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Extension 1 Syllabus

Functions

ME-F1 Further Work with FunctionsYear 11

F1.1: Graphical relationships

• examine the relationship between the graph of $y=f(x)$ and the graph of $y=\frac{1}{f(x)}$ and hence sketch the graphs
• examine the relationship between the graph of $y=f(x)$ and the graphs of $y^2=f(x)$ and $y=\sqrt{f(x)}$ and hence sketch the graphs
• examine the relationship between the graph of $y=f(x)$ and the graphs of $y=\left|f(x)\right|$ and $y=f\left(\left|x\right|\right)$ and hence sketch the graphs
• examine the relationship between the graphs of $y=f(x)$ and $y=g(x)$ and the graphs of $y=f(x)+g(x)$ and $y=f(x)g(x)$ and hence sketch the graphs
• apply knowledge of graphical relationships to solve problems in practical and abstract contexts

F1.2: Inequalities

• solve quadratic inequalities using both algebraic and graphical techniques
• solve inequalities involving rational expressions, including those with the unknown in the denominator
• solve absolute value inequalities of the form $\left|ax+b\right|\geq k$, $\left|ax+b\right|\leq k$, $\left|ax+b\right|< k$ and $\left|ax+b\right|>k$

F1.3: Inverse functions

• define the inverse relation of a function $y=f(x)$ to be the relation obtained by reversing all the ordered pairs of the function
• examine and use the reflection property of the graph of a function and the graph of its inverse
  • understand why the graph of the inverse relation is obtained by reflecting the graph of the function in the line $y=x$
  • using the fact that this reflection exchanges horizontal and vertical lines, recognise that the horizontal line test can be used to determine whether the inverse relation of a function is again a function
• write the rule or rules for the inverse relation by exchanging $x$ and $y$ in the function rules, including any restrictions, and solve for $y$, if possible
• when the inverse relation is a function, use the notation $f^{-1}(x)$ and identify the relationships between the domains and ranges of $f(x)$ and $f^{-1}(x)$
• when the inverse relation is not a function, restrict the domain to obtain new functions that are one-to-one, and compare the effectiveness of different restrictions
• solve problems based on the relationship between a function and its inverse function using algebraic or graphical techniques

F1.4: Parametric form of a function or relation

• understand the concept of parametric representation and examine lines, parabolas and circles expressed in parametric form
  • understand that linear and quadratic functions, and circles can be expressed in either parametric form or Cartesian form
  • convert linear and quadratic functions, and circles from parametric form to Cartesian form and vice versa
  • sketch linear and quadratic functions, and circles expressed in parametric form

ME-F2 PolynomialsYear 11

F2.1: Remainder and factor theorems

• define a general polynomial in one variable, 𝑥, of degree 𝑛 with real coefficients to be the expression: $a_nx^n+a_{n-1}x^{n-1}+\dots+a_2x^2+a_1x+a_0$, where $a_n\neq0$
  • understand and use terminology relating to polynomials including degree, leading term, leading coefficient and constant term
• use division of polynomials to express $P(x)$ in the form $P(x)=A(x).Q(x)+R(x)$ where $\text{deg}R(x)<\text{deg}A(x)$ and $A(x)$ is a linear or quadratic divisor, $Q(x)$ the quotient and $R(x)$ the remainder
  • review the process of division with remainders for integers
  • describe the process of division using the terms: dividend, divisor, quotient, remainder
• prove and apply the factor theorem and the remainder theorem for polynomials and hence solve simple polynomial equations

F2.2: Sums and products of roots of polynomials

• solve problems using the relationships between the roots and coefficients of quadratic, cubic and quartic equations
  • consider quadratic, cubic and quartic equations, and derive formulae as appropriate for the sums and products of roots in terms of the coefficients
• determine the multiplicity of a root of a polynomial equation
  • prove that if a polynomial equation of the form $P(x)=0$ has a root of multiplicity $r>1$, then $P'(x)=0$ has a root of multiplicity $r-1$
• graph a variety of polynomials and investigate the link between the root of a polynomial equation and the zero on the graph of the related polynomial function
  • examine the sign change of the function and shape of the graph either side of roots of varying multiplicity

