Pure Maths

[01 Algebra, Logs, Quadratics & Partial Fractions] What are the three main index laws?

a^m a^n=a^(m+n); a^m/a^n=a^(m-n); (a^m)^n=a^(mn). Also a^(-n)=1/a^n and a^(p/q)=qth root of a^p.

[01 Algebra, Logs, Quadratics & Partial Fractions] What are the three main logarithm laws?

log_a(xy)=log_a x+log_a y; log_a(x/y)=log_a x-log_a y; log_a(x^k)=k log_a x.

[01 Algebra, Logs, Quadratics & Partial Fractions] What is the change-of-base formula?

log_a x = log_b x / log_b a. On a calculator you can use ln x / ln a.

[01 Algebra, Logs, Quadratics & Partial Fractions] How do you recognise a factor-theorem question?

A factor such as (x-a) is given or must be tested. Use f(a)=0; the remainder on division by (x-a) is f(a).

[01 Algebra, Logs, Quadratics & Partial Fractions] What are the sum and product of roots α,β of ax²+bx+c=0?

α+β=-b/a and αβ=c/a.

[01 Algebra, Logs, Quadratics & Partial Fractions] How does the discriminant classify quadratic roots?

Δ=b²-4ac. Δ>0: two distinct real roots; Δ=0: repeated real root; Δ<0: no real roots (complex conjugate pair).

[01 Algebra, Logs, Quadratics & Partial Fractions] What is the safest procedure for a polynomial or rational inequality?

Move everything to one side; factor; mark roots and denominator zeros on a number line; test intervals; include/exclude endpoints correctly.

[01 Algebra, Logs, Quadratics & Partial Fractions] When must you do polynomial division before partial fractions?

When the rational fraction is improper: degree of numerator ≥ degree of denominator.

[01 Algebra, Logs, Quadratics & Partial Fractions] Partial fractions: distinct linear factors?

A/linear1 + B/linear2 (+ C/linear3 if needed).

[01 Algebra, Logs, Quadratics & Partial Fractions] Partial fractions: repeated linear factor (ax+b)²?

Include A/(ax+b) + B/(ax+b)².

[01 Algebra, Logs, Quadratics & Partial Fractions] Partial fractions: linear × irreducible quadratic?

Use A/(linear) + (Bx+C)/(quadratic).

[01 Algebra, Logs, Quadratics & Partial Fractions] What is the main trick when rationalising a denominator containing surds?

Multiply numerator and denominator by the conjugate so the denominator becomes rational.

[02 Sequences, Series & Binomial] AP: what is the nth term?

u_n=a+(n-1)d.

[02 Sequences, Series & Binomial] GP: what is the nth term?

u_n=ar^(n-1).

[02 Sequences, Series & Binomial] What condition is required for a GP to have a finite sum to infinity?

|r|<1, not merely r<1.

[02 Sequences, Series & Binomial] What is the sum to infinity of a convergent GP?

S∞=a/(1-r), valid only when |r|<1.

[02 Sequences, Series & Binomial] What does Σ notation tell you to do?

Substitute integer values of the index from the lower limit to the upper limit and add the resulting terms.

[02 Sequences, Series & Binomial] How do you recognise a GP written in sigma notation?

Look for a constant multiplied by a fixed ratio raised to a linear expression in the index, e.g. ar^(k-1).

[02 Sequences, Series & Binomial] When should you use the general binomial series rather than Pascal's triangle?

When the power is negative or fractional (or otherwise not a non-negative integer).

[02 Sequences, Series & Binomial] State the start of the general binomial expansion.

(1+x)^n = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + …

[02 Sequences, Series & Binomial] How do you prepare (a+bx)^n for a general binomial expansion?

Factor out a^n: (a+bx)^n=a^n(1+(b/a)x)^n.

[02 Sequences, Series & Binomial] What is the convergence condition for the general binomial series?

The inner variable must have modulus <1. If the inner variable is kx, require |kx|<1.

[03 Counting & Probability] When do you multiply numbers of choices in counting?

When choices occur in successive stages: if stage 1 has m choices and stage 2 has n choices, total=mn.

[03 Counting & Probability] When do you add numbers of choices in counting?

For mutually exclusive alternatives: do case 1 OR case 2, then add the counts.

[03 Counting & Probability] Permutation or combination: how do you choose?

Permutation if order matters; combination if only the selected group matters.

[03 Counting & Probability] State nPr and nCr.

nPr=n!/(n-r)!; nCr=n!/[r!(n-r)!].

[03 Counting & Probability] How do you count arrangements when some objects are repeated?

n! divided by the factorial of each repetition count.

[03 Counting & Probability] How do you handle 'these objects must stay together'?

Treat the required group as one block, arrange the blocks, then multiply by the internal arrangements of the block.

[03 Counting & Probability] How do you handle 'A and B must not be adjacent'?

Count all arrangements, then subtract arrangements in which A and B are treated as one block.

