LearnAQA7357First teaching 2017Paper 2 · Pure and MechanicsPure content (sections A–J)

Paper 2 · Module

Pure content (sections A–J)

66 subtopics≈ 2710 min in total

Start the module: Completing the square and the vertex Open graphing calculator

Completing the square and the vertex
Rewrite $ax^2+bx+c$ as $a(x+p)^2+q$ , read off the vertex, line of symmetry and least or greatest value, and use it to sketch parabolas and solve real problems about maximum height or area.
≈ 45 min
Solving quadratic equations
Three ways to solve $ax^2+bx+c=0$ (factorising, completing the square and the formula), choosing the quickest, giving exact answers, and spotting quadratics in disguise such as $x^4-5x^2+4=0$ .
≈ 45 min
The discriminant and the nature of roots
How $b^2-4ac$ tells you whether a quadratic has two, one or no real roots without solving it, how to find unknown constants from that, and how to show an expression is always positive.
≈ 40 min
Linear and quadratic inequalities
Solve linear inequalities, then quadratic ones using critical values and a sketch, write the answer correctly (and, or, set notation), and combine two conditions.
≈ 35 min
Simultaneous equations and lines meeting curves
Solve one linear and one quadratic equation together by substitution, find the points where a line meets a curve, and use the discriminant to decide whether a line crosses, touches or misses a curve.
≈ 40 min
Gradient, length, midpoint and equations of lines
The three measurements of a line segment (gradient, length and midpoint), the three forms of the equation of a straight line, and how to find where lines cross.
≈ 35 min
Parallel and perpendicular lines
Why parallel lines share a gradient and perpendicular gradients multiply to $-1$ , and how to use that for perpendicular bisectors, the foot of a perpendicular and right-angled shapes.
≈ 35 min
The equation of a circle
Why $(x-a)^2+(y-b)^2=r^2$ is a circle, how to find the centre and radius from the expanded form by completing the square, and how to use diameters and right angles to find a circle's equation.
≈ 40 min
Tangents, chords and intersections
The two right angles that unlock circle questions (tangent and radius, centre and chord), and how substituting a line into a circle or curve turns geometry into a quadratic whose discriminant tells you how they meet.
≈ 45 min
Radians
What a radian is, why mathematicians prefer it to the degree, and how to convert between the two, with exact values in terms of $\pi$ .
≈ 30 min
Arc length and sector area
Why the formulae $s=r\theta$ and $A=\tfrac12r^2\theta$ work, how to use them for perimeters, and how to work backwards from a length or an area to the radius or angle.
≈ 35 min
Segments, chords and composite shapes
Cutting a sector with a chord: the segment formula, chord lengths, and the exam-style shaded regions built from sectors, triangles and tangents.
≈ 45 min
Exact values and the unit circle
How sine, cosine and tangent are defined for any angle using the unit circle, the CAST rule for signs, and the exact values at $30^\circ$ , $45^\circ$ and $60^\circ$ that you need without a calculator.
≈ 40 min
Graphs and transformations
The shapes of $y=\sin x$ , $y=\cos x$ and $y=\tan x$ , their symmetries, and how $a$ , $b$ , $c$ and $d$ in $y=a\sin(b(x-c))+d$ stretch and shift them. Then use the graphs to count solutions and model things that repeat.
≈ 45 min
Inverse trigonometric functions
Why $\sin x=\tfrac12$ has infinitely many solutions but $\sin^{-1}\tfrac12$ has only one, the restricted domains that make sine, cosine and tangent one-to-one, and the principal values your calculator gives.
≈ 35 min
Trigonometric identities
The two identities at the heart of AS trigonometry, where they come from, and how to use them to simplify expressions, find exact values and write clear proofs.
≈ 40 min
Solving trigonometric equations
How to find every solution of a trig equation in an interval: principal value, symmetry, and period. Then equations with $2x$ or $x+30^\circ$ inside, quadratics in $\sin x$ or $\cos x$ , and equations that need an identity first.
≈ 55 min
Binomial expansion
How to expand $(a+b)^n$ without multiplying out $n$ brackets, using Pascal's triangle or $\binom{n}{r}$ , and how to pick out one term, such as the coefficient of $x^3$ or the term independent of $x$ .
≈ 45 min
Arithmetic progressions
Sequences that go up (or down) by the same amount each time: the $n$ th term, the sum of the first $n$ terms and why it works, and how to find unknowns from given terms or sums.
≈ 40 min
Geometric progressions
Sequences that multiply by the same number each time: the $n$ th term, the sum of the first $n$ terms and its proof, finding unknowns, and the growth and decay problems they model.
≈ 45 min
Sum to infinity
How adding infinitely many numbers can give a finite answer, the condition $|r|<1$ that makes it happen, and how to use $S_\infty=\dfrac{a}{1-r}$ for recurring decimals, bouncing balls and exam problems.
≈ 35 min
Gradients and the power rule
What the gradient of a curve means, how chords close in on the tangent, and the power rule $\dfrac{d}{dx}x^n=nx^{n-1}$ that makes differentiating fast, including roots, fractions and second derivatives.