Trigonometric Functions

ME-T1 Inverse Trigonometric FunctionsYear 11

• define and use the inverse trigonometric functions
  • understand and use the notation $\arcsin x$ and $\sin^{-1} x$ for the inverse function of $\sin x$ when $-\frac{\pi}{2}\leq x\leq\frac{\pi}{2}$ (and similarly for $\cos x$ and $\tan x$) and understand when each notation might be appropriate to avoid confusion with the reciprocal functions
  • use the convention of restricting the domain of $\sin x$ to $-\frac{\pi}{2}\leq x\leq\frac{\pi}{2}$, so the inverse function exists. The inverse of this restricted sine function is defined by: $y=\sin^{-1} x$, $-1\leq x\leq1$ and $-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}$
  • use the convention of restricting the domain of $\cos x$ to $0\leq x\leq\pi$, so the inverse function exists. The inverse of this restricted cosine function is defined by: $y=\cos^{-1}x$, $-1\leq x\leq1$ and $0\leq y\leq\pi$
  • use the convention of restricting the domain of $\tan x$ to $-\frac{\pi}{2}<x<\frac{\pi}{2}$,so the inverse function exists. The inverse of this restricted tangent function is defined by: $y=\tan^{-1}x$, $x$ is a real number and $-\frac{\pi}{2}<y<\frac{\pi}{2}$
  • classify inverse trigonometric functions as odd, even or neither odd nor even
• sketch graphs of the inverse trigonometric functions
• use the relationships $\sin\left(\sin^{-1}x\right)=x$ and $\sin^{-1}\left(\sin x\right)=x$, $\cos\left(\cos^{-1}x\right)=x$ and $\cos^{-1}\left(\cos x\right)=x$, and $\tan\left(\tan^{-1}x\right)=x$ and $\tan^{-1}\left(\tan x\right)=x$ where appropriate, and state the values of $x$ for which these relationships are valid
• prove and use the properties: $\sin^{-1}(-x)=-\sin^{-1}x$, $\cos^{-1}=\pi-\cos^{-1}x$, $\tan^{-1}(-x)=-\tan^{-1}x$ and $cos^{-1}x+\sin^{-1}x=\frac{\pi}{2}$
• solve problems involving inverse trigonometric functions in a variety of abstract and practical situations

ME-T2 Further Trigonometric IdentitiesYear 11

• derive and use the sum and difference expansions for the trigonometric functions $\sin(A\pm B)$, $\cos(A\pm B)$ and $\tan(A\pm B)$
  • $\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$
  • $\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$
  • $\tan(A\pm B)=\frac{\tan A\,\pm\,\tan B}{1\,\mp\,\tan A\tan B}$
• derive and use the double angle formulae for $\sin2A$, $\cos2A$ and $\tan2A$
  • $\sin2A=2\sin A\cos A$
  • $\cos2A=\cos^2A-\sin^2A$
    $\qquad\quad=2\cos^2A-\sin^2A$
    $\qquad\quad=1-2\sin^2A$
  • $\tan2A=\frac{2\tan A}{1-\tan^2A}$
• derive and use expressions for $\sin A$, $\cos A$ and $\tan A$ in terms of $t$ where $t=\tan\frac{A}{2}$ (the $t$-formulae)
  • $\sin A=\frac{2t}{1+t^2}$
  • $\cos A=\frac{1-t^2}{1+t^2}$
  • $\tan A=\frac{2t}{1-t^2}$
• derive and use the formulae for trigonometric products as sums and differences for $\cos A\cos B$, $\sin A\sin B$, $\sin A\cos B$ and $\cos A\sin B$
  • $\cos A\cos B=\frac{1}{2}\left[\cos(A-B)+\cos(A+B)\right]$
  • $\sin A\sin B=\frac{1}{2}\left[\cos(A-B)-\cos(A+B)\right]$
  • $\sin A\cos B=\frac{1}{2}\left[\sin(A+B)+\sin(A-B)\right]$
  • $\cos A\sin B=\frac{1}{2}\left[\sin(A+B)-\sin(A-B)\right]$

ME-T3 Trigonometric EquationsYear 12

• convert expressions of the form $a\cos x+b\sin x$ to $R\cos(x\pm\alpha)$ or $R\sin(x\pm\alpha)$ and apply these to solve equations of the form $a\cos x+b\sin x=c$, sketch graphs and solve related problems
• solve trigonometric equations requiring factorising and/or the application of compound angle, double angle formulae or the $t$-formulae
• prove and apply other trigonometric identities, for example $\cos3x=4\cos^3x-3\cos x$
• solve trigonometric equations and interpret solutions in context using technology or otherwise

