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]

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


 i^2 = -1 \,\!

Square root of any negative real number[edit]

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


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

General form of a complex number[edit]

 z = x + i y \,\!

where  x \,\! and  y \,\! are real numbers

Modulus of a complex number[edit]

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

Argument of a complex number[edit]

The argument of  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])

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

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

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

Polar form of a complex number[edit]

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


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


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


 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[edit]

In general, if  z_1 = a_1 + ib_1 and  z_2 = a_2 + ib_2 ,

 z_1 + z_2 = (a_1 + a_2) + i(b_1 + b_2)
 z_1 - z_2 = (a_1 - a_2) + i(b_1 - b_2)
 z_1 z_2 = a_1 a_2 - b_1 b_2 + i(a_2 b_1 + a_1 b_2)

Complex conjugates[edit]

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


 zz^* = |z|^2 \,\!

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

 \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[edit]

If  z_1 = (r_1,\mbox{ } \theta_1) and  z_2 = (r_2,\mbox{ } \theta_2) then  z_1 z_2 = (r_1 r_2,\mbox{ } \theta_1+\theta_2) , with the proviso that  2 \pi may have to be added to, or subtracted from,  \theta_1 + \theta_2 if  \theta_1 + \theta_2 is outside the permitted range for  \theta .

If  z_1 = (r_1,\mbox{ } \theta_1) and  z_2 = (r_2,\mbox{ } \theta_2) then  \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  \theta_1 - \theta_2 .

Equating real and imaginary parts[edit]

\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[edit]

If the complex number z_1 is represented by the point A, and the complex number z_2 is represented by the point B in an Argand diagram, then |z_2 - z_1| = AB \,\!, and \arg{(z_2 - z_1)} is the angle between \overrightarrow{AB} and the positive direction of the x-axis.

Loci on Argand diagrams[edit]

|z| = k represents a circle with centre O and radius k

|z-z_1| = k represents a circle with centre z_1 and radius k

|z-z_1| = |z-z_2| represents a straight line — the perpendicular bisector of the line joining the points z_1 and z_2

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

\mbox{arg}(z-z_1) = \alpha represents the half line through the point z_1 inclined at an angle \alpha to the positive direction of Ox

De Moivre's theorem and its applications[edit]

De Moivre's theorem[edit]

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

De Moivre's theorem for integral n[edit]

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


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

Exponential form of a complex number[edit]

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


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


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


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


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

The cube roots of unity[edit]

The cube roots of unity are 1, w and w^2, where

 w^3 = 1 \,\!

 1 + w + w^2 = 0 \,\!

and the non-real roots are

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


The nth roots of unity[edit]

The equation  z^n = 1 has roots

 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[edit]

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

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

Hyperbolic functions[edit]

Definitions of hyperbolic functions[edit]

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


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


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


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


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


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

Hyperbolic identities[edit]

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


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


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

Addition formulae[edit]

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


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

Double angle formulae[edit]

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


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

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]

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


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


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



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


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


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

Integration of hyperbolic functions[edit]

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


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


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



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


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

Inverse hyperbolic functions[edit]

Logarithmic form of inverse hyperbolic functions[edit]

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


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


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

Derivatives of inverse hyperbolic functions[edit]

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


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


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



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


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


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

Integrals which integrate to inverse hyperbolic functions[edit]

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


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


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

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]

 s = \int^{x_2}_{x_1} \sqrt{ 1 + \left ( \frac{dy}{dx} \right )^2 } dx = \int^{t_2}_{t_1} \sqrt{ \left ( \frac{dx}{dt} \right )^2 + \left ( \frac{dy}{dt} \right )^2 }  dt


 S = 2 \pi \int^{x_2}_{x_1} y \sqrt{ 1 + \left ( \frac{dy}{dx} \right )^2 } dx = 2 \pi \int^{t_2}_{t_1} y \sqrt{ \left ( \frac{dx}{dt} \right )^2 + \left ( \frac{dy}{dt} \right )^2 }  dt

Further reading[edit]

The AQA's free textbook [2]