PUREEE

[Algebra] What are the three main index laws?

aᵐ × aⁿ = aᵐ⁺ⁿ • aᵐ⁄aⁿ = aᵐ⁻ⁿ • (aᵐ)ⁿ = aᵐⁿ. Also, a⁻ⁿ means 1⁄aⁿ.

[Algebra] What are the three main logarithm laws?

logₐ(xy) = logₐx + logₐy • logₐ(x divided by y) = logₐx − logₐy • logₐ(xᵏ) = k logₐx.

[Algebra] What is the change-of-base formula?

To calculate logₐx on your calculator, use ln(x)⁄ln(a).

[Algebra] 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).

[Algebra] What are the sum and product of roots α,β of ax²+bx+c=0?

Sum of roots: α + β = −b⁄a. Product of roots: αβ = c⁄a.

[Algebra] How does the discriminant classify quadratic roots?

Δ = b² − 4ac. • Δ > 0: two dif and only iferent real roots • Δ = 0: one repeated real root • Δ < 0: no real roots.

[Algebra] 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 or exclude endpoints correctly.

[Algebra] When must you do polynomial division before partial fractions?

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

[Algebra] Partial fractions: distinct linear factors?

If the denominator has dif and only iferent linear factors, use one constant numerator over each factor: A⁄(first factor) + B⁄(second factor).

[Algebra] Partial fractions: repeated linear factor (ax+b)²?

For a repeated factor (ax+b)², include BOTH: A⁄(ax+b) and B⁄(ax+b)².

[Algebra] Partial fractions: linear × irreducible quadratic?

Use a constant over the linear factor, and a linear numerator over the quadratic factor: A⁄(linear factor) + (Bx+C)⁄(quadratic factor).

[Algebra] What is the main trick when rationalising a denominator containing surds?

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

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

uₙ = a + (n−1)d.

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

uₙ = arⁿ⁻¹.

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

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

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

S∞ = a⁄(1−r). Use this only when |r| < 1.

[Sequences & 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.

[Sequences & 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, for example arᵏ⁻¹.

[Sequences & 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).

[Sequences & 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!] + …

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

Factor out a first: (a+bx)ⁿ = aⁿ(1 + bx⁄a)ⁿ. Then expand the bracket.

[Sequences & 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.

[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.

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

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

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

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

[Probability] State nPr and nCr.

ⁿPᵣ = n!⁄(n−r)! • ⁿCᵣ = n!⁄[r!(n−r)!].

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

n! divided by the factorial of each repetition count.

[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.

[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.

[Probability] What is the complement rule?

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

[Probability] State the addition rule for two events.

P(A∪B)=P(A)+P(B)-P(A∩B).

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

Probability = favourable outcomes⁄total possible outcomes.

[Coordinate Geometry] Gradient between two points?

m = (change in y)⁄(change in x). In symbols: (y₂−y₁)⁄(x₂−x₁).

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

y-y₁=m(x-x₁).

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

Parallel lines have the same gradient. Perpendicular lines satisfy m₁m₂ = −1.

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

Solve their two equations simultaneously.

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

Use tanθ = absolute value of [(m₂−m₁)⁄(1+m₁m₂)]. Then use tan⁻¹.

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

(x−a)² + (y−b)² = r². Centre = (a,b). Radius = r.

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

Complete the square in x and y. The centre is (−D⁄2, −E⁄2).

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

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

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

For x²+y²=r², the tangent at (x₁, y₁) is xx₁ + yy₁ = r².

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

Distance between centres = r₁+r₂.

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

Distance between the centres = absolute value of r₁−r₂.

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

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

[Functions] What are domain and range?

Domain: permitted inputs. Range: outputs actually produced.

[Functions] What does one-to-one (injective) mean?

Dif and only iferent inputs give dif and only iferent outputs • equivalently each output is produced by at most one input.

[Functions] What does onto (surjective) mean?

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

[Functions] When does a function have an inverse function?

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

[Functions] How do you find an inverse algebraically?

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

[Functions] How do you form (f∘g)(x)?

Substitute g(x) everywhere x appears in f.

[Functions] What does y=f(x)+a do?

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

[Functions] What does y=f(x+a) do?

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

[Functions] What does y=af(x) do?

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

[Functions] What does y=f(ax) do?

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

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

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

[Trig] State the reciprocal definitions.

sec x = 1⁄cos x • cosec x = 1⁄sin x • cot x = 1⁄tan x.

[Trig] What exact angles should be known instantly?

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

[Trig] State the main Pythagorean identities.

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

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

sin 2x = 2 sin x cos x • cos 2x = cos²x − sin²x = 1 − 2sin²x = 2cos²x − 1.

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

sin²x = (1−cos2x)⁄2 • cos²x = (1+cos2x)⁄2.

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

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

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

R = √(A²+B²).

[Trig] 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.

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

Maximum R • minimum -R.

[Trig] What are the small-angle approximations?

For small angles in radians: sin x ≈ x • tan x ≈ x • cos x ≈ 1 − x²⁄2.

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

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

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

If z = a+bi, then z̄ = a−bi.

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

If z = a+bi, then |z| = √(a²+b²).

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

Multiply numerator and denominator by the conjugate of the denominator.

[Complex Numbers] What identity links z and z̄?

z × z̄ = |z|².

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

First calculate tanθ = b⁄a. Then use the signs of a and b to choose the correct quadrant.

[Complex Numbers] What is the principal argument convention?

-π < arg z ≤ π.

[Complex Numbers] What is polar form?

z = r(cosθ + i sinθ), where r is the modulus and θ is the argument.

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

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

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

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

[Differentiation] When do you use the product rule?

When two non-constant functions are multiplied.

[Differentiation] When do you use the quotient rule?

When one non-constant function is divided by another.

[Differentiation] When do you use the chain rule?

When one function is inside another, for example (3x²+1)⁵, e²ˣ, sin(x²).

[Differentiation] Chain rule in words?

Dif and only iferentiate the outside, keep the inside, multiply by the derivative of the inside.

[Differentiation] When is implicit differentiation needed?

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

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

Dif and only iferentiate the y-term normally, then multiply by y′. Example: derivative of y³ is 3y²y′.

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

Use the product rule: derivative of xy = xy′ + y.

[Differentiation] How do you find a tangent equation?

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

[Differentiation] How do you find a normal gradient?

Normal gradient = −1⁄the tangent gradient.

[Differentiation] How do you find stationary points?

Solve y′ = 0. Find the matching y-values. Then decide whether each point is a maximum, minimum or neither.

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

At a stationary point: y″ > 0 means minimum • y″ < 0 means maximum • y″ = 0 is inconclusive.

[Integration] What must every indefinite integral include?

+C, the constant of integration.

[Integration] How do you recognise substitution?

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

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

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

[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.

[Integration] How do you recognise integration by parts?

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

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

∫u dv = uv − ∫v du.

[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.

[Integration] When do partial fractions help in integration?

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

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

Rewrite the square first: sin²x = (1−cos2x)⁄2, or cos²x = (1+cos2x)⁄2.

[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.

[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.

[Integration] How do you recognise ∫a function divided by itself after differentiation dx?

If the numerator is the derivative of the denominator, then ∫f′(x)⁄f(x) dx = ln|f(x)| + C.

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

You can rearrange it so every y-term is beside dy and every x-term is beside dx.

[Differential Equations] What is the procedure for a separable equation of the form y′ = f(x)g(y)?

1) Move all y-terms with dy. 2) Move all x-terms with dx. 3) Integrate both sides. 4) Add a constant. 5) Use the initial condition if given. 6) Isolate y if asked.