A-level Mathematics/AQA/MFP2

Roots of polynomials

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

Square root of minus one

${\displaystyle {\sqrt {-1}}=i\,\!}$

${\displaystyle i^{2}=-1\,\!}$

Square root of any negative real number

${\displaystyle {\sqrt {-2}}={\sqrt {2\times -1}}={\sqrt {2}}\times {\sqrt {-1}}={\sqrt {2}}\times i=i{\sqrt {2}}\,\!}$

${\displaystyle {\sqrt {-n}}=i{\sqrt {n}}\,\!}$

General form of a complex number

${\displaystyle z=x+iy\,\!}$

where ${\displaystyle x\,\!}$ and ${\displaystyle y\,\!}$ are real numbers

Modulus of a complex number

${\displaystyle |z|={\sqrt {x^{2}+y^{2}}}\,\!}$

Argument of a complex number

The argument of ${\displaystyle z\,\!}$ is the angle between the positive x-axis and a line drawn between the origin and the point in the complex plane (see [1])

${\displaystyle \tan {\theta }={\frac {y}{x}}\,\!}$

${\displaystyle \arg {z}=\theta \,\!}$

${\displaystyle \arg {z}=\tan ^{-1}{\left({\frac {y}{x}}\right)}\,\!}$

Polar form of a complex number

${\displaystyle x+iy=z=|z|e^{i\theta }=\left({\sqrt {x^{2}+y^{2}}}\right)e^{i\theta }\,\!}$

${\displaystyle e^{i\theta }=\cos {\theta }+i\sin {\theta }\,\!}$

${\displaystyle z=|z|e^{i\theta }=|z|\left(\cos {\theta }+i\sin {\theta }\right)\,\!}$

${\displaystyle e^{i\theta }={\frac {z}{|z|}}={\frac {x+iy}{\sqrt {x^{2}+y^{2}}}}\,\!}$

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

In general, if ${\displaystyle z_{1}=a_{1}+ib_{1}}$ and ${\displaystyle z_{2}=a_{2}+ib_{2}}$,

${\displaystyle z_{1}+z_{2}=(a_{1}+a_{2})+i(b_{1}+b_{2})}$
${\displaystyle z_{1}-z_{2}=(a_{1}-a_{2})+i(b_{1}-b_{2})}$
${\displaystyle z_{1}z_{2}=a_{1}a_{2}-b_{1}b_{2}+i(a_{2}b_{1}+a_{1}b_{2})}$

Complex conjugates

${\displaystyle {\mbox{If }}z=x+iy{\mbox{, then }}z^{*}=x-iy\,\!}$

${\displaystyle zz^{*}=|z|^{2}\,\!}$

Division of complex numbers of the form x + iy

${\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {z_{1}}{z_{2}}}{\frac {z_{2}^{*}}{z_{2}^{*}}}={\frac {z_{1}z_{2}^{*}}{|z_{2}|^{2}}}}$

Products and quotients of complex numbers in their polar form

If ${\displaystyle z_{1}=(r_{1},{\mbox{ }}\theta _{1})}$ and ${\displaystyle z_{2}=(r_{2},{\mbox{ }}\theta _{2})}$ then ${\displaystyle z_{1}z_{2}=(r_{1}r_{2},{\mbox{ }}\theta _{1}+\theta _{2})}$, with the proviso that ${\displaystyle 2\pi }$ may have to be added to, or subtracted from, ${\displaystyle \theta _{1}+\theta _{2}}$ if ${\displaystyle \theta _{1}+\theta _{2}}$ is outside the permitted range for ${\displaystyle \theta }$.

If ${\displaystyle z_{1}=(r_{1},{\mbox{ }}\theta _{1})}$ and ${\displaystyle z_{2}=(r_{2},{\mbox{ }}\theta _{2})}$ then ${\displaystyle {\frac {z_{1}}{z_{2}}}=\left({\frac {r_{1}}{r_{2}}},{\mbox{ }}\theta _{1}-\theta _{2}\right)}$, with the same proviso regarding the size of the angle ${\displaystyle \theta _{1}-\theta _{2}}$.