Calculus

ME-C1 Rates of ChangeYear 11

C1.1: Rates of change with respect to time

• describe the rate of change of a physical quantity with respect to time as a derivative
  • investigate examples where the rate of change of some aspect of a given object with respect to time can be modelled using derivatives
  • use appropriate language to describe rates of change, for example 'at rest', 'initially', 'change of direction' and 'increasing at an increasing rate'
• find and interpret the derivative $\dfrac{dQ}{dt}$, given a function in the form $Q=f(t)$, for the amount of a physical quantity present at time $t$
• describe the rate of change with respect to time of the displacement of a particle moving along the $x$-axis as a derivative $\frac{dx}{dt}$ or $\dot{x}$
• describe the rate of change with respect to time of the velocity of a particle moving along the $x$-axis as a derivative $\frac{d^2x}{dt^2}$ or $\ddot{x}$

C1.2: Exponential growth and decay

• construct, analyse and manipulate an exponential model of the form $N(t)=Ae^{kt}$ to solve a practical growth or decay problem in various contexts (for example population growth, radioactive decay or depreciation)
  • establish the simple growth model, $\frac{dN}{dt}=kN$, where $N$ is the size of the physical quantity, $N=N(t)$ at time $t$ and $k$ is the growth constant
  • verify (by substitution) that the function $N(t)=Ae^{kt}$ satisfies the relationship $\frac{dN}{dt}=kN$, with $A$ being the initial value of $N$
  • sketch the curve $N(t)=Ae^{kt}$ for positive and negative values of $k$
  • recognise that this model states that the rate of change of a quantity varies directly with the size of the quantity at any instant
• establish the modified exponential model, $\frac{dN}{dt}=k(N-P)$, for dealing with problems such as 'Newton's Law of Cooling' or an ecosystem with a natural 'carrying capacity'
  • verify (by substitution) that a solution to the differential equation $\frac{dN}{dt}=k(N-P)$ is $N(t)=P+Ae^{kt}$, for an arbitrary constant $A$, and $P$ a fixed quantity, and that the solution is $N=P$ in the case when $A=0$
  • sketch the curve $N(t)=P+Ae^{kt}$ for positive and negative values of $k$
  • note that whenever $k<0$, the quantity $N$ tends to the limit $P$ as $t\to\infty$, irrespective of the initial conditions
  • recognise that this model states that the rate of change of a quantity varies directly with the difference in the size of the quantity and a fixed quantity at any instant
• solve problems involving situations that can be modelled using the exponential model or the modified exponential model and sketch graphs appropriate to such problems

C1.3: Related rates of change

• solve problems involving related rates of change as instances of the chain rule
• develop models of contexts where a rate of change of a function can be expressed as a rate of change of a composition of two functions, and to which the chain rule can be applied

ME-C2 Further Calculus SkillsYear 12

• find and evaluate indefinite and definite integrals using the method of integration by substitution, using a given substitution
  • change an integrand into an appropriate form using algebra
• prove and use the identities $\sin^2nx=\frac{1}{2}(1-\cos2nx)$ and $\cos^2nx=\frac{1}{2}(1+\cos2nx)$ to solve problems
• solve problems involving $\int\sin^2nx\,dx$ and $\int\cos^2nxdx$
• find derivatives of inverse functions by using the relationship $\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}$
• solve problems involving the derivatives of inverse trigonometric functions
• integrate expressions of the form $\frac{1}{\sqrt{a^2-x^2}}$ or $\frac{a}{a^2+x^2}$

ME-C3 Applications of CalculusYear 12

C3.1: Further area and volumes of solids of revolution

• calculate area of regions between curves determined by functions
• sketch, with and without the use of technology, the graph of a solid of revolution whose boundary is formed by rotating an arc of a function about the $x$-axis or $y$-axis
• calculate the volume of a solid of revolution formed by rotating a region in the plane about the $x$-axis or $y$-axis, with and without the use of technology
• determine the volumes of solids of revolution that are formed by rotating the region between two curves about either the $x$-axis or $y$-axis in both real-life and abstract contexts

C3.2: Differential equations

• recognise that an equation involving a derivative is called a differential equation
• recognise that solutions to differential equations are functions and that these solutions may not be unique
• sketch the graph of a particular solution given a direction field and initial conditions
  • form a direction field (slope field) from simple first-order differential equations
  • recognise the shape of a direction field from several alternatives given the form of a differential equation, and vice versa
  • sketch several possible solution curves on a given direction field
• solve simple first-order differential equations
  • solve differential equations of the form $\frac{dy}{dx}=f(x)$
  • solve differential equations of the form $\frac{dy}{dx}=g(y)$
  • solve differential equations of the form $\frac{dy}{dx}=f(x)g(y)$ using separation of variables
• recognise the features of a first-order linear differential equation and that exponential growth and decay models are first-order linear differential equations, with known solutions
• model and solve differential equations including to the logistic equation that will arise in situations where rates are involved, for example in chemistry, biology and economics

