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C4 Revision Notes
Binomial expansions
• General binomial expansion, for           :




• When the series is infinite, this is only valid if |x|<1

• (a+x)n should be written as
Partial fractions
• A proper algebraic fraction with a denominator which factorises can be
  decomposed into a sum of partial fractions. Following forms should be
  used:




• e.g.
• e.g.




• E.g
Trigonometry
Reciprocal trig functions:
•                  ;              ;



•                        ;

Compound-angle formulae
•
•
•
•

•

•
Double-angle formulae
•



•



•



•



•
The r, α formulae
•

•

•

•
Small-angle approximations

•

•

•


    Keep Θ in radians
Parametric Equations
• To draw a graph from parametric equations, plot the points on the curve
  given by different values of the parameter.
• Eliminate the parameter to give the cartesian equation of the curve.
• Parametric equations of circles:

    - Circle centre (0,0) and radius r




    - Circle centre (a,b) and radius r




•
Techniques for integration
Volumes of revolution
• About the x axis:




• About the y axis:




• Trapezium rule, with n strips of width h:




Note that this gives an overestimation of the area under the curve.
Vectors
• Magnitude-direction form:            (2 dimensions)
                                                          measured
                                                        anticlockwise
• Component form:                                           from
                                                         horizontal




• Position vector      is from origin to point P

• Vector

• ‘r’ denotes position vector of a general point
Vector equations
• Vector equation of the line through A with direction u is given by:



• Vector equation through points A and B is given by:




• Equation of line through            in direction       is given by:




                                               Cartesian form
               Vector form
Angle between two vectors
• The angle between a and b is given by




• Where

                              in two dimensions



                                      in three dimensions
• Cartesian equation of a plane perpendicular to             is:




• Equation of the plane through the point with position vector a, and
  perpendicular to n, is given by

                                 (r-a).n=0.
Differential equations
• A differential equation involves derivatives such as




• First-order differential equation involves only a first derivative

• Some first-order differential equations can be solved by separating the
  variables

• In a general solution you leave the constant of integration in the
  solution, and in a particular solution you use additional information to
  calculate the constant of integration.
Example

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C4 EDEXCEL HELP

  • 2. Binomial expansions • General binomial expansion, for : • When the series is infinite, this is only valid if |x|<1 • (a+x)n should be written as
  • 3. Partial fractions • A proper algebraic fraction with a denominator which factorises can be decomposed into a sum of partial fractions. Following forms should be used: • e.g.
  • 5. Trigonometry Reciprocal trig functions: • ; ; • ; Compound-angle formulae • • • • • •
  • 7. The r, α formulae • • • •
  • 9. Parametric Equations • To draw a graph from parametric equations, plot the points on the curve given by different values of the parameter. • Eliminate the parameter to give the cartesian equation of the curve. • Parametric equations of circles: - Circle centre (0,0) and radius r - Circle centre (a,b) and radius r •
  • 10. Techniques for integration Volumes of revolution • About the x axis: • About the y axis: • Trapezium rule, with n strips of width h: Note that this gives an overestimation of the area under the curve.
  • 11. Vectors • Magnitude-direction form: (2 dimensions) measured anticlockwise • Component form: from horizontal • Position vector is from origin to point P • Vector • ‘r’ denotes position vector of a general point
  • 12. Vector equations • Vector equation of the line through A with direction u is given by: • Vector equation through points A and B is given by: • Equation of line through in direction is given by: Cartesian form Vector form
  • 13. Angle between two vectors • The angle between a and b is given by • Where in two dimensions in three dimensions
  • 14. • Cartesian equation of a plane perpendicular to is: • Equation of the plane through the point with position vector a, and perpendicular to n, is given by (r-a).n=0.
  • 15. Differential equations • A differential equation involves derivatives such as • First-order differential equation involves only a first derivative • Some first-order differential equations can be solved by separating the variables • In a general solution you leave the constant of integration in the solution, and in a particular solution you use additional information to calculate the constant of integration.