AS & A Level Mathematics — Complete Revision Notes

Completing the square, the discriminant, and functions with their domains, ranges, inverses and transformations — the toolkit the whole of Pure 1 is written on top of.

3 sections·~11 min·จำนวนและพีชคณิต (Number and Algebra)·CC A-SSE, A-REI, F-IF, F-BF

Straight lines from coordinates, the circle in both of its forms, and the tangent property that reduces most circle questions to line questions you can already do.

3 sections·~8 min·การวัดและเรขาคณิต (Measurement and Geometry)·CC G-GPE, G-C

Radians and the arc and sector formulas they make possible, the exact values, the basic identities, and solving trigonometric equations across a stated interval.

3 sections·~9 min·การวัดและเรขาคณิต (Measurement and Geometry)·CC F-TF, G-C

The binomial expansion for a positive integer power, arithmetic and geometric progressions, and the condition that decides whether an infinite sum exists at all.

3 sections·~8 min·จำนวนและพีชคณิต (Number and Algebra)·CC A-SSE, A-APR, F-BF

The derivative as a limit, the rules that follow from it, and the three things Pure 1 asks you to do with one: tangents and normals, stationary points, and connected rates of change.

3 sections·~10 min·แคลคูลัส (Calculus)

Integration as the reverse of differentiation, definite integrals as areas, and volumes of revolution — plus the two situations where a naive integral gives the wrong area.

3 sections·~10 min·แคลคูลัส (Calculus)

Modulus equations and inequalities, polynomial division with the factor and remainder theorems, partial fractions in all three of their forms, and what to do when an equation cannot be solved exactly.

4 sections·~16 min·จำนวนและพีชคณิต (Number and Algebra)·CC A-APR, A-REI

The laws of logarithms, solving equations where the unknown is an index, the graphs of $e^x$ and $\ln x$, and turning experimental data into a straight line to measure its constants.

3 sections·~10 min·จำนวนและพีชคณิต (Number and Algebra)·CC F-LE, F-BF

The addition and double-angle formulas, the reciprocal identities, and the R-formula — which rewrites $a\sin\theta + b\cos\theta$ as a single wave and thereby answers every question about its maximum.

3 sections·~11 min·การวัดและเรขาคณิต (Measurement and Geometry)·CC F-TF