Combinatorics

ME-A1 Working with CombinatoricsYear 11

A1.1: Permutations and combinations

• list and count the number of ways an event can occur
• use the fundamental counting principle (also known as the multiplication principle)
• use factorial notation to describe and determine the number of ways $n$ different items can be arranged in a line or a circle
  • solve problems involving cases where some items are not distinct (excluding arrangements in a circle)
• solve simple problems and prove results using the pigeonhole principle
  • understand that if there are $n$ pigeonholes and $n+1$ pigeons to go into them, then at least one pigeonhole must hold 2 or more pigeons
  • generalise to: If $n$ pigeons are sitting in $k$ pigeonholes, where $n>k$, then there is at least one pigeonhole with at least $\frac{n}{k}$ pigeons in it
  • prove the pigeonhole principle
• understand and use permutations to solve problems
  • understand and use the notation ${}^nP_r$ and the formula ${}^nP_r=\frac{n!}{(n-r)!}$
• solve problems involving permutations and restrictions with or without repeated objects
• understand and use combinations to solve problems
  • understand and use the notations $\begin{pmatrix}n\\r\end{pmatrix}$ and ${}^nC_r$ and the formula ${}^nC_r=\frac{n!}{r!(n-r)!}$
• solve practical problems involving permutations and combinations, including those involving simple probability situations

A1.2: The binomial expansion and Pascal's triangle

• expand $(x+y)^n$ for small positive integers $n$
  • note the pattern formed by the coefficients of $x$ in the expansion of $(1+x)^n$ and recognise links to Pascal’s triangle
  • recognise the numbers $\begin{pmatrix}n\\r\end{pmatrix}$ (as denoted ${}^nC_r$) as binomial coefficients
• derive and use simple identities associated with Pascal’s triangle
  • establish combinatorial proofs of the Pascal’s triangle relations ${}^nC_0=1$, ${}^nC_n=1$; ${}^nC_r={}^{n-1}C_{r-1}+{}^{n-1}C_r$ for $1\leq r\leq n-1$; and ${}^nC_r={}^nC_{n-r}$

Proof

ME-P1 Proof by Mathematical InductionYear 12

• understand the nature of inductive proof, including the 'initial statement' and the inductive step
• prove results using mathematical induction
  • prove results for sums, for example $1+4+9+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for any positive integer $n$
  • prove divisibility results, for example $3^{2n}-1$ is divisible by 8 for any positive integer $n$
• identify errors in false 'proofs by induction', such as cases where only one of the required two steps of a proof by induction is true, and understand that this means that the statement has not been proved
• recognise situations where proof by mathematical induction is not appropriate

Vectors

ME-V1 Introduction to VectorsYear 12

V1.1: Introduction to vectors

• define a vector as a quantity having both magnitude and direction, and examine examples of vectors, including displacement and velocity
  • explain the distinction between a position vector and a displacement (relative) vector
• define and use a variety of notations and representations for vectors in two dimensions
  • use standard notations for vectors, for example: $\utilde{a}$, $\overrightarrow{AB}$, $\textbf{a}$
  • represent vectors graphically in two dimensions as directed line segments
  • define unit vectors as vectors of magnitude 1, and the standard two-dimensional perpendicular unit vectors $\utilde{i}$ and $\utilde{j}$
  • express and use vectors in two dimensions in a variety of forms, including component form, ordered pairs and column vector notation
• perform addition and subtraction of vectors and multiplication of a vector by a scalar algebraically and geometrically, and interpret these operations in geometric terms
  • graphically represent a scalar multiple of a vector
  • use the triangle law and the parallelogram law to find the sum and difference of two vectors
  • define and use addition and subtraction of vectors in component form
  • define and use multiplication by a scalar of a vector in component form