Equating real and imaginary parts

${\displaystyle {\mbox{If }}a+ib=c+id{\mbox{, where }}a{\mbox{, }}b{\mbox{, }}c{\mbox{ and }}d{\mbox{ are real, then }}a=c{\mbox{ and }}b=d\,\!}$

Coordinate geometry on Argand diagrams

If the complex number ${\displaystyle z_{1}}$ is represented by the point ${\displaystyle A}$, and the complex number ${\displaystyle z_{2}}$ is represented by the point ${\displaystyle B}$ in an Argand diagram, then ${\displaystyle |z_{2}-z_{1}|=AB\,\!}$, and ${\displaystyle \arg {(z_{2}-z_{1})}}$ is the angle between ${\displaystyle {\overrightarrow {AB}}}$ and the positive direction of the x-axis.

Loci on Argand diagrams

${\displaystyle |z|=k}$ represents a circle with centre ${\displaystyle O}$ and radius ${\displaystyle k}$

${\displaystyle |z-z_{1}|=k}$ represents a circle with centre ${\displaystyle z_{1}}$ and radius ${\displaystyle k}$

${\displaystyle |z-z_{1}|=|z-z_{2}|}$ represents a straight line — the perpendicular bisector of the line joining the points ${\displaystyle z_{1}}$ and ${\displaystyle z_{2}}$

${\displaystyle {\mbox{arg }}z=\alpha }$ represents the half line through ${\displaystyle O}$ inclined at an angle ${\displaystyle \alpha }$ to the positive direction of ${\displaystyle Ox}$

${\displaystyle {\mbox{arg}}(z-z_{1})=\alpha }$ represents the half line through the point ${\displaystyle z_{1}}$ inclined at an angle ${\displaystyle \alpha }$ to the positive direction of ${\displaystyle Ox}$

De Moivre's theorem and its applications

De Moivre's theorem

${\displaystyle \left(\cos {\theta }+i\sin {\theta }\right)^{n}=\cos {n\theta }+i\sin {n\theta }\,\!}$

De Moivre's theorem for integral n

${\displaystyle z+{\frac {1}{z}}=2\cos {\theta }}$

${\displaystyle z-{\frac {1}{z}}=2i\sin {\theta }}$

Exponential form of a complex number

${\displaystyle {\mbox{If }}z=r(\cos {\theta }+i\sin {\theta }){\mbox{, }}\,\!}$

${\displaystyle {\mbox{then }}z=re^{i\theta }\,\!}$

${\displaystyle {\mbox{and }}z^{n}=\left(re^{i\theta }\right)^{n}=r^{n}e^{ni\theta }\,\!}$

${\displaystyle \cos {\theta }={\frac {e^{i\theta }+e^{-i\theta }}{2}}}$

${\displaystyle \sin {\theta }={\frac {e^{i\theta }-e^{-i\theta }}{2i}}}$

The cube roots of unity

The cube roots of unity are ${\displaystyle 1}$, ${\displaystyle w}$ and ${\displaystyle w^{2}}$, where

${\displaystyle w^{3}=1\,\!}$

${\displaystyle 1+w+w^{2}=0\,\!}$

and the non-real roots are

${\displaystyle {\frac {-1\pm i{\sqrt {3}}}{2}}}$

The nth roots of unity

The equation ${\displaystyle z^{n}=1}$ has roots

${\displaystyle z=e^{\frac {2k\pi i}{n}}{\mbox{ where }}k=0,1,2,\dots ,(n-1)}$

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

The equation ${\displaystyle z^{n}=\alpha }$, where ${\displaystyle \alpha =re^{i\theta }}$, has roots

${\displaystyle z=r^{\frac {1}{n}}e^{\frac {i(\theta +2k\pi )}{n}}{\mbox{ where }}k=0,1,2,\dots ,(n-1)}$

