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Extension 2 Syllabus

Proof

MEX-P1 The Nature of ProofYear 12

• use the formal language of proof, including the terms statement, implication, converse, negation and contrapositive
  • use the symbols for implication $(\implies)$, equivalence ($\iff$) and equality ($=$), demonstrating a clear understanding of the difference between them
  • use the phrases ‘for all’ ($\forall$), ‘if and only if’ (iff) and ‘there exists’ ($\exists$)
  • understand that a statement is equivalent to its contrapositive but that the converse of a true statement may not be true
• prove simple results involving numbers
• use proof by contradiction including proving the irrationality for numbers such as $\sqrt2$ and $\log_25$
• use examples and counter-examples
• prove results involving inequalities. For example:
  • prove inequalities by using the definition of $a>b$ for real $a$ and $b$
  • prove inequalities by using the property that squares of real numbers are non-negative
  • prove and use the triangle inequality $|x|+|y|\geq|x+y|$ and interpret the inequality geometrically
  • establish and use the relationship between the arithmetic mean and geometric mean for two non-negative numbers
• prove further results involving inequalities by logical use of previously obtained inequalities

MEX-P2 Further Proof by Mathematical InductionYear 12

• prove results using mathematical induction where the initial value of $n$ is greater than 1, and/or $n$ does not increase strictly by 1, for example prove that $n^2+2n$ is a multiple of 8 if $n$ is an even positive integer
• understand and use sigma notation to prove results for sums, for example:
  $\displaystyle{\sum_{n=1}^N\frac{1}{(2n+1)(2n-1)}=\frac{N}{2N+1}}$
• understand and prove results using mathematical induction, including inequalities and results in algebra, calculus, probability and geometry. For example:
  • prove inequality results, eg $2n>n^2$, for positive integers $n>4$
  • prove divisibility results, eg $3^{2n+4}-2^{2n}$ is divisible by 5 for any positive integer $n$
  • prove results in calculus, eg prove that for any positive integer $n$, $\frac{d}{dx}\left(x^n\right)=nx^{n-1}$
  • prove results related to probability, eg the binomial theorem:
    $\displaystyle{(x+a)^n=\sum_{r=0}^n {}^nC_rx^{n-r}a^r}$
  • prove geometric results, eg prove that the sum of the exterior angles of an $n$-sided plane convex polygon is 360°
• use mathematical induction to prove first-order recursive formulae

Vectors

MEX-V1 Further Work with VectorsYear 12

V1.1: Introduction to three-dimensional vectors

• understand and use a variety of notations and representations for vectors in three dimensions
  • define the standard unit vectors $\utilde{i}$, $\utilde{j}$ and $\utilde{k}$
  • express and use a vector in three dimensions in a variety of forms, including component form, ordered triples and column vector notation
• perform addition and subtraction of three-dimensional vectors and multiplication of three- dimensional vectors by a scalar algebraically and geometrically, and interpret these operations in geometric terms

V1.2: Further operations with three-dimensional vectors

• define, calculate and use the magnitude of a vector in three dimensions
  • establish that the magnitude of a vector in three dimensions can be found using:
    $\left|x\utilde{i}+y\utilde{j}+z\utilde{k}\right|=\sqrt{x^2+y^2+z^2}$
  • convert a non-zero vector $\utilde{u}$ into a unit vector $\hat{\utilde{u}}$ by dividing by its length: $\displaystyle{\hat{\utilde{u}}=\frac{\utilde{u}}{\left|\utilde{u}\right|}}$
• define and use the scalar (dot) product of two vectors in three dimensions
  • define and apply the scalar product $\utilde{u}\cdot\utilde{v}$ to vectors expressed in component form, where $\utilde{u}\cdot\utilde{v}=x_1x_2+y_1y_2+z_1z_2$, $\utilde{u}=x_1\utilde{i}+y_1\utilde{j}+z_1\utilde{k}$ and $\utilde{v}=x_2\utilde{i}+y_2\utilde{j}+z_2\utilde{k}$
  • extend the formula $\utilde{u}\cdot\utilde{v}=\left|\utilde{u}\right|\left|\utilde{v}\right|\cos\theta$ for three dimensions and use it to solve problems
• prove geometric results in the plane and construct proofs in three dimensions

V1.3: Vectors and vector equations of lines

• use Cartesian coordinates in two and three-dimensional space
• recognise and find the equations of spheres
• use vector equations of curves in two or three dimensions involving a parameter, and determine a corresponding Cartesian equation in the two-dimensional case, where possible
• understand and use the vector equation $\utilde{r}=\utilde{a}+\lambda\utilde{b}$ of a straight line through points $A$ and $B$ where $R$ is a point on $AB$ and $\utilde{a}=\overrightarrow{OA}$, $\utilde{b}=\overrightarrow{AB}$, $\lambda$ is a parameter and $\utilde{r}=\overrightarrow{OR}$
• make connections in two dimensions between the equation $\utilde{r}=\utilde{a}+\lambda\utilde{b}$ and $y=mx+c$
• determine a vector equation of a straight line or straight-line segment, given the position of two points or equivalent information, in two and three dimensions
• determine when two lines in vector form are parallel
• determine when intersecting lines are perpendicular in a plane or three dimensions
• determine when a given point lies on a given line in vector form

