Core Mathematics A-Level

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Laws of Indices

Completing the Square

Quadratic Graphs: Positive

Quadratic Graph: Negative

Exponential Graphs, Positive

Exponential Graph, Negative.

Reciprocal Graphs x and X²

Cubic Graphs

Domain, Range

Range: All valid Y values. Domain: All valid x values.

Composite Functions

With two functions when you feed one into another.
I.e if you have f(x) and g(x)
f(g(x)) means sub G into F.

Inverse Functions

How to find the inverse of a function
1. Replace f(x) with y.
2. Swap positions of x and y in the equation.
3. Rearrange/solve to make y the subject
4. Replace y with f(x)-1

Domain and range values swap when function inversed.
Must be one-to-one to have an inverse

Partial Fractions

• Check if the fraction is proper; if not, perform polynomial division.

• Factor the denominator fully.

• Write the partial fraction form using constants (A, B, C…) in the correct structure.

• Multiply both sides by the full denominator to clear fractions.

• Substitute convenient x‑values to find constants where possible.

• Equate coefficients to find any remaining constants.

• Rewrite the original fraction using the constants found.

• Proceed with the required operation (usually integration or simplification).

Gradient, Midpoint, distance

Equation of straight line with two known coordinates.

y-y1=m(x-x1)

Equation of a circle

(x-a)²+(y-b)²=r²

where a is x coordinate of centre and b is y coordinate of centre

Gradient conditions parallel and perpendicular.

Parallel: Two gradients equal
Perpendicular lines: Negative reciprocal gradients or they multiply to -1.

Circle Theorems.

Increasing, Decreasing, Periodic Sequences.

Increasing sequence — a sequence where each term is greater than or equal to the previous term.
Decreasing sequence — a sequence where each term is less than or equal to the previous term.
Periodic sequence — a sequence where terms repeat in a fixed cycle after a certain number of steps.

Arithmetic Sequence Equations

For nth term: Un=a+(n-1)d
For sum: Sn=n/2(a+l)

Where a is first term, d is common difference and l is last term.

Geometric Sequence Equations

Nth term: Tn=ar^n-1
Sum: Sn=a(1-r^n)/1-r

Sine rule, cosine rule, area of triangle.

In all three triangle formulas (Sine Rule, Cosine Rule, Area Formula), the letters a, b, and c always mean the same thing:

a, b, c (side lengths)

• a = side opposite angle A

• b = side opposite angle B

• c = side opposite angle C

Radian Formulae.

Sin cos tan sketching.

Cot, Cosec, Sec.

Arcsin Arctan Arccos

ALL Trig Identities

Harmonic Form

• Write the target form

• Expand using the compound‑angle identity

• Match coefficients

• Find R using Pythagoras

• Find α using tanα = b/a

ALL Log Solving Question.

Simplify both sides to one log using known laws then drop logs and solve.

SOME Logs solving.

Put all logs on one side and constant terms on the other.
Use law of logs to simplify to a single log.
Change from log form to power form.

NO logs solving

Method 1: Change to power form.
OR
Method 2: Take logs of both sides,

Reduction to linear form

1. Use trig identities to rewrite powers or products

2. Convert everything into sin x and cos x

3. Factorise where possible

4. Use identities to simplify to a single trig function

5. Solve the resulting linear trig equation

Differentiation from first principles

• Write the limit definition of the derivative

• Substitute f(x+h) and f(x) into the formula

• Expand brackets or expressions fully

• Simplify the numerator

• Factor out h from the numerator

• Cancel the h with the denominator

• Take the limit as h → 0

Chain Rule differentiation

Differentiate outer function, then differentiate inner function and multiply the two.

Product Rule differentiation

Differentiate u and v to u’ and v’ then multiply uv’ and u’v and add them together.

Implicit Differentiation

1. Differentiate both sides with respect to x

2. Differentiate y‑terms using the chain rule

3. Attach dy/dx to every differentiated y‑term

4. Collect all dy/dx terms on one side

5. Factor out dy/dx

6. Solve for dy/dx

Parametric Differentiation

• Differentiate x with respect to t

• Differentiate y with respect to t

• Form dy/dx as (dy/dt) ÷ (dx/dt)

• Simplify the expression

• Substitute a given t‑value if required

Parametric to Cartesian Steps

• Isolate t in one equation

• Substitute into the other equation

• Use identities or algebra to remove t

• Rearrange into Cartesian form

• State domain restrictions if needed

Connected Rates of Change Steps

• Identify all given relationships between variables

• Differentiate each relationship with respect to time

• Substitute known values into the differentiated equations

• Link the required rate to known rates using the equations

• Solve for the unknown rate

Reverse Chain Rule

• Spot an inside function and its derivative

• Let u equal the inside function

• Differentiate u to get du

• Rewrite the integral fully in terms of u

• Integrate with respect to u

• Substitute back to x

Ln integrals

• Rewrite the integrand so ln(x) is the “inside” function

• Let u = ln(x)

• Differentiate to get du = 1/x dx

• Rewrite the whole integral in terms of u

• Integrate with respect to u

• Substitute back to x

Integration by substitution

• Choose a substitution u = inside function

• Differentiate to get du

• Rewrite dx in terms of du

• Rewrite the entire integral in u

• Integrate with respect to u

• Substitute back to x

Integration by parts

• Choose u and dv

• Differentiate u to get du

• Integrate dv to get v

• Apply the formula uv − ∫v du

• Integrate the remaining integral

• Simplify the final expression

Integrating partial fractions

• Split the expression into partial fractions

• Integrate each fraction separately

• Use ln for 1/(x−a) terms

• Use power rule for (x−a)ⁿ terms

• Use arctan form for irreducible quadratics

• Add +C

Differential equations

• Separate variables if possible

• Integrate both sides

• Add the constant of integration

• Use initial conditions to find the constant

• Rewrite the solution in terms of y

If the equation is not separable, the A Level method switches to:

• Rewrite into integrating factor form dy/dx + P(x)y = Q(x)

• Find the integrating factor e∫P(x)dx

• Multiply the whole equation by the integrating factor

• Recognise the left side as a product derivative

• Integrate and solve for y

Integration of parametric functions

• Write the integral as ∫ y (dx/dt) dt

• Differentiate x with respect to t

• Substitute x(t) and y(t) into the integral

• Integrate with respect to t

• Apply limits by converting x‑limits to t‑limits

• Evaluate the definite integral

newton-raphson

1) Start with an equation f(x) = 0

This is the equation whose root you want to approximate.

2) Choose a starting value x₀

You need an initial guess close to the root so the method converges.

3) Use the formula xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ)

This is the Newton–Raphson iteration.
It uses the tangent line at xₙ to jump closer to the root.

4) Repeat the iteration until values settle

Each iteration moves you closer to the actual root.
Stop when xₙ stops changing to the required accuracy.

5) State the root to required decimal places

This is your final numerical approximation.

Definition of population, census and sample

Population- Whole set of items that are of interest
Census- observation of whole population

Sample- Observations from a subset of the population

Advantages and disadvantages of census and samples

Census advantages: Gives completely accurate result

Census disadvantages: Time consuming and expensive. Cannot be used when testing process destroys item. Difficult to process massive data amounts.

Sample advantages: Quicker and cheaper. Less people respond. Less data to process
Sample disadvantages: Data may not be accurate, and may not be large enough to give information about subgroups.

Definition of sampling unit, frames and parameters

Sampling unit- Individual units of population

Sampling frame- When units are named or numbered to form a list

Parameter- A number that describes the entire population