Further Maths Pure

Question 1

what is the determinant of a 2x2 matrix?

Answer 1

ad-bc

Question 2

what is the determinant of a 3x3 matrix?

Answer 2

a(ei-fh)-b(di-fg)+c(dh-ge)

Question 3

what is the formula for volume of revolution around the axes?

Answer 3

around x-axis: π∫y²dx

around y-axis: π∫x²dy

Question 4

if M is a 2x2 matrix, what is inverse M?

Answer 4

swap a and d
times b and c by -1
times by 1/detM

Question 5

what is (AB)^-1 equal to?

Answer 5

B^-1 * A^-1

Question 6

how do you invert a 3x3 matrix?

Answer 6

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

Question 7

what is consistent/ inconsistent?

Answer 7

consistent is when there is a set of values that satisfies all equations (at least one place where they all cross)

Question 8

what are invariant lines/points?

Answer 8

lines or points that stay in the same place after a transformation

Question 9

what is the dot product and cross product formulas for two lines?

Answer 9

a.b=|a| |b| cosθ

axb=|a| |b| sinθ |n̂|

Question 10

what is the dot product formula for two planes?

Answer 10

n₁.n₂=|n₁| |n₂| cosθ

Question 11

what is the dot product formula for a line and a plane?

Answer 11

b.n=|b| |n| sinθ

Question 12

what is r.n̂ equal to?

Answer 12

the shortest distance from the origin to the plane.

Question 13

what is the vector form of an imaginary number?

Answer 13

r(cosθ+isinθ)

Question 14

how do you work out the matrix of a rotation around an axis when the matrix is 3x3?

Answer 14

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

Question 15

what is the equation of area before and after and determinant?

Answer 15

area before = area after / |determinant|

Question 16

what is L’Hospital’s Rule and when can it be used?

Answer 16

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)

Question 17

how does volumes of revolution work with parametric equations?

Answer 17

V = π∫y² dx/dt dt
V = π∫x² dy/dt dt
with the parameters as a value for t not x or y

Question 18

how do you find the tangent parallel or perpendicular to the initial line of a polar equation?

Answer 18

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.

Question 19

what is the equation for sinh and cosh?

Answer 19

cosh(x) = 1/2 (e^x + e^-x)
sinh(x) = 1/2 (e^x - e^-x)

Question 20

how do you solve an equation in the form:

dy/dx + f(x)*y = g(x)

Answer 20

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

Question 21

when a question in core pure has acceleration/ velocity/displacement in an equation, what will the question most likely be asking you to do?

Answer 21

Solving a first or second order differential equation.

Question 22

what are the two equations that apply to simple harmonic motion?

Answer 22

ẍ = -ω²x where ω is angular velocity
ẍ = v dv/dx
where the dots are order of differential in terms of t

Question 23

what information can be obtained from the equation:

ẍ + kẋ + ω²x = 0 if the type of damping is known?

Answer 23

heavy damping: k²-4ω² > 0
critical damping: k²-4ω² = 0
light damping: k²-4ω² < 0

Question 24

what is the area of a triangle and a parallelogram and volume of a tetrahedron and parallelpiped using vectors?

Answer 24

triangle area = 1/2 |axb|
parallelogram area = |axb|
tetrahedron volume = 1/6 |a.(bxc)|
parallelepiped volume = |a.(bxc)|

Question 25

what is the scalar triple product and what is it used for?

Answer 25

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) |

Question 26

how can the shortest distance between two skew lines be calculated? R1 = a + λb, R2 = c +μd

Answer 26

| ( (a-c)·(bxd) )/ |bxd| |

Question 27

what is eccentricity?

Answer 27

constant ratio of distances between the focus and directrix and a point on the line.

Question 28

what is t equal to and what is sin, cos and tan in terms of t?

Answer 28

``` t = tan(x/2) sinx = 2t/ (1+t^2) cosx = (1-t^2)/(1+t^2) tanx = 2t/(1-t^2) ```

Question 29

what is the Maclaurin and Taylor series formulae?

Answer 29

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

Question 30

what is Leibnitz's formula?

Answer 30

dⁿy/dxⁿ = Σ nCk dᵏy/dxᵏ * dⁿ⁻ᵏy/dxⁿ⁻ᵏ

Question 31

how do you find the Nth root of a number?

Answer 31

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

Question 32

what is Simpson's Rule? what does it do? when can it be used?

Answer 32

∫f(x) ≈ h/3 (y₀+ yₙ+ 4ΣY.odd+ 2ΣY.even) estimates an integral (5 y values is 4 intervals) only works with even intervals

Question 33

what are the three methods for finding a general solution?

Answer 33

``` separating the variables (first order) integrating factor (first order) lambda method (second order) ```

Question 34

how do you prove that sin(nθ) or cos(nθ) equals something?

Answer 34

start with sin(nθ) is the imaginary part of | ( cos(θ)+isin(θ) )^n or the non-imaginary part for cos

Question 35

how do you prove that (sinθ)^n equals something?

Answer 35

make z = cosθ +isinθ and use the (z±1/z) = 2cosθ or 2isinθ. then put both sides to the power of n

Question 36

what is the formula for volumes of revolution?

Answer 36

V = π∫y^2 dx