Core Pure 2 Flashcards
What is Eulers relation
eiθ = cos θ + isin θ
How to write a complex number in exponential form
z = reiθ
θ = arg z
r = |z|
What is de Moivre’s theorem
(r(cosθ + isinθ))n = rn(cos nθ + i sin nθ)
How to find the mean value of a function f(x) over the interval [a,b]
1/(b-a) ∫ba f(x) dx
d/dx (arcsin x) =
= 1/ √(1-x2)
d/dx (arccos x) =
-1 / √ (1-x2)
d/dx (arctanx) =
= 1 / 1+x2
i3 = ?
-i
i4 = ?
1
i2 = ?
-1
For a complex number z = a+bi, what is the complex conjugate
z* = a-bi
If the roots of a quadratic equation are α and β, what are 2 ways you can write the equation in terms of α and β
What is the relationship between α and β
0 = (z-α)(z-β)
or
0 = z2 - (α+β)z + αβ
α and β are a conjugate pair
Where z = a+bi
z + z* = ?
2a
Where z = a+bi
zz* = ?
a2 + b2
For a cubic equation, describe the possible combinations of real and imaginary roots
3 real roots
or
2 imaginary roots(complex conjugate pair) and 1 real root
For a quartic equation, describe the possible combinations of real and imaginary roots
4 real roots
2 real roots, 2 imaginary roots (complex conjugate pair)
4 imaginary roots (2 complex conjugate pairs)
What is an argand diagram
A diagram used to show complex numbers
y axis = Im
x axis = Re
What is the argument of a complex number (arg z)
The angle made between the line joining the number to the origin and the x axis
It is measured in the anitclockwise direction
For a complex no. z with |z| = r and arg z = θ, what is the modulus argument form
z = r(cos θ + isin θ)
arg (z1 / z2) = ?
arg (z1z2) = ?
arg (z1 / z2) = arg z1 - arg z2
arg (z1z2) = arg z1 + arg z2
Given that z1 = x1 + iy1 , what is the locus of point z, such that |z-z1| = r
A circle of centre (x1,y1) and radius r
Given that z1 = x1 + iy1 and z2 = x2 + iy2 ,what is the locus of points z on an argand diagram such that
|z-z1| = |z-z2|
The perpendicular bisector of the line segment joining z1 and z2
Given that z1 = x1 + iy1 ,what is the locus of points z such that arg(z-z1) = θ
A half line (line from a point that extends infinitely in 1 direction) from the fixed point z1 ,making an angle θ with a line from z1 ,parallel to the real axis