Level 3 International Foundation Diploma for Higher Education Studies (Part 7)

5. Syllabus

l. Further Mathematics

Title Further Mathematics
Unit reference number H/615/2415
Credits 10
Level 3

 

Guided Learning Hours 60 hours Total Qualification Time 100 hours

 

Learning Outcomes;
The Learner will:
Assessment Criteria;
The Learner can:
1. Understand different types of businesses and their functions Understand different techniques to solve cubic equations and write expressions in terms of their partial fractions 1.1 Find the quotient of a cubic equation
when divided by a linear factor, using
algebraic long division
1.2 Use the factor theorem to find roots of
cubic equations
1.3 Convert rational functions into their
partial fractions
1.4 Express improper fractions as partial
fractions
2. Be able to work with complex
numbers, perform arithmetic
calculations using complex numbers,
solve higher order polynomials with
complex roots and sketch regions in
the complex plane
2.1 Solve simple quadratic equations with
complex roots by completing the square or using the quadratic formula
2.2 Represent complex numbers on an
Argand diagram
2.3 Add, subtract, multiply and divide
complex numbers
2.4 Calculate the modulus and argument
of a complex number
2.5 Solve polynomial equations with real
coefficients and complex roots, appreciating that such roots occur in conjugate pairs
2.6 Identify regions on Argand diagrams
showing the area that represents solutions to inequalities involving complex numbers
3. Be able to perform arithmetic
operations using matrices, understand basic transformations
using matrices and, in addition,
understand which matrices represent linear transformations and calculate the inverse of a matrix
3.1 Add and subtract matrices of the same
dimension
3.2 Perform matrix multiplication, demonstrating an understanding of
non-commutativity and associativity
3.3 Find the image of points in the x-y
plane under given matrix transformations
3.4 State whether a given transformation is a linear transformation and describe a
transformation in terms of its effect on a column vector in two dimensions
3.5 Find the 2×2 matrix which represents a
given linear transformation or find the linear transformation represented by a given matrix
3.6 Use matrix products to find matrices
that represent combinations of two
transformations
3.7 Calculate the determinant of a 2×2
matrix and find the inverse if it exists
4. Understand the properties of rational functions and understand conic sections 4.1 Sketch the basic shape of quadratics,
cubics, quartics, trigonometric functions and reciprocals, and understand the effect transformations have on the equations
4.2 Sketch rational functions with a linear
numerator and denominator, finding asymptotes and points of intersections
with coordinate axes
4.3 Sketch rational functions with two
distinct linear factors in the denominator and repeated factors in the denominator
4.4 Find stationary points on the graphs of
rational functions
4.5 Recognise the standard equations of
parabolas, ellipses and hyperbolas in both Cartesian and parametric form and sketch the given equations, understanding the effects of transformations on the given equations
4.6 Find the Cartesian equations of
parabolas, given their focus and directrix
4.7 Find the coordinates of the focus and
an equation for the directrix of a parabola
5. Understand how to use sigma
notation to calculate the sum of
simple finite series, and appreciate
the relationship between the roots of
polynomials and their coefficients
5.1 Use the sigma notation, ∑ , to calculate
the sum of simple finite series
5.2 Use the formula for the sum of the first
n natural numbers, and the sum of the
squares and cubes of the first n natural
numbers
5.3 Use the method of differences to find
the sum of a series
5.4 Find the sum and product of the roots
of a quadratic equation, and derive a quadratic equation given information
about its roots
6. Understand further techniques in
calculus to differentiatecombinations
of functions, how to use these
techniques to solve problems
involving functions given parametrically and how to derive
Maclaurin and Taylor series
6.1 Use the chain rule, product rule or
quotient rule to differentiate functions
6.2 Covert parametric equations into
Cartesian form
6.3 Differentiate a curve whose equation is
given parametrically
6.4 Find the equations of tangents and
normals of curves whose equations are
given parametrically
6.5 Use the chain and product rule to find
second, third and higher order
derivatives
6.6 Derive and find the Maclaurin
expansion of a given function in
ascending powers of x
6.7 Derive and use Taylor’s series to
expand a given function in ascending
powers of x
7. Understand further trigonometry and hyperbolic functions 7.1 Solve problems involving trigonometric
identities
7.2 Understand and use compound angle
formulae
7.3 Understand and use the double angle
formulae
7.4 Write down the definitions of the hyperbolic functions, including the reciprocal hyperbolic functions
7.5 Sketch the graphs of the main hyperbolic functions, including the reciprocal hyperbolic functions
7.6 Solve equations using hyperbolic
functions
8. Understand Euler’s relation and De
Moivre’s theorem and derive relations
between trigonometric functions and
hyperbolic functions
8.1 Calculate the product and quotient of
two complex numbers in polar coordinate form
8.2 Derive Euler’s relation and write
complex numbers in exponential form
8.3 Derive de Moivre’s theorem and obtain
formulae for sinnθ and cosnθ in terms of sinθ and cosθ
8.4 Use the exponential form of a complex
number to derive relations between
trigonometric functions and hyperbolic
functions