Complex Numbers

MEX-N1 Introduction to Complex NumbersYear 12

N1.1: Arithmetic of complex numbers

• use the complex number system
  • develop an understanding of the classification of numbers and their associated properties, symbols and representations
  • define the number, $i$ as a root of the equation $x^2=-1$
  • use the symbol $i$ to solve quadratic equations that do not have real roots
• represent and use complex numbers in Cartesian form
  • use complex numbers in the form $z=a+ib$, where $a$ and $b$ are real numbers and $a$ is the real part $\text{Re}(z)$ and $b$ is the imaginary part $\text{Im}(z)$ of the complex number
  • identify the condition for $z_1=a+ib$ and $z_2=c+id$ to be equal
  • define and perform complex number addition, subtraction and multiplication
  • define, find and use complex conjugates, and denote the complex conjugate of $z$ as $\bar{z}$
  • divide one complex number by another complex number and give the result in the form $a+ib$
  • find the reciprocal and two square roots of complex numbers in the form $z=a+ib$

N1.2: Geometric representation of a complex number

• represent and use complex numbers in the complex plane
  • use the fact that there exists a one-to-one correspondence between the complex number $z=a+ib$ and the ordered pair $(a,b)$
  • plot the point corresponding to $z=a+ib$
• represent and use complex numbers in polar or modulus-argument form, $\displaystyle{z=r(\cos\theta+i\sin\theta)}$, where $r$ is the modulus of $z$ and $\theta$ is the argument of $z$
  • define and calculate the modulus of a complex number $z=a+ib$ as $|z|=\sqrt{a^2+b^2}$
  • define and calculate the argument of a non-zero complex number $z=a+ib$ as $\displaystyle{\text{arg}(z)=\theta}$, where $\tan\theta=\frac{b}{a}$
  • define, calculate and use the principal argument $\text{Arg}(z)$ of a non-zero complex number $z$ as the unique value of the argument in the interval $(-\pi, \pi]$
• prove and use the basic identities involving modulus and argument
  • $\left|z_1z_2\right|=\left|z_1\right|\left|z_2\right|$ and $\text{arg}\left(z_1z_2\right)=\text{arg}\,z_1+\text{arg}\,z_2$
  • $\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}$ and $\text{arg}\left(\frac{z_1}{z_2}\right)=\text{arg}\,z_1-\text{arg}\,z_2, z_2\neq0$
  • $\left|z^n\right|=\left|z\right|^n$ and $\text{arg}\left(z^n\right)=n\,\text{arg}\,z$
  • $\bar{z_1}+\bar{z_2}=\overline{z_1+z_2}$
  • $\bar{z_1}\bar{z_2}=\overline{z_1z_2}$
  • $z\bar{z}=\left|z\right|^2$
  • $z+\bar{z}=2\text{Re}(z)$
  • $z-\bar{z}=2i\text{Im}(z)$

N1.3: Other representations of complex numbers

• understand Euler’s formula, $e^{ix}=\cos x+i\sin x$, for real $x$
• represent and use complex numbers in exponential form, $z=re^{i\theta}$, where $r$ is the modulus of $z$ and $\theta$ is the argument of $z$
• use Euler’s formula to link polar form and exponential form
• convert between Cartesian, polar and exponential forms of complex numbers
• find powers of complex numbers using exponential form
• use multiplication, division and powers of complex numbers in polar form and interpret these geometrically
• solve problems involving complex numbers in a variety of forms

MEX-N2 Using Complex NumbersYear 12

N2.1: Solving equations with complex numbers

• use De Moivre’s theorem with complex numbers in both polar and exponential form
  • prove De Moivre’s theorem for integral powers using proof by induction
  • use De Moivre’s theorem to derive trigonometric identities such as $\displaystyle{\sin3\theta=3\cos^2\theta\sin\theta-\sin^3\theta}$
• determine the solutions of real quadratic equations
• define and determine complex conjugate solutions of real quadratic equations
• determine conjugate roots for polynomials with real coefficients
• solve problems involving real polynomials with conjugate roots
• solve quadratic equations of the form $ax^2+bx+c=0$, where $a$, $b$, $c$ are complex numbers

N2.2: Geometrical implications of complex numbers

• examine and use addition and subtraction of complex numbers as vectors in the complex plane
  • given the points representing $z_1$ and $z_2$, find the position of the points representing ${z_1+z_2}$ and $z_1-z_2$
  • describe the vector representing $z_1+z_2$ or $z_1-z_2$ as corresponding to the relevant diagonal of a parallelogram with vectors representing $z_1$ and $z_2$ as adjacent sides
• examine and use the geometric interpretation of multiplying complex numbers, including rotation and dilation in the complex plane
• recognise and use the geometrical relationship between the point representing a complex number $z=a+ib$, and the points representing $\bar{z}$, $cz$ (where $c$ is real) and $iz$
• determine and examine the $n^\text{th}$ roots of unity and their location on the unit circle
• determine and examine the $n^\text{th}$ roots of complex numbers and their location in the complex plane
• solve problems using $n^\text{th}$ roots of complex numbers
• identify subsets of the complex plane determined by relations, for example $\left|z-3i\right|\leq4$, $\frac{\pi}{4}\leq\text{Arg}(z)\leq\frac{3\pi}{4}$, $\text{Re}(z)>\text{Im}(z)$ and $\left|z-1\right|=2\left|z-i\right|$

