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 |
Learning Outcome: 1 |
| Complex Numbers I |
Learning Outcome: 2 |
| Complex Numbers II |
Learning Outcome: 2 |
| Matrices |
Learning Outcome: 3 |
| Graphs of Rational Functions |
Learning Outcome: 4 |
| Series |
Learning Outcome: 5 |
| Further Calculus Techniques I |
Learning Outcome: 6 |
| Further Calculus Techniques II & Maclaurin and Taylor Series |
Learning Outcome: 6 |
| Trigonometric Identities & Hyperbolic Functions |
Learning Outcome: 7 |
| Euler’s Relation and De Moivre’s Theorem |
Learning Outcome: 8 |
| Parametric Equations |
Learning Outcomes: 6 |
| Coordinate Systems |
Learning Outcome: 4 |
| Assessment Type |
|
| 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 |
Learning Outcome: 1 |
| Forces |
Learning Outcome: 2 |
| Work and energy |
Learning Outcome: 3 |
| Momentum and collisions |
Learning Outcome: 4 |
| Periodic motion |
Learning Outcome: 5 |
| Thermal physics |
Learning Outcome: 6 |
| Electrostatics I |
Learning Outcome: 7 |
| Electrostatics II |
Learning Outcome: 7 |
| Electrodynamics I |
Learning Outcomes: 8 |
| Electrodynamics II |
Learning Outcome: 8 |
| Magnetism I |
Learning Outcomes: 9 |
| Magnetism II |
Learning Outcome: 9 |
| Assessment Type |
|
| See also Section 3 above |
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