 

Syllabus Content
Topic Course coverage
Cubic
Polynomials &
Partial
Fractions
  • Products of polynomials and equating coefficients
  • Algebraic long division
  • Factor theorem
  • Factorising cubic polynomials
  • Expressing rational functions in terms of their partial fractions, given:
    (a) Two linear factors in the denominator
    (b) A repeated root
  • How to express improper algebraic fractions in terms of their partial fractions

Learning Outcome: 1

Complex
Numbers I
  • Completing the square of quadratic trinomials
  • An introduction to complex numbers
  • Solving quadratic equations with complex roots
  • Representing complex numbers on an Argand diagram

Learning Outcome: 2

Complex
Numbers II
  • The modulus-argument form of a complex number
  • Solve further problems involving complex numbers
  • Solve polynomial equations with real coefficients
  • Loci in the complex plane
  • Inequalities with complex numbers

Learning Outcome: 2

Matrices
  • An introduction to matrices including performing basic operations on
    matrices
  • Properties of matrix multiplication including non-commutativity and
    associativity
  • Finding and using the inverse of a matrix when it exists
  • Linear transformations

Learning Outcome: 3

Graphs of
Rational
Functions
  • Sketching the basic shape of quadratics, cubics, trigonometric functions and reciprocals, understanding the effects of transformations of these graphs
  • Sketching rational functions with linear numerators and
    denominators, calculating any asymptotes
  • Finding any turning points on graphs of rational functions without
    using calculus

Learning Outcome: 4

Series
  • Calculating basic arithmetic series
  • Use of sigma notation to calculate the sum of given series
  • Use of the formula for the sum of the first n natural numbers
    (including squares and cubes)
  • Method of differences

Learning Outcome: 5

Further
Calculus
Techniques I
  • Further techniques in differentiation of more complex rational functions
  • Use of the chain rule, the product rule and quotient rule
  • An introduction to trigonometric identities and techniques to differentiate the trigonometric functions and their reciprocals

Learning Outcome: 6

Further
Calculus
Techniques II &
Maclaurin and
Taylor Series
  • Binomial series expansion for (1 + 𝑥) ^𝑛
  • Use of the chain and product rule to find second, third and higher order derivatives
  • Maclaurin series expansion of a given function in ascending powersof 𝑥
  • Taylor’s series to expand a given function in ascending powers of 𝑥

Learning Outcome: 6

Trigonometric
Identities &
Hyperbolic
Functions
  • Solving trigonometric equations including solving problems using trigonometric identities
  • Definitions of hyperbolic functions and their graphs
  • Osborn’s rule
  • Differentiating hyperbolic functions
  • Solving equations involving hyperbolic functions