Hyperbolic functions

Definitions of hyperbolic functions

${\displaystyle \sinh {x}={\frac {e^{x}-e^{-x}}{2}}}$

${\displaystyle \cosh {x}={\frac {e^{x}+e^{-x}}{2}}}$

${\displaystyle \tanh {x}={\frac {\sinh {x}}{\cosh {x}}}}$

${\displaystyle \operatorname {cosech} {x}={\frac {1}{\sinh {x}}}}$

${\displaystyle \operatorname {sech} ={\frac {1}{\cosh {x}}}}$

${\displaystyle \coth {x}={\frac {1}{\tanh {x}}}}$

Hyperbolic identities

${\displaystyle \cosh ^{2}{x}-\sinh ^{2}{x}=1\,\!}$

${\displaystyle 1-\tanh ^{2}{x}=\operatorname {sech} ^{2}{x}\,\!}$

${\displaystyle \coth ^{2}{x}-1=\operatorname {cosech} ^{2}{x}\,\!}$

${\displaystyle \sinh {(x+y)}=\sinh {x}\cosh {y}+\cosh {x}\sinh {y}\,\!}$

${\displaystyle \cosh {(x+y)}=\cosh {x}\cosh {y}+\sinh {x}\sinh {y}\,\!}$

Double angle formulae

${\displaystyle \sinh {2x}=2\sinh {x}\cosh {y}\,\!}$

{\displaystyle {\begin{aligned}\cosh {2x}&=\cosh ^{2}{x}+\sinh ^{2}{x}\\&=2\cosh ^{2}{x}-1\\&=1+2\sinh ^{2}{x}\end{aligned}}\,\!}

Osborne's rule

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

${\displaystyle {\frac {d}{dx}}\sinh {x}=\cosh {x}}$

${\displaystyle {\frac {d}{dx}}\cosh {x}=\sinh {x}}$

${\displaystyle {\frac {d}{dx}}\tanh {x}=\operatorname {sech} ^{2}{x}}$

${\displaystyle {\frac {d}{dx}}\sinh {kx}=k\cosh {kx}}$

${\displaystyle {\frac {d}{dx}}\cosh {kx}=k\sinh {kx}}$

${\displaystyle {\frac {d}{dx}}\tanh {kx}=k\operatorname {sech} ^{2}{kx}}$

Integration of hyperbolic functions

${\displaystyle \int \sinh {x}\,dx=\cosh {x}+c}$

${\displaystyle \int \cosh {x}\,dx=\sinh {x}+c}$

${\displaystyle \int \operatorname {sech} ^{2}{x}\,dx=\tanh {x}+c}$

${\displaystyle \int \tanh {x}\,dx=\ln {\cosh {x}}+c}$

${\displaystyle \int \coth {x}\,dx=\ln {\sinh {x}}+c}$

Inverse hyperbolic functions

Logarithmic form of inverse hyperbolic functions

${\displaystyle \sinh ^{-1}{x}=\ln {\left(x+{\sqrt {x^{2}+1}}\right)}}$

${\displaystyle \cosh ^{-1}{x}=\ln {\left(x+{\sqrt {x^{2}-1}}\right)}}$

${\displaystyle \tanh ^{-1}{x}={\frac {1}{2}}\ln {\left({\frac {1+x}{1-x}}\right)}}$

Derivatives of inverse hyperbolic functions

${\displaystyle {\frac {d}{dx}}\sinh ^{-1}{x}={\frac {1}{\sqrt {1+x^{2}}}}}$

${\displaystyle {\frac {d}{dx}}\cosh ^{-1}{x}={\frac {1}{\sqrt {x^{2}-1}}}}$

${\displaystyle {\frac {d}{dx}}\tanh ^{-1}{x}={\frac {1}{1-x^{2}}}}$

${\displaystyle {\frac {d}{dx}}\sinh ^{-1}{\frac {x}{a}}={\frac {1}{\sqrt {a^{2}+x^{2}}}}}$

${\displaystyle {\frac {d}{dx}}\cosh ^{-1}{\frac {x}{a}}={\frac {1}{\sqrt {x^{2}-a^{2}}}}}$

${\displaystyle {\frac {d}{dx}}\tanh ^{-1}{\frac {x}{a}}={\frac {1}{a^{2}-x^{2}}}}$

Integrals which integrate to inverse hyperbolic functions

${\displaystyle \int {\frac {1}{\sqrt {a^{2}+x^{2}}}}\,dx=\sinh ^{-1}{\frac {x}{a}}+c}$

${\displaystyle \int {\frac {1}{\sqrt {x^{2}-a^{2}}}}\,dx=\cosh ^{-1}{\frac {x}{a}}+c}$

${\displaystyle \int {\frac {1}{a^{2}-x^{2}}}\,dx=\tanh ^{-1}{\frac {x}{a}}+c}$

Arc length and area of surface of revolution

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

${\displaystyle s=\int _{x_{1}}^{x_{2}}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}dx=\int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}dt}$

${\displaystyle S=2\pi \int _{x_{1}}^{x_{2}}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}dx=2\pi \int _{t_{1}}^{t_{2}}y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}dt}$