≈ 45 min
The chain rule
How to differentiate a function of a function, such as $(2x-5)^4$ or $\sqrt{x^2+5}$ , without multiplying out: differentiate the outside, keep the inside, and multiply by the derivative of the inside.
≈ 40 min
Tangents and normals
Using $\dfrac{dy}{dx}$ to find the equation of the tangent and the normal to a curve at a point, points where the tangent has a given gradient, and the follow-on problems examiners love.
≈ 40 min
Stationary points and increasing functions
Where a curve turns: finding stationary points with $\dfrac{dy}{dx}=0$ , deciding whether each is a maximum or minimum with the second derivative, and finding where a function is increasing or decreasing.
≈ 45 min
Optimisation and rates of change
Using derivatives to answer real questions: the largest area, the smallest surface, the cheapest design, and how fast one quantity changes when another one does.
≈ 50 min
Indefinite integration
Integration as differentiation in reverse: integrating powers of $x$ , why there is always a $+c$ , and how one point pins down the exact curve.
≈ 40 min
Integrating (ax + b)ⁿ
How to integrate a power of a linear bracket in one step by reversing the chain rule: raise the power, divide by the new power, and divide by the coefficient of $x$ .
≈ 35 min
Definite integrals
What $\displaystyle\int_a^b f(x)\,dx$ means and how to evaluate it, why there is no $+c$ , how to find an unknown limit, and how some integrals with an infinite limit still give a finite answer.
≈ 40 min
Areas under and between curves
Using definite integrals to find areas: under a curve, regions below the axis, between a curve and a line, between two curves, and against the $y$ -axis.
≈ 50 min
Modulus graphs and equations
What $|x|$ means, how to sketch $y=|ax+b|$ by reflecting the negative part, and two reliable ways to solve modulus equations: squaring both sides, or splitting into cases and checking.
≈ 40 min
Modulus inequalities
Solving inequalities with moduli: the "between" and "outside" patterns, squaring when both sides are moduli, and using a sketch when a modulus is compared with a line.
≈ 40 min
Polynomials: division, factor and remainder theorems
Long division for polynomials, why the remainder on dividing by $(x-a)$ is just $p(a)$ , and how the factor theorem lets you factorise cubics and find unknown coefficients.
≈ 45 min
Partial fractions
Splitting a complicated fraction into simpler ones, which you need for integrating and for binomial expansions: the right form for each kind of denominator, and quick ways to find the constants.
≈ 45 min
Binomial expansion for any rational n
Extending $(1+x)^n$ to negative and fractional powers: an infinite series that only works for $|x|<1$ , how to handle $(a+bx)^n$ , and how to use the expansion with partial fractions and for approximations.
≈ 45 min
Laws of logarithms
What a logarithm is (the power you need), the three laws that come straight from the index laws, and how to combine and simplify logarithms, including $\ln$ and $\lg$ .
≈ 40 min
The functions eˣ and ln x
Meet $e\approx2.718$ , the most natural base for growth, and its inverse $\ln x$ : their graphs, asymptotes and intercepts, transformations, inverses and exact answers.
≈ 40 min
Exponential equations and inequalities
Bringing an unknown power down with logarithms: solving $a^x=b$ , equations with $x$ in two powers, disguised quadratics, and inequalities like "the smallest $n$ such that $0.8^n<0.1$ ".
≈ 40 min
Logarithmic equations
Solving equations with the unknown inside a logarithm: combine with the log laws, remove the logs by raising to a power, and always check that the answers are in the domain.
≈ 35 min
Exponential models and linear forms
Using logarithms in the real world: modelling growth and decay with $Ae^{kt}$ , and turning curves like $y=kx^n$ and $y=ka^x$ into straight lines so their constants can be read from a graph.
≈ 45 min
Reciprocal functions and their identities
Three new functions that are just $1$ over the old ones: their values, their graphs with asymptotes, the two Pythagorean identities they satisfy, and how to use them to solve equations.
≈ 40 min
Derivatives of eˣ, ln x and trigonometric functions
The derivatives of $e^x$ , $\ln x$ , $\sin x$ , $\cos x$ and $\tan x$ , why they need radians, and how to combine them with the chain rule for functions like $e^{3x}$ , $\ln(x^2+1)$ and $\sin^2x$ .
≈ 40 min
The product and quotient rules
How to differentiate two functions multiplied together, $uv$ , or divided, $\dfrac uv$ , and how to tidy the result by factorising, so it is ready for finding stationary points.
≈ 45 min
Implicit differentiation
Differentiating curves like $x^2+xy+y^2=12$ that can't easily be written as $y=\ldots$ : differentiate every term with respect to $x$ , remembering that $y$ terms pick up a factor $\dfrac{dy}{dx}$ .
≈ 45 min
Parametric differentiation
Curves described by a third variable $t$ : $x=f(t)$ , $y=g(t)$ . How to find their gradient with $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$ , and how to find tangents and special points.
≈ 40 min
Stationary points and tangents