Learning Outcome: 7

Euler’s Relation
and De
Moivre’s
Theorem
  • Compound angle identities
  • Products and quotients of complex numbers in polar form
  • Exponential form of complex numbers and Euler’s formula
  • De Moivre’s theorem
  • Relationships between trigonometric and hyperbolic functions

Learning Outcome: 8

Parametric
Equations
  • Drawing equations given parametrically by plotting points on the graph
  • Converting functions between their Cartesian form and parametric form
  • Differentiating curves given in parametric form
  • Tangents and normals to curves given parametrically
  • The second derivative

Learning Outcomes: 6

Coordinate
Systems
  • An introduction to conic sections
  • The parabola and its transformations, including finding the equation of the parabola given its focus and directrix
  • The ellipse and its transformations
  • The hyperbola and its transformations

Learning Outcome: 4

 

Assessment Type 
  • Global Assignment (100%)
See also Section 3 above

 

m. Physics

Title Physics
Unit reference number K/615/2416
Credits 10
Level 3

 

Guided Learning Hours 48 hours Total Qualification Time 100 hours

 

Learning Outcomes;
The Learner will:
Assessment Criteria;
The Learner can:
1. Understand the mechanics of motion 1.1 Define and explain the relationships of
displacement, velocity and acceleration
1.2 Calculate average and instantaneous
velocity and acceleration
1.3 Solve problems involving equations of
motion
1.4 Demonstrate the use of motion
equations for non-constant acceleration
1.5 Describe the motion of objects in free
fall and calculate their position and velocity
1.6 Explain the importance of circular
motion
2. Understand the mechanics of forces 2.1 Explain the concept of force and how it
causes change in motion
2.2 State and apply Newton’s three laws of
motion
2.3 Apply Newton’s laws in onedimensional and circular motion
2.4 Describe the conditions and calculate
the forces necessary for equilibrium
3. Understand the mechanics of energy 3.1 Explain the meaning of work and find
out the work done by constant forces
3.2 Evaluate the work done by variable
forces with position
3.3 Define the concept of kinetic energy
and state its relation to work
3.4 Find out the relation between energy
and power
3.5 Define potential energy and calculate it
dependent on conservative force as a
function of position
4. Understand the mechanics of
momentum
4.1 Explain the principle of momentum and
conservation of momentum
4.2 Describe the difference between
inelastic and elastic collisions
4.3 Find out the centre of mass for
individual particles
4.4 Calculate rotational kinetic energy
5. Understand the mechanics of periodic motion 5.1 Explain the simple harmonic oscillator
5.2 Determine the maximum speed of an
oscillator system
5.3 Measure the acceleration of a simple
pendulum due to gravity
6. Understand the basic principles of
thermal physics
6.1 Explain the meaning of temperature
and heat
6.2 Describe the three phases of matter
and find out the energies for phase
change
6.3 Calculate thermal expansion effects in
solids, liquids and gases
6.4 State the first law of thermodynamics
and explain how thermal energy is involved in the conservation of energy principle
6.5 Describe the effects of thermodynamic
processes
6.6 Define the specific heat of an ideal gas
6.7 Explain the second law of thermodynamics and its limitations
6.8 Calculate the efficiencies of heat
engines and refrigerators
6.9 Explain the meaning of, or calculate,
entropy
7. Understand the fundamentals of
electrostatics
7.1 Examine the behaviour of electric
charge using Coulomb’s law
7.2 Explain the meaning of, or calculate,
an electric field
7.3 Explain Gauss’s law for electric fields
7.4 Explain the concept of electric potential
difference
7.5 Calculate the potential difference
between two points in a simple electric
field
7.6 Calculate the electric potential for a
point in the electric field of a point charge
7.7 Describe charge distribution on
conductors
7.8 Explain the concept of capacitance
7.9 Find out the capacitance of a parallel
plate capacitor
7.10 Calculate the equivalent capacitance of a combination of capacitors consisting
of parallel and series capacitors
7.11 Demonstrate how dielectrics make
capacitors more effective
8. Understand the fundamentals of
electrodynamics
8.1 Describe electric current and current
density
8.2 Describe electrical resistance
8.3 Relate electrical current, voltage and
resistance using Ohm’s law
8.4 Calculate electric power
8.5 Draw a circuit with resistors in parallel
and in series
8.6 Explain the reason why the total
resistance of a parallel circuit is less than smallest resistance of any of the resistors in the circuit
8.7 Analyse a complex circuit using
Kirchhoff’s rules
8.8 State the main functions of voltmeters
and ammeters
9. Understand the fundamentals of
magnetism
9.1 Describe the meaning of magnetic
field, magnetic field lines and magnetic
flux
9.2 Calculate the motion of a charged
particle in a magnetic field
9.3 Explain the relation between magnetic
fields and magnetic forces
9.4 Calculate the magnetic field of a
moving charge
9.5 Calculate the magnetic field of a
current element
9.6 Calculate the force between parallel
conductors
9.7 Understand Ampere’s law
9.8 Calculate a magnetic field using
Ampere’s law
9.9 Explain electromagnetic induction
9.10 Calculate an induced electric field
using Faraday’s law

