A-level Mathematics/AQA/MFP2

Roots of polynomialsEdit

The relations between the roots and the coefficients of a polynomial equation; the occurrence of the non-real roots in conjugate pairs when the coefficents of the polynomials are real.

Complex numbersEdit

Square root of minus oneEdit

 

 

Square root of any negative real numberEdit

 

 

General form of a complex numberEdit

 

where   and   are real numbers

Modulus of a complex numberEdit

 

Argument of a complex numberEdit

The argument of   is the angle between the positive x-axis and a line drawn between the origin and the point in the complex plane (see [1])

 

 

 

Polar form of a complex numberEdit

 

 

 

 

Addition, subtraction and multiplication of complex numbers of the form x + iyEdit

In general, if   and  ,

 
 
 

Complex conjugatesEdit

 

 

Division of complex numbers of the form x + iyEdit

 

Products and quotients of complex numbers in their polar formEdit

If   and   then  , with the proviso that   may have to be added to, or subtracted from,   if   is outside the permitted range for  .

If   and   then  , with the same proviso regarding the size of the angle  .

Equating real and imaginary partsEdit

 

Coordinate geometry on Argand diagramsEdit

If the complex number   is represented by the point  , and the complex number   is represented by the point   in an Argand diagram, then  , and   is the angle between   and the positive direction of the x-axis.

Loci on Argand diagramsEdit

  represents a circle with centre   and radius  

  represents a circle with centre   and radius  

  represents a straight line — the perpendicular bisector of the line joining the points   and  

  represents the half line through   inclined at an angle   to the positive direction of  

  represents the half line through the point   inclined at an angle   to the positive direction of  

De Moivre's theorem and its applicationsEdit

De Moivre's theoremEdit

 

De Moivre's theorem for integral nEdit

 

 

Exponential form of a complex numberEdit

 

 

 

 

 

The cube roots of unityEdit

The cube roots of unity are  ,   and  , where

 

 

and the non-real roots are

 

The nth roots of unityEdit

The equation   has roots

 

The roots of zn = α where α is a non-real numberEdit

The equation  , where  , has roots

 

Hyperbolic functionsEdit

Definitions of hyperbolic functionsEdit

 

 

 

 

 

 

Hyperbolic identitiesEdit

 

 

 

Addition formulaeEdit

 

 

Double angle formulaeEdit

 

 

Osborne's ruleEdit

Osborne's rule states that:

to change a trigonometric function into its corresponding hyperbolic function, where a product of two sines appears, change the sign of the corresponding hyperbolic form

Note that Osborne's rule is an aide mémoire, not a proof.

Differentiation of hyperbolic functionsEdit

 

 

 

 

 

 

Integration of hyperbolic functionsEdit

 

 

 

 

 

Inverse hyperbolic functionsEdit

Logarithmic form of inverse hyperbolic functionsEdit

 

 

 

Derivatives of inverse hyperbolic functionsEdit

 

 

 

 

 

 

Integrals which integrate to inverse hyperbolic functionsEdit

 

 

 

Arc length and area of surface of revolutionEdit

Calculation of the arc length of a curve and the area of a surface using Cartesian or parametric coordinatesEdit

 

 

Further readingEdit

The AQA's free textbook [2]