A-level Mathematics/AQA/MFP2

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Roots of polynomials[edit]

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 numbers[edit]

Square root of minus one[edit]


Square root of any negative real number[edit]


General form of a complex number[edit]

where and are real numbers

Modulus of a complex number[edit]

Argument of a complex number[edit]

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 number[edit]




Addition, subtraction and multiplication of complex numbers of the form x + iy[edit]

In general, if and ,

Complex conjugates[edit]


Division of complex numbers of the form x + iy[edit]

Products and quotients of complex numbers in their polar form[edit]

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 parts[edit]

Coordinate geometry on Argand diagrams[edit]

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 diagrams[edit]

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 applications[edit]

De Moivre's theorem[edit]

De Moivre's theorem for integral n[edit]


Exponential form of a complex number[edit]





The cube roots of unity[edit]

The cube roots of unity are , and , where

and the non-real roots are


The nth roots of unity[edit]

The equation has roots

The roots of zn = α where α is a non-real number[edit]

The equation , where , has roots

Hyperbolic functions[edit]

Definitions of hyperbolic functions[edit]






Hyperbolic identities[edit]



Addition formulae[edit]


Double angle formulae[edit]


Osborne's rule[edit]

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 functions[edit]







Integration of hyperbolic functions[edit]






Inverse hyperbolic functions[edit]

Logarithmic form of inverse hyperbolic functions[edit]



Derivatives of inverse hyperbolic functions[edit]







Integrals which integrate to inverse hyperbolic functions[edit]



Arc length and area of surface of revolution[edit]

Calculation of the arc length of a curve and the area of a surface using Cartesian or parametric coordinates[edit]


Further reading[edit]

The AQA's free textbook [2]