Further Maths Pure Flashcards
what is the determinant of a 2x2 matrix?
ad-bc
what is the determinant of a 3x3 matrix?
a(ei-fh)-b(di-fg)+c(dh-ge)
what is the formula for volume of revolution around the axes?
around x-axis: π∫y²dx
around y-axis: π∫x²dy
if M is a 2x2 matrix, what is inverse M?
swap a and d
times b and c by -1
times by 1/detM
what is (AB)^-1 equal to?
B^-1 * A^-1
how do you invert a 3x3 matrix?
1: find matrix of minors (determinants of 2x2 matrices when row and column is crossed out)
2: multiply every other number by -1 (the ones in the middle diamond)
3: reflect all values through the leading diagonal (bottom left to top right)
4: multiply by 1/detM
what is consistent/ inconsistent?
consistent is when there is a set of values that satisfies all equations (at least one place where they all cross)
what are invariant lines/points?
lines or points that stay in the same place after a transformation
what is the dot product and cross product formulas for two lines?
a.b=|a| |b| cosθ
axb=|a| |b| sinθ |n̂|
what is the dot product formula for two planes?
n₁.n₂=|n₁| |n₂| cosθ
what is the dot product formula for a line and a plane?
b.n=|b| |n| sinθ
what is r.n̂ equal to?
the shortest distance from the origin to the plane.
what is the vector form of an imaginary number?
r(cosθ+isinθ)
how do you work out the matrix of a rotation around an axis when the matrix is 3x3?
for around x-axis: replace the 2x2 rotation matrix with the numbers that aren’t in the column or row of the x-value (bottom right 2x2)
same for y and z axes
what is the equation of area before and after and determinant?
area before = area after / |determinant|
what is L’Hospital’s Rule and when can it be used?
If lim x→a of f(x) = lim x→a of g(x) = 0 OR ± ∞ THEN
the lim x→a of f(x)/g(x) = lim x→a of f’(x)/g’(x)
how does volumes of revolution work with parametric equations?
V = π∫y² dx/dt dt
V = π∫x² dy/dt dt
with the parameters as a value for t not x or y
how do you find the tangent parallel or perpendicular to the initial line of a polar equation?
parallel: dy/dθ = 0
perpendicular: dx/dθ = 0
must be found by differentiating the equation for y and x after r is substituted out for θ.
what is the equation for sinh and cosh?
cosh(x) = 1/2 (e^x + e^-x) sinh(x) = 1/2 (e^x - e^-x)
how do you solve an equation in the form:
dy/dx + f(x)*y = g(x)
multiply by the integrating factor:
e^( ∫f(x) dx)
the left side will then be in the form left after differentiating by product rule.
integrate both sides and make y the subject.
DONT FORGET THE +C
when a question in core pure has acceleration/ velocity/displacement in an equation, what will the question most likely be asking you to do?
Solving a first or second order differential equation.
what are the two equations that apply to simple harmonic motion?
ẍ = -ω²x where ω is angular velocity
ẍ = v dv/dx
where the dots are order of differential in terms of t
what information can be obtained from the equation:
ẍ + kẋ + ω²x = 0 if the type of damping is known?
heavy damping: k²-4ω² > 0
critical damping: k²-4ω² = 0
light damping: k²-4ω² < 0
what is the area of a triangle and a parallelogram and volume of a tetrahedron and parallelpiped using vectors?
triangle area = 1/2 |axb|
parallelogram area = |axb|
tetrahedron volume = 1/6 |a.(bxc)|
parallelepiped volume = |a.(bxc)|
what is the scalar triple product and what is it used for?
a·(bxc) or determinant of a 3x3 matrix of rows a, b and c.
area of a tetrahedron = 1/6 | a·(bxc) |
area of a parallelpiped = | a·(bxc) |
how can the shortest distance between two skew lines be calculated?
R1 = a + λb, R2 = c +μd
( (a-c)·(bxd) )/ |bxd| |
what is eccentricity?
constant ratio of distances between the focus and directrix and a point on the line.
what is t equal to and what is sin, cos and tan in terms of t?
t = tan(x/2) sinx = 2t/ (1+t^2) cosx = (1-t^2)/(1+t^2) tanx = 2t/(1-t^2)
what is the Maclaurin and Taylor series formulae?
Taylor series: f(x)= f(a) + f’(a)x + f’‘(a)/2! x^2 + f’’‘(a)/3! x^3
near x=a. Maclaurin series is where a=0
what is Leibnitz’s formula?
dⁿy/dxⁿ = Σ nCk dᵏy/dxᵏ * dⁿ⁻ᵏy/dxⁿ⁻ᵏ
how do you find the Nth root of a number?
rearrange for zⁿ=a(cosα+isinα)
rⁿ[cos(θ+2πk/n) + isin(θ+2πk/n)] = a(cosα+isinα)
solve for k=0 to n-1
what is Simpson’s Rule?
what does it do?
when can it be used?
∫f(x) ≈ h/3 (y₀+ yₙ+ 4ΣY.odd+ 2ΣY.even)
estimates an integral
(5 y values is 4 intervals) only works with even intervals
what are the three methods for finding a general solution?
separating the variables (first order) integrating factor (first order) lambda method (second order)
how do you prove that sin(nθ) or cos(nθ) equals something?
start with sin(nθ) is the imaginary part of
( cos(θ)+isin(θ) )^n or the non-imaginary part for cos
how do you prove that (sinθ)^n equals something?
make z = cosθ +isinθ and use the (z±1/z) = 2cosθ or 2isinθ. then put both sides to the power of n
what is the formula for volumes of revolution?
V = π∫y^2 dx