V1.2: Further operations with vectors

• define, calculate and use the magnitude of a vector in two dimensions and use the notation $\left|\utilde{u}\right|$ for the magnitude of a vector $\utilde{u}=x\utilde{i}+y\utilde{j}$
  • prove that the magnitude of a vector, $\utilde{u}=x\utilde{i}+y\utilde{j}$, can be found using: $\left|\utilde{u}\right|=\left|x\utilde{i}+y\utilde{j}\right|=\sqrt{x^2+y^2}$
  • identify the magnitude of a displacement vector $\overrightarrow{AB}$ as being the distance between the points $A$ and $B$
  • convert a non-zero vector $\utilde{u}$ into a unit vector $\utilde{\hat{u}}$ by dividing by its length: $\utilde{\hat{u}}=\frac{\utilde{u}}{\left|\utilde{u}\right|}$
• define and use the direction of a vector in two dimensions
• define, calculate and use the scalar (dot) product of two vectors $\utilde{u}=x_1\utilde{i}+y_1\utilde{j}$ and $\utilde{v}=x_2\utilde{i}+y_2\utilde{j}$
  • apply the scalar product, $\utilde{u}\cdot\utilde{v}$, to vectors expressed in component form, where $\utilde{u}\cdot\utilde{v}=x_1x_2+y_1y_2$
  • use the expression for the scalar (dot) product, $\utilde{u}\cdot\utilde{v}=\left|\utilde{u}\right|\left|\utilde{v}\right|\cos\theta$ where $\theta$ is the angle between vectors $\utilde{u}$ and $\utilde{v}$ to solve problems
  • demonstrate the equivalence, $\utilde{u}\cdot\utilde{v}=\left|\utilde{u}\right|\left|\utilde{v}\right|\cos\theta=x_1x_2+y_1y_2$ and use this relationship to solve problems
  • establish and use the formula $\utilde{v}\cdot\utilde{v}=\left|\utilde{v}\right|^2$
  • calculate the angle between two vectors using the scalar (dot) product of two vectors in two dimensions
• examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular
• define and use the projection of one vector onto another
• solve problems involving displacement, force and velocity involving vector concepts in two dimensions
• prove geometric results and construct proofs involving vectors in two dimensions including to proving that:
  • the diagonals of a parallelogram meet at right angles if and only if it is a rhombus
  • the midpoints of the sides of a quadrilateral join to form a parallelogram
  • the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides

V1.3: Projectile motion

• understand the concept of projectile motion, and model and analyse a projectile’s path assuming that:
  • the projectile is a point
  • the force due to air resistance is negligible
  • the only force acting on the projectile is the constant force due to gravity, assuming that the projectile is moving close to the Earth’s surface
• model the motion of a projectile as a particle moving with constant acceleration due to gravity and derive the equations of motion of a projectile
  • represent the motion of a projectile using vectors
  • recognise that the horizontal and vertical components of the motion of a projectile can be represented by horizontal and vertical vectors
  • derive the horizontal and vertical equations of motion of a projectile
  • understand and explain the limitations of this projectile model
• use equations for horizontal and vertical components of velocity and displacement to solve problems on projectiles
• apply calculus to the equations of motion to solve problems involving projectiles

Statistical Analysis

ME-S1 The Binomial DistributionYear 12

S1.1: Bernoulli and binomial distributions

• use a Bernoulli random variable as a model for two-outcome situations
  • identify contexts suitable for modelling by Bernoulli random variables
• use Bernoulli random variables and their associated probabilities to solve practical problems
  • understand and apply the formulae for the mean, $E(X)=\bar{x}=p$, and variance, $\text{Var}(X)=p(1-p)$, of the Bernoulli distribution with parameter $p$, and $X$ defined as the number of successes
• understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’ in $n$ independent Bernoulli trials, with the same probability of success $p$ in each trial
  • calculate the expected frequencies of the various possible outcomes from a series of Bernoulli trials
• use binomial distributions and their associated probabilities to solve practical problems
  • identify contexts suitable for modelling by binomial random variables
  • identify the binomial parameter $p$ as the probability of success
  • understand and use the notation $X\sim\text{Bin}(n,p)$ to indicate that the random variable $X$ is distributed binomially with parameters $n$ and $p$
  • apply the formulae for probabilities $P(X=r)={}^nC_rp^r(1-p)^{n-r}$ associated with the binomial distribution with parameters $n$ and $p$ and understand the meaning of ${}^nC_r$ as the number of ways in which an outcome with $r$ successes can occur
  • understand and apply the formulae for the mean, $E(X)=\bar{x}=np$, and the variance, $\text{Var}(X)=np(1-p)$, of a binomial distribution with parameters $n$ and $p$

S1.2: Normal approximation for the sample proportion

• use appropriate graphs to explore the behaviour of the sample proportion on collected or supplied data
  • understand the concept of the sample proportion $\hat{p}$ as a random variable whose value varies between samples
• explore the behaviour of the sample proportion using simulated data
  • examine the approximate normality of the distribution of $\hat{p}$ for large samples
• understand and use the normal approximation to the distribution of the sample proportion and its limitations