A-level Further Maths

sum to n, r=1: n

1

sum to n, r=1: k

kn

sum to n, r=1: r

1/2n(n+1)

sum of the roots of a quadratic equation

A+B = -b/a

product of the roots of a quadratic equation

AB = c/a

sum of the roots of a cubic equation

A + B + Y = -b/a

sum of products of roots of a cubic

AB + BY + AY = c/a

Products of roots of a cubic equation

ABY = -d/a

reflection in y-axis

-1 0

0 1

reflection in x-axis

1 0

0 -1

reflection in y=x

0 1

1 0

reflection in y=-x

0 -1

-1 0

rotation θ anticlockwise about origin

cos(θ) -sin(θ)

sin(θ) cos(θ)

what does detM tell you about the transformation M

area scale factor

transformation A followed by transformation B

BA

reflection in x=0 plane

-1 0 0

0 1 0

0 0 1

reflection in y=0 plane

1 0 0

0 -1 0

0 0 1

reflection in z=0 plane

1 0 0

0 1 0

0 0 -1

rotation θ anticlockwise about x-axis

1 0 0

0 cosθ -sinθ

0 sinθ cosθ

rotation θ anticlockwise about y-axis

cosθ 0 sinθ

0 1 0

-sinθ 0 cosθ

rotation θ anticlockwise about z-axis

cosθ -sinθ 0

sinθ cosθ 0

0 0 1

square matrix

one with the same number of rows and columns

zero matrix

all elements are zero

identity matrix

ones down leading diagonal, zeros everywhere else. Multiplying doesnt change

associative law for multiplication

Ax(BxC) = (AxB)xC

singular matrix

determinant is zero, with no inverse

inverse of a 3x3

- determinant

- matrix of minors

- matrix of cofactors

-transpose it

- multiply in determinant

if determinant is zero...

the equations have no unique solutions.

consistent linear equations

have ATLEAST one set of values that satisfies all the equations. They ALL meet at ATLEAST 1 point

(could be unique [non-singular], could be infinite [singular])

3D unique solution geometrically

planes meet at a single point of intersection

3D dependent and consistent solution

happens when each equation is a linear combination of the other two, (including the constants).

This gives an infinite number of solutions, so planes form a sheaf.

Two types of 3D inconsistent solutions geometrically

two cases:

1. no solutions and the planes for a prism (Two planes are linear combinations of each other- not including constants, but one is not)

2. the planes are parallel (The planes are linear combinations of each other except for the RHS)

reflection in y = mx

θ = arctan(m) [angle between the line and the y axis]

cos(2θ) sin(2θ)

sin(2θ) -cos(2θ)

invariant points

a point which is mapped to itself by the transformation (for linear transformations, either the origin is the only one, or they lie on a straight line through the origin)

invariant lines

a line that is mapped to itself by the transformation

conclusion for proof by induction

if the result is true for n=k, it is true for n=k+1, as the result is true for n =1 , it must therefore be true for all natural numbers by induction

chi squared

(observed-expected)^2/expected

work energy principle

Net Work = Change in KE + Change in PE

Power

Work done/time or Driving force x Velocity

mod arg form of a complex number

z = a + ib, z = r(cos(θ) + isin(θ))

where r is modulus of z and θ is arg(z)

Multiplying mod arg form of two complex numbers

z = r(cos(θ) + isin(θ))

w = t(cos(∅) + isin(∅))

zw = rt(cos(θ+∅) + isin(θ+∅))

Dividing mod arg form of two complex numbers

z = r(cos(θ) + isin(θ))

w = t(cos(∅) + isin(∅))

z/w = r-t(cos(θ-∅) + isin(θ-∅))

Derivative of arcsin(x)

1/√(1−x²)

Derivative of arccos(x)

-1/√(1-x²)

Derivative of arctan(x)

1/(1+x²)

Negative binomial

the number of trials (X) needed to obtain r sucesses

how to check for a linear combination of matrices

ax + by + cz = d

ex + fy + gz = h

ix + jy + kz = m

αa + βe = i

αb + βf = j

αc + βg = k

αd + βh = m

rule for matrix multiplication

AB = C

A = mxn

B= nxp

and

C = mxp

mean of f(x)

1/(b-a)∫f(x)

and integral is between a and b

When can PGFs be used

discrete distributions that take non-negative integer values.

Formula

Gx(t) = ∑P(X = x)t^x

Mean/Expectance of a distribution

G'(1)

Variance of a distribution

G"(x) + G'(x) - (G'(1))²

X + Y = Z, how do I make a PGF for z

Gx+y(t) = Gx(t) x Gy(t)

Y = aX +b , make a PGF for Y

Gy(t) = t^bGx(t^a)

Formula used for finding PGFs from first principles

Gx(t) = E(t^x)

Type I Error

Rejecting H₀ when you should have accepted it (false positive)

Probability of a Type I Error

the actual significance level of the test

Type II Error

Incorrectly accepting H₀ when you should have rejected it (false negative)

Probability of a Type II Error

Probability that you are not in the critical region, using the actual probability (it would be given to you)

Impact of reducing significance level

reduction in P(Type I Error), but increase in P(Type II Error)

Size of a test

P(Type I Error)/ Actual Sig level

Power of a test

1-P(Type II Error)

x in polar coords

x = rcos(θ)

y in polar coords

y= rsin(θ)

r in polar coords

r =√(x²+y²)

integration of a polar curve

½∫r²dθ

Eq for a polar egg/dimple

r= a(p+qcos(θ))

egg

p>=2p

dimple

q

cardioid

p = q

When can you use integrating factor?

dy/dy + P(x)y = Q(x)

What is the integrating factor and how to use it?

I =e^∫P(x)dx

Multiply equation throughout by I

Then use reverse product rule on LHS and integrate RHS

When a polar curve is parallel to initial line?

dy/dθ = 0

When a polar curve is perpendicular to initial line?

dx/dθ = 0

How to calculate Degrees of Freedom for Goodness of Fit Test

Number of Columns (After combining) - 1 (take away another if you used your theoretical distribution to form the expected values)

What are possible constraints for Degrees of Freedom

- a limited/set number of trials ( almost always true): -1

- If you estimate an expected based on the observed: Like if you calculate the parameters of your distribution with you observed frequencies,then use that distribution to find expected: -1

One way stretch parallel to x-axis, scale factor a

a 0

01

One way stretch parallel to y-axis, scale factor b

1 0

0 b

Two way stretch, factor p in x direction, q in y direction

p 0

0 q

Conditions for a geometric distribution

- fixed prob of success

- independent trials

- exactly two outcomes

- indefinite number of successive trials

Geometric: P(X≤x)

- The probability that NOT all the first x trials are failures

1 - 1(1-p)^x

Conditions for negative binomial

- fixed prob of success

- independent trials

- exactly 2 outcomes

- indefinite number of successive trials

When to combine in a Chi squared test?

When the EXPECTED values are less than 5

How to calculate Degrees of Freedom for Contingency Table

(rows after combining -1)(columns after combining - 1)