Putting the Pure 2/3 rules to work: finding exact stationary points of curves like $y=xe^{2x}$ and $y=\dfrac{\ln x}{x}$ , deciding their nature, and finding tangents and normals.
≈ 45 min
Integrating eˣ, 1/x and trigonometric functions
Reversing the new derivatives: $\int e^{ax+b}\,dx$ , $\int\dfrac{1}{ax+b}\,dx$ (the missing case $n=-1$ ), and $\int\sin$ , $\cos$ and $\sec^2$ , always remembering to divide by $a$ .
≈ 40 min
Integration using trigonometric identities
Some trigonometric expressions can't be integrated as they stand. Rewriting them with identities, such as $\cos^2x=\tfrac12(1+\cos2x)$ and $\tan^2x=\sec^2x-1$ , turns them into standard integrals.
≈ 40 min
Exact definite integrals and logarithms
Evaluating definite integrals exactly when the answer involves $e$ , $\ln$ or $\pi$ ; simplifying logarithmic answers; spotting $\dfrac{f'(x)}{f(x)}$ ; and integrating with partial fractions.
≈ 45 min
The trapezium rule
When an integral can't be found exactly, estimate it: replace the curve by straight lines, add up the trapezia, and use the shape of the curve to say whether the answer is too big or too small.
≈ 35 min
Integration by substitution
The reverse of the chain rule: replace part of the integrand by $u$ , change $dx$ into $du$ , change the limits, and turn a hard integral into an easy one.
≈ 50 min
Integration by parts
The reverse of the product rule: $\displaystyle\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx$ . How to choose $u$ , when to use it twice, and the trick for $\int\ln x\,dx$ .
≈ 50 min
Compound-angle formulae
$\sin(A+B)$ is not $\sin A+\sin B$ . The correct formulae for $\sin(A\pm B)$ , $\cos(A\pm B)$ and $\tan(A\pm B)$ , how to use them for exact values, and how they turn awkward equations into $\tan\theta=k$ .
≈ 45 min
Double-angle formulae
Setting $B=A$ in the compound-angle formulae gives formulae for $\sin2A$ , $\cos2A$ and $\tan2A$ . Three versions of $\cos2A$ let you match the function in an equation, and they are essential for integration later.
≈ 45 min
The R-form (harmonic form)
Any combination $a\sin\theta+b\cos\theta$ is a single sine wave in disguise: $R\sin(\theta+\alpha)$ . How to find $R$ and $\alpha$ , and how this makes equations and maximum and minimum questions easy.
≈ 45 min
Locating roots: sketches and sign changes
Most equations have no exact solution. Before you can approximate a root you need to know how many there are and roughly where: sketch two graphs to count them, then trap each one with a change of sign.
≈ 35 min
Fixed-point iteration
Feed a number into a formula, feed the answer back in, and repeat. When the values settle down they settle on a root. This lesson is the engine of every P2 and P3 numerical question.
≈ 40 min
Rearrangements, staircases and cobwebs
One equation can be rearranged into $x=F(x)$ in many ways, and they do not all work. See why some iterations close in on a root, some zigzag towards it, and some run away, and learn to find the equation a limit satisfies.
≈ 40 min
Newton–Raphson, bisection and interpolation
Slide down the tangent to the axis and repeat: the Newton–Raphson method roughly doubles the number of correct digits at every step. Also the two simpler interval methods in Edexcel FP1, and the ways all of them can go wrong. For AQA and Edexcel Further Pure 1; not examined in Cambridge 9709.
≈ 40 min
Forming differential equations
Turn words about change into mathematics. "Grows in proportion to", "decreases at a rate inversely proportional to", "flows in and leaks out": each becomes an equation for a derivative, ready to solve.
≈ 30 min
Separating the variables
Gather every $y$ with $dy$ on one side and every $x$ with $dx$ on the other, integrate, and add one constant. The method that solves almost every Paper 3 differential equation.
≈ 35 min
Particular solutions with harder integrals
Exam differential equations are integration questions in disguise. Once the variables are separated, one side often needs partial fractions, integration by parts or a trigonometric identity. Bring all of Paper 3 integration to bear, then pin down the constant.
≈ 45 min
Modelling with differential equations
Populations, cooling drinks, draining tanks and spreading rumours. Solve the equation, use two pieces of data to fix the constants, and then ask the real questions: when, how much, and what happens in the long run?
≈ 45 min
Position vectors, magnitude and unit vectors
Vectors are arrows that carry a size and a direction. Learn to move between points with displacement vectors, measure them with three-dimensional Pythagoras, scale them to length $1$ , and use them to find midpoints, ratios and missing corners of shapes.
≈ 35 min
Deduction, exhaustion and counterexample
A proof must work for every case, not just the ones you tried. Deduction uses algebra for a general case; exhaustion splits all possibilities into a few cases and checks each; and one single counterexample is enough to show a statement is false.
≈ 30 min
Proof by contradiction
To prove a statement, assume it is false and show that this leads to something impossible. The assumption must have been wrong, so the statement is true. The two famous examples are that $\sqrt2$ is irrational and that there are infinitely many primes.
≈ 35 min

