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#### Unit 1: Combinatorics, vectors and proof

##### Topic 1: Combinatorics

The inclusion–exclusion principle for the union of two sets and three sets

Unit 1: Combinatorics, vectors and proof > Topic 1: Combinatorics > The inclusion–exclusion principle for the union of two sets and three sets

- Determine and use the formulas (including the addition principle) for finding the number of elements in the union of two and the union of three sets

- Use the multiplication principle

Permutations (ordered arrangements) and combinations (unordered selections)

Unit 1: Combinatorics, vectors and proof > Topic 1: Combinatorics > Permutations (ordered arrangements) and combinations (unordered selections)

- Solve problems involving permutations

- Use factorial notation

- Use the notation nPr

- Solve problems involving permutations with restrictions

- Solve problems involving combinations

- Use the notation nCr

- Derive and use simple identities associated with Pascal's triangle

- Solve problems involving combinations with restrictions

- Apply permutations and combinations to probability problems

The pigeon-hole principle

Unit 1: Combinatorics, vectors and proof > Topic 1: Combinatorics > The pigeon-hole principle

- Solve problems and prove results using the pigeon-hole principle

##### Topic 2: Vectors in the plane

Representing vectors in the plane by directed line segments

Unit 1: Combinatorics, vectors and proof > Topic 2: Vectors in the plane > Representing vectors in the plane by directed line segments

- Examine examples of vectors

- Understand the difference between a scalar and a vector

- Define and use the magnitude and direction of a vector

- Understand and use vector equality

- Understand and use both the Cartesian form and polar form of a vector

- Represent a scalar multiple of a vector

- Use the triangle rule to find the sum and difference of two vectors

Algebra of vectors in the plane

Unit 1: Combinatorics, vectors and proof > Topic 2: Vectors in the plane > Algebra of vectors in the plane

- Use ordered pair notation and column vector notation to represent a vector

- Understand and use vector notation

- Convert between Cartesian form and polar form

- Determine a vector between two points

- Define and use unit vectors and the perpendicular unit vectors i and j

- Express a vector in component form using the unit vectors i and j

- Examine and use addition and subtraction of vectors in component form

- Define and use multiplication by a scalar of a vector in component form

- Define and use a vector representing the midpoint of a line segment

- Define and use scalar (dot) product

- Apply the scalar product to vectors expressed in component form

- Examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular

- Define and use projections of vectors

- Solve problems involving displacement, force, velocity, equilibrium and relative velocity involving the above concepts

##### Topic 3: Introduction to proof

The nature of proof

Unit 1: Combinatorics, vectors and proof > Topic 3: Introduction to proof > The nature of proof

- Use implication, converse, equivalence, negation, contrapositive

- Use proof by contradiction

- Use the symbols for implication, equivalence, and equality

- Use the quantifiers ‘for all' and ‘there exists'

- Use examples and counterexamples

Rational and irrational numbers

Unit 1: Combinatorics, vectors and proof > Topic 3: Introduction to proof > Rational and irrational numbers

- Prove simple results involving numbers

- Express rational numbers as terminating or eventually recurring decimals and vice versa

- Prove irrationality by contradiction

Circle properties and their proofs

Unit 1: Combinatorics, vectors and proof > Topic 3: Introduction to proof > Circle properties and their proofs

- Prove circle properties, such as an angle in a semicircle is a right angle, the angle at the centre subtended by an arc of a circle is twice the angle at the circumference subtended by the same arc, angles at the circumference of a circle subtended by the same arc are equal, the opposite angles of a cyclic quadrilateral are supplementary, chords of equal length subtend equal angles at the centre and conversely chords subtending equal angles at the centre of a circle have the same length, a tangent drawn to a circle is perpendicular to the radius at the point of contact, the alternate segment theorem, when two chords of a circle intersect, the product of the lengths of the intervals on one chord equals the product of the lengths of the intervals on the other chord and its converse, when a secant (meeting the circle at A and B) and a tangent (meeting the circle at T) are drawn to a circle from an external point M, the square of the length of the tangent equals the product of the lengths to the circle on the secant and its converse

- Solve problems finding unknown angles and lengths and prove further results using the circle properties listed above

Geometric proofs using vectors

Unit 1: Combinatorics, vectors and proof > Topic 3: Introduction to proof > Geometric proofs using vectors

- Prove the diagonals of a parallelogram meet at right angles if and only if it is a rhombus

- Prove midpoints of the sides of a quadrilateral join to form a parallelogram

- Prove the sum of the squares of the lengths of a parallelogram's diagonals is equal to the sum of the squares of the lengths of the sides

- Prove an angle in a semicircle is a right angle

#### Unit 2: Complex numbers, trigonometry, functions and matrices

##### Topic 1: Complex numbers 1

Complex numbers

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 1: Complex numbers 1 > Complex numbers

- Define the imaginary number i as a root of the equation x^2 = -1

- Use complex numbers in the form a + bi where a and b are the real and imaginary parts

- Determine and use complex conjugates

- Perform complex-number arithmetic: addition, subtraction, multiplication and division

The complex plane (the Argand plane)

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 1: Complex numbers 1 > The complex plane (the Argand plane)

- Consider complex numbers as points in a plane with real and imaginary parts as Cartesian coordinates

- Examine and use addition of complex numbers as vector addition in the complex plane

- Understand and use location of complex conjugates in the complex plane

- Examine and use multiplication as a linear transformation in the complex plane

Complex arithmetic using polar form

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 1: Complex numbers 1 > Complex arithmetic using polar form

- Use the modulus of a complex number z and the argument Arg(z) of a non-zero complex number z

- Convert between Cartesian form and polar form

- Define and use multiplication, division and powers of complex numbers in polar form and the geometric interpretation of these

Roots of equations

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 1: Complex numbers 1 > Roots of equations

- Use the general solution of real quadratic equations

- Determine complex conjugate solutions of real quadratic equations

- Determine linear factors of real quadratic polynomials

##### Topic 2: Trigonometry and functions

The basic trigonometric functions

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 2: Trigonometry and functions > The basic trigonometric functions

- Find all solutions of f(a(x-b)) = c where f(?) is one of sin(?), cos(?) or tan(?)

