 # 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

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