Pure content (sections A–J)

Completing the square and the vertex

Solving quadratic equations

The discriminant and the nature of roots

Linear and quadratic inequalities

Simultaneous equations and lines meeting curves

Gradient, length, midpoint and equations of lines

Parallel and perpendicular lines

The equation of a circle

Tangents, chords and intersections

Radians

Arc length and sector area

Segments, chords and composite shapes

Exact values and the unit circle

Graphs and transformations

Inverse trigonometric functions

Trigonometric identities

Solving trigonometric equations

Binomial expansion

Arithmetic progressions

Geometric progressions

Sum to infinity

Gradients and the power rule

The chain rule

Tangents and normals

Stationary points and increasing functions

Optimisation and rates of change

Indefinite integration

Integrating (ax + b)ⁿ

Definite integrals

Areas under and between curves

Modulus graphs and equations

Modulus inequalities

Polynomials: division, factor and remainder theorems

Partial fractions

Binomial expansion for any rational n

Laws of logarithms

The functions eˣ and ln x

Exponential equations and inequalities

Logarithmic equations

Exponential models and linear forms

Reciprocal functions and their identities

Derivatives of eˣ, ln x and trigonometric functions

The product and quotient rules

Implicit differentiation

Parametric differentiation

Stationary points and tangents

Integrating eˣ, 1/x and trigonometric functions

Integration using trigonometric identities

Exact definite integrals and logarithms

The trapezium rule

Integration by substitution

Integration by parts

Compound-angle formulae

Double-angle formulae

The R-form (harmonic form)

Locating roots: sketches and sign changes

Fixed-point iteration

Rearrangements, staircases and cobwebs

Newton–Raphson, bisection and interpolation

Forming differential equations

Separating the variables

Particular solutions with harder integrals

Modelling with differential equations

Position vectors, magnitude and unit vectors

Deduction, exhaustion and counterexample

Proof by contradiction