 

Syllabus Content
Topic Course coverage
Motion
  • Definition of kinematics and dynamics
  • Displacement, time, velocity and acceleration
  • Equations of motion
  • Non-uniform motion
  • Free falling bodies and projectile motion
  • Circular motion

Learning Outcome: 1

Forces
  • Types of forces
  • Newton’s first law
  • Newton’s second law
  • Newton’s third law
  • Newton’s second law applied in circular motion
  • Equilibrium

Learning Outcome: 2

Work and energy
  • Work and kinetic energy
  • The work-energy conservation law
  • Power
  • Potential energy

Learning Outcome: 3

Momentum and
collisions
  • Linear momentum
  • Conservation of momentum
  • Collisions
  • Elastic collisions
  • Inelastic collisions
  • Centre of mass frame
  • Rotational kinetic energy

Learning Outcome: 4

Periodic motion
  • Simple harmonic motion
  • Total energy of a harmonic oscillator
  • Importance of simple harmonic motion
  • Motion of a simple pendulum

Learning Outcome: 5

Thermal physics
  • Temperature and heat
  • Thermal properties of matter
  • The first law of thermodynamics
  • The second law of thermodynamics

Learning Outcome: 6

Electrostatics I
  • Electric charge and Coulomb’s law
  • Electric field
  • Charge and electric flux
  • Gauss’s law

Learning Outcome: 7

Electrostatics II
  • Electric potential
  • Conductors, capacitors and capacitance
  • Capacitors in series and parallel connection
  • Dielectrics

Learning Outcome: 7

Electrodynamics I
  • Electric current
  • Resistivity and resistance
  • Electromotive force in electric circuits
  • Energy and power in electric circuits

Learning Outcomes: 8

Electrodynamics II
  • Direct current circuits
  • Resistors in series and parallel
  • Kirchhoff’s laws
  • Electrical measuring instruments

Learning Outcome: 8

Magnetism I
  • Reasons for interpreting statements
  • Magnetic field, magnetic field lines and magnetic flux
  • Motion of a charged particle in a magnetic field
  • Magnetic force on a current-carrying conductor
  • Magnetic field of a moving charge
  • Magnetic field of a current element

Learning Outcomes: 9

Magnetism II
  • Magnetic field of a current-carrying conductor
  • Force between parallel conductors
  • Ampere’s law
  • Induction and Faraday’s law
  • Induced electric field

Learning Outcome: 9

 

Assessment Type 
  • Global Assignment (100%)
See also Section 3 above

 

Download: Level 3 International Foundation Diploma for Higher Education Studies Pdf: Here

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