The document provides revision notes on various mathematics topics including: 1) Binomial expansions, partial fractions, trigonometry formulas, and techniques for integration like volumes of revolution. 2) Parametric equations, vectors, vector equations, planes, and differential equations. 3) Details are given for solving problems involving these topics, such as using compound angle formulas, rewriting algebraic fractions as partial fractions, and separating variables to solve first-order differential equations.

1. C4 Revision Notes

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.

4. • e.g.
• E.g

5. Trigonometry
Reciprocal trig functions:
• ; ;
• ;
Compound-angle formulae
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6. Double-angle formulae
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7. The r, α formulae
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8. Small-angle approximations
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Keep Θ in radians

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
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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.

16. Example