- Sketch and graph functions with rules of the form y = f(a(x-b)) where f(?) is one of sin(?), cos(?) or tan(?)

Sketching graphs

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 2: Trigonometry and functions > Sketching graphs

- Use and apply the notation for the absolute value for the real number x and the graph of y = |x|

- Examine the relationship between the graph of y = f(|x|) and the graphs of y = 1/f(x), y = |f(x)| and y = f(|x|)

- Sketch the graphs of simple rational functions where the numerator and denominator are polynomials of low degree

The reciprocal trigonometric functions, secant, cosecant and cotangent

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 2: Trigonometry and functions > The reciprocal trigonometric functions, secant, cosecant and cotangent

- Define the reciprocal trigonometric functions, sketch their graphs, and graph simple transformations of them

Trigonometric identities

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 2: Trigonometry and functions > Trigonometric identities

- Prove and apply the Pythagorean identities

- Prove and apply the angle sum, difference and double-angle identities for sines and cosines

- Prove and apply the identities for products of sines and cosines expressed as sums and differences

- Convert sums acos(x) + bsin(x) to Rcos(x ± a) or Rsin(x ± a) and apply these to sketch graphs, solve equations of the form a cos(x)+ b sin(x) = c and solve real-world problems

- Use the binomial theorem to prove and apply multi-angle trigonometric identities up to sin(4x) and cos(4x)

Applications of trigonometric functions to model periodic phenomena

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 2: Trigonometry and functions > Applications of trigonometric functions to model periodic phenomena

- Model periodic motion using sine and cosine functions, and understand the relevance of the period and amplitude of these functions in the model

##### Topic 3: Matrices Matrix arithmetic

Matrix arithmetic

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 3: Matrices Matrix arithmetic > Matrix arithmetic

- Understand the matrix definition and notation

- Define and use addition and subtraction of matrices, scalar multiplication, matrix multiplication, multiplicative identity and multiplicative inverse

- Calculate the determinant and inverse of 2 x 2 matrices algebraically and solve matrix equations of the form AX = B, where A is a 2 x 2 matrix and X and B are column vectors

- Calculate the determinant and inverse of higher order matrices and solve matrix equations using technology

Transformations in the plane

Unit 2: Complex numbers, trigonometry, functions and matrices > Topic 3: Matrices Matrix arithmetic > Transformations in the plane

- Understand translations and their representation as column vectors

- Define and use basic linear transformations: dilations of the form (x, y) -> (ax, by), rotations about the origin and reflection in a line that passes through the origin, and the representations of these transformations by 2 x 2 matrices

- Apply these transformations to points in the plane and geometric objects

- Define and use composition of linear transformations and the corresponding matrix products

- Define and use inverses of linear transformations and the relationship with the matrix inverse

- Examine the relationship between the determinant and the effect of a linear transformation on area

- Establish geometric results by matrix multiplications

#### Unit 3: Mathematical induction, and further vectors, matrices and complex numbers

view_agenda query_stats##### Topic 1: Proof by mathematical induction

view_agenda query_statsMathematical induction

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 1: Proof by mathematical induction > Mathematical induction

##### Topic 2: Vectors and matrices

view_agenda query_statsThe algebra of vectors in three dimensions

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 2: Vectors and matrices > The algebra of vectors in three dimensions

Vector and Cartesian equations

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 2: Vectors and matrices > Vector and Cartesian equations

Systems of linear equations

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 2: Vectors and matrices > Systems of linear equations

Applications of matrices

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 2: Vectors and matrices > Applications of matrices

Vector calculus

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 2: Vectors and matrices > Vector calculus

##### Topic 3: Complex numbers 2

view_agenda query_statsCartesian forms

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 3: Complex numbers 2 > Cartesian forms

Complex arithmetic using polar form

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 3: Complex numbers 2 > Complex arithmetic using polar form

The complex plane (the Argand plane)

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 3: Complex numbers 2 > The complex plane (the Argand plane)

Roots of complex numbers

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 3: Complex numbers 2 > Roots of complex numbers

Factorisation of polynomials

view_agenda query_statsUnit 3: Mathematical induction, and further vectors, matrices and complex numbers > Topic 3: Complex numbers 2 > Factorisation of polynomials

#### Unit 4: Further calculus and statistical inference

view_agenda query_stats##### Topic 1: Integration and applications of integration

view_agenda query_statsIntegration techniques

view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 1: Integration and applications of integration > Integration techniques

Applications of integral calculus

view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 1: Integration and applications of integration > Applications of integral calculus

##### Topic 2: Rates of change and differential equations

view_agenda query_statsRates of change

view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 2: Rates of change and differential equations > Rates of change

Modelling motion

view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 2: Rates of change and differential equations > Modelling motion

##### Topic 3: Statistical inference

view_agenda query_statsSample means

view_agenda query_statsUnit 4: Further calculus and statistical inference > Topic 3: Statistical inference > Sample means

- Simulate repeated random sampling from a variety of distributions and a range of sample sizes to illustrate properties of the distribution of X^ across samples of a fixed size n, including its mean ?, its standard deviation ? /√n (where ? and ? are the mean and standard deviation of X^) and its approximate normality if n is large

Confidence intervals for means

view_agenda query_stats