Calculus

MEX-C1 Further IntegrrationYear 12

• find and evaluate indefinite and definite integrals using the method of integration by substitution, where the substitution may or may not be given
• integrate rational functions involving a quadratic denominator by completing the square or otherwise
• decompose rational functions whose denominators have simple linear or quadratic factors, or a combination of both, into partial fractions
• use partial fractions to integrate functions
• evaluate integrals using the method of integration by parts
  • develop the method for integration by parts, expressed as $\int uv'\,dx=uv-\int vu'\,dx$ or $\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx$, where $u$ and $v$ are both functions of $x$
• derive and use recurrence relationships
• apply these techniques of integration to practical and theoretical situations

Mechanics

MEX-M1 Applications of Calculus to MechanicsYear 12

M1.1: Simple harmonic motion

• derive equations for displacement, velocity and acceleration in terms of time, given that a motion is simple harmonic and describe the motion modelled by these equations
  • establish that simple harmonic motion is modelled by equations of the form: $x=a\cos(nt+\alpha)+c$ or $x=a\sin(nt+\alpha)+c$, where $x$ is displacement from a fixed point, $a$ is the aimplitued, $\frac{2\pi}{n}$ is the period, $\frac{\alpha}{n}$ is the phase shift and $c$ is the central point of motion
  • establish that when a particle moves in simple harmonic motion about $c$, the central point of motion, then $\ddot{x}=-n^2(x-c)$
• prove that motion is simple harmonic when given an equation of motion for acceleration, velocity or displacement and describe the resulting motion
• sketch graphs of $x$, $\dot{x}$ and $\ddot{x}$ as functions of $t$ and interpret and describe features of the motion
• prove that motion is simple harmonic when given graphs of motion for acceleration, velocity or displacement and determine equations for the motion and describe the resulting motion
• derive $v^2=g(x)$ and the equations for velocity and displacement in terms of time when given $\ddot{x}=f(x)$ and initial conditions, and describe the resulting motion
• use relevant formulae and graphs to solve problems involving simple harmonic motion

M1.2: Modelling motion without resistance

• examine force, acceleration, action and reaction under constant and non-constant force
• examine motion of a body under concurrent forces
• consider and solve problems involving motion in a straight line with both constant and non-constant acceleration and derive and use the expressions $\frac{dv}{dt}$, $v\frac{dv}{dx}$ and $\frac{d}{dx}\left(\frac{1}{2}v^2\right)$ for acceleration
• use Newton’s laws to obtain equations of motion in situations involving motion other than projectile motion or simple harmonic motion
  • use $F=m\ddot{x}$ where $F$ is the force acting on a mass, $m$, with acceleration $\ddot{x}$
• describe mathematically the motion of particles in situations other than projectile motion and simple harmonic motion
  • interpret graphs of displacement-time and velocity-time to describe the motion of a particle, including the possible direction of a force which acts on the particle
• derive and use the equations of motion of a particle travelling in a straight line with both constant and variable acceleration

M1.3: Resisted motion

• solve problems involving resisted motion of a particle moving along a horizontal line
  • derive, from Newton’s laws of motion, the equation of motion of a particle moving in a single direction under a resistance proportional to a power of the speed
  • derive an expression for velocity as a function of time
  • derive an expression for velocity as a function of displacement
  • derive an expression for displacement as a function of time
  • solve problems involving resisted motion along a horizontal line
• solve problems involving the motion of a particle moving vertically (upwards or downwards) in a resisting medium and under the influence of gravity
  • derive, from Newton’s laws of motion, the equation of motion of a particle moving vertically in a medium, with a resistance 𝑅 proportional to the first or second power of its speed
  • derive an expression for velocity as a function of time and for velocity as a function of displacement (or vice versa)
  • derive an expression for displacement as a function of time
  • determine the terminal velocity of a falling particle from its equation of motion
  • solve problems by using the expressions derived for acceleration, velocity and displacement including obtaining the maximum height reached by a particle, and the time taken to reach this maximum height and obtaining the time taken for a particle to reach ground level when falling

M1.4: Projectiles and resisted motion

• solve problems involving projectiles in a variety of contexts
  • use parametric equations of a projectile to determine a corresponding Cartesian equation for the projectile
  • use the Cartesian equation of the trajectory of a projectile, including problems in which the initial speed and/or angle of projection may be unknown
• solve problems involving projectile motion in a resisting medium and under the influence of gravity which include consideration of the complete motion of a particle projected vertically upwards or at an angle to the horizontal