[03 Counting & Probability] What is the complement rule?

P(A')=1-P(A). Often fastest for 'at least one'.

[03 Counting & Probability] State the addition rule for two events.

P(A∪B)=P(A)+P(B)-P(A∩B).

[03 Counting & Probability] What does 'equally likely' probability reduce to?

Probability = number of favourable outcomes / total number of possible outcomes.

[04 Coordinate Geometry & Circles] Gradient between two points?

m=(y₂-y₁)/(x₂-x₁).

[04 Coordinate Geometry & Circles] Equation of a line through (x₁,y₁) with gradient m?

y-y₁=m(x-x₁).

[04 Coordinate Geometry & Circles] Conditions for parallel and perpendicular non-vertical lines?

Parallel: same gradient. Perpendicular: m₁m₂=-1.

[04 Coordinate Geometry & Circles] How do you find the intersection of two Cartesian lines?

Solve their two equations simultaneously.

[04 Coordinate Geometry & Circles] How do you find the angle between two lines from gradients?

Use tanθ=|(m₂-m₁)/(1+m₁m₂)|, or use direction vectors and the dot product if given.

[04 Coordinate Geometry & Circles] What is the centre-radius equation of a circle?

(x-a)²+(y-b)²=r²; centre (a,b), radius r.

[04 Coordinate Geometry & Circles] How do you find the centre/radius from x²+y²+Dx+Ey+F=0?

Complete the square in x and y. Centre is (-D/2,-E/2).

[04 Coordinate Geometry & Circles] What geometric fact is used for a tangent to a circle?

The tangent is perpendicular to the radius at the point of contact.

[04 Coordinate Geometry & Circles] Tangent to x²+y²=r² at (x₁,y₁)?

xx₁+yy₁=r².

[04 Coordinate Geometry & Circles] Condition for two circles to touch externally?

Distance between centres = r₁+r₂.

[04 Coordinate Geometry & Circles] Condition for two circles to touch internally?

Distance between centres = |r₁-r₂|.

[04 Coordinate Geometry & Circles] How do you begin a locus problem?

Let the moving point be (x,y), translate the geometric condition into algebra (distance/gradient/etc.), then simplify.

[05 Functions & Graph Transformations] What are domain and range?

Domain: permitted inputs. Range: outputs actually produced.

[05 Functions & Graph Transformations] What does one-to-one (injective) mean?

Different inputs give different outputs; equivalently each output is produced by at most one input.

[05 Functions & Graph Transformations] What does onto (surjective) mean?

Every element of the stated codomain is reached by at least one input.

[05 Functions & Graph Transformations] When does a function have an inverse function?

When it is one-to-one on the stated domain; restrict the domain if necessary.

[05 Functions & Graph Transformations] How do you find an inverse algebraically?

Write y=f(x), swap x and y, solve for y, then state the inverse's domain/range as required.

[05 Functions & Graph Transformations] How do you form (f∘g)(x)?

Substitute g(x) everywhere x appears in f.

[05 Functions & Graph Transformations] What does y=f(x)+a do?

Translate the graph vertically up by a (down if a<0).

[05 Functions & Graph Transformations] What does y=f(x+a) do?

Translate the graph left by a (right if a<0).

[05 Functions & Graph Transformations] What does y=af(x) do?

Vertical scale factor |a|; if a<0, also reflect in the x-axis.

[05 Functions & Graph Transformations] What does y=f(ax) do?

Horizontal scale factor 1/|a|; if a<0, also reflect in the y-axis.

[06 Trigonometry] Why must radians be used in calculus and small-angle formulas?

The standard derivative/integral and small-angle formulas assume the angle is measured in radians.

[06 Trigonometry] State the reciprocal definitions.

sec x=1/cos x; cosec x=1/sin x; cot x=1/tan x.

[06 Trigonometry] What exact angles should be known instantly?

0, π/6, π/4, π/3, π/2 for sin, cos and tan.

[06 Trigonometry] State the main Pythagorean identities.

sin²x+cos²x=1; 1+tan²x=sec²x; 1+cot²x=cosec²x.

[06 Trigonometry] State the double-angle identities you used most.

sin2x=2sinx cosx; cos2x=cos²x-sin²x=1-2sin²x=2cos²x-1.

[06 Trigonometry] How do you rewrite sin²x and cos²x for integration?

sin²x=(1-cos2x)/2; cos²x=(1+cos2x)/2.

[06 Trigonometry] How do you recognise an R-method question?

A linear combination A cosθ + B sinθ (or A sinθ+B cosθ), often followed by maximum/minimum, equation solving or sketching.

[06 Trigonometry] For A cosx + B sinx, what is R?

R=√(A²+B²).

[06 Trigonometry] Sign memory for R cos(x±α): positive sine term? negative sine term?

R cos(x-α) gives +R sinα sinx; R cos(x+α) gives -R sinα sinx.

