \tan x}{1 - \tan^2 x}$ is expected. Use in simple identities. For example, $\sin 3x = \sin (2x+x) = \sin x(3-4 \sin^2x)$ Solution of trigonometric equations in a given interval. For example, solve $3 \sin 2x = \cos x, \space 0 \leq x \leq 4 \pi .$ Use in integration. For example $\int \cos^2 x \mbox{d} x$

### 15.5 Exponentials and Logarithms

 Exponential growth and decay The use of exponential functions as models.

### 15.6 Differentiation and Integration

 Formation of simple differential equations. To include the context of growth and decay. Analytical solution of simple first order differential equations with separable variables. To include applications to practical problems. Differentiation of simple functions defined implicitly or parametrically. The second derivative of curves defined implicitly or parametrically is not required. Equations of tangents and normals for curves specified implicitly or in parametric form. Simple cases of integrationusing partial fractions. Greatest level of difficulty $\int \dfrac{(1-4x)}{(3x-4)(x+3)^2}\mathrm{d}x$; $\int \dfrac{x^2}{(x+5)(x-3)}\mathrm{d}x$.

### 15.7 Vectors

 Vectors in two and three dimensions. Column vectors will be used in questions but candidates may use $\mathbf{i},\space \mathbf{j}, \space \mathbf{k}$ notation if they wish. Magnitude of a vector. Algebraic operations of vector addition and multiplication by scalars, and their geometrical interpretations. The result $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$ Parallel vectors Position vectors. The distance between two points. Vector equation of lines. Equations of lines in the form ${\bf r}={\bf a} + t{\bf b}.$ Eg $\begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 2 \\ \end{bmatrix} + t \begin{bmatrix} -1 \\ 2 \\ 3 \\ \end{bmatrix}$ To include the intersection of two straight lines in two and three dimensions. Parallel lines. Skew lines in three dimensions. The scalar product. Its use for calculating the angle between two lines. To include finding the coordinates of the foot of the perpendicular from a point to a line and hence the perpendicular distance from a point to a line.

## 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.