[06 Trigonometry] If y=R cos(…), what are its maximum and minimum values?

Maximum R; minimum -R.

[06 Trigonometry] What are the small-angle approximations?

sin x≈x; tan x≈x; cos x≈1-x²/2 for small x in radians.

[06 Trigonometry] What is the safe method for solving a trig equation on an interval?

Find reference angle; determine correct quadrants from signs/CAST; list every solution in the required interval and units.

[07 Complex Numbers] Conjugate of z=a+bi?

z̄=a-bi.

[07 Complex Numbers] Modulus of z=a+bi?

|z|=√(a²+b²).

[07 Complex Numbers] How do you divide complex numbers in Cartesian form?

Multiply numerator and denominator by the conjugate of the denominator.

[07 Complex Numbers] What identity links z and z̄?

zz̄=|z|².

[07 Complex Numbers] How do you find the argument safely?

Start with tanθ=b/a, then use the signs of a and b to select the correct quadrant.

[07 Complex Numbers] What is the principal argument convention?

-π < arg z ≤ π.

[07 Complex Numbers] What is polar form?

z=r(cosθ+i sinθ), where r=|z| and θ=arg z.

[07 Complex Numbers] How do modulus and argument behave under multiplication?

Moduli multiply; arguments add (then adjust by 2π if needed).

[07 Complex Numbers] How do modulus and argument behave under division?

Moduli divide; arguments subtract (then adjust to the required range).

[08 Differentiation] When do you use the product rule?

When two non-constant functions are multiplied.

[08 Differentiation] When do you use the quotient rule?

When one non-constant function is divided by another.

[08 Differentiation] When do you use the chain rule?

When one function is inside another, e.g. (3x²+1)^5, e^(2x), sin(x²).

[08 Differentiation] Chain rule in words?

Differentiate the outside, keep the inside, multiply by the derivative of the inside.

[08 Differentiation] When is implicit differentiation needed?

When x and y are mixed in an equation and y is not isolated conveniently.

[08 Differentiation] What happens when differentiating a y-term implicitly?

Differentiate with respect to y, then multiply by dy/dx. Example d(y³)/dx=3y² dy/dx.

[08 Differentiation] How do you differentiate xy in an implicit equation?

Use product rule: d(xy)/dx = x dy/dx + y.

[08 Differentiation] How do you find a tangent equation?

Find dy/dx at the point, then use y-y₁=m(x-x₁).

[08 Differentiation] How do you find a normal gradient?

m_normal=-1/m_tangent when the tangent gradient is finite and non-zero.

[08 Differentiation] How do you find stationary points?

Solve dy/dx=0, find the corresponding y-values, then determine their nature.

[08 Differentiation] Second-derivative test for a stationary point?

At f'=0: f''>0 minimum; f''<0 maximum; f''=0 inconclusive.

[09 Integration] What must every indefinite integral include?

+C, the constant of integration.

[09 Integration] How do you recognise substitution?

A composite expression appears together with something proportional to its derivative.

[09 Integration] If du differs from the integral by only a constant factor, what do you do?

Compensate by multiplying/dividing by that constant. A constant mismatch is fine.

[09 Integration] If your substitution leaves an unwanted non-constant x-expression, what does that suggest?

The substitution is probably unsuitable unless that x-expression can also be rewritten in terms of u.

[09 Integration] How do you recognise integration by parts?

A product where differentiating one factor simplifies it and integrating the other is manageable, e.g. x ln x or polynomial × exponential/trig.

[09 Integration] What is the integration-by-parts formula?

∫u dv = uv - ∫v du.

[09 Integration] What is a common choice for u in ∫x ln x dx?

Choose u=ln x because differentiating ln x simplifies it; integrate x for dv.

[09 Integration] When do partial fractions help in integration?

When integrating a rational function whose denominator factors into manageable linear/quadratic factors.

[09 Integration] What trig identity is useful for ∫sin²x dx or ∫cos²x dx?

Use sin²x=(1-cos2x)/2 or cos²x=(1+cos2x)/2.

[09 Integration] What is the inner-factor rule for ∫cos(ax)dx and ∫sin(ax)dx?

∫cos(ax)dx=sin(ax)/a+C; ∫sin(ax)dx=-cos(ax)/a+C.

[09 Integration] For a definite integral with substitution, what should you do with limits?

Either change the limits to u-values and stay in u, or substitute back to x before using the original limits. Do not mix systems.

[09 Integration] How do you recognise ∫f'(x)/f(x) dx?

It gives ln|f(x)|+C.

[10 Separable Differential Equations] How do you recognise a separable differential equation?

It can be rearranged so all y-terms are with dy and all x-terms are with dx.

[10 Separable Differential Equations] What is the procedure for dy/dx=f(x)g(y)?

Rearrange to [1/g(y)]dy=f(x)dx; integrate both sides; include a constant; use any initial condition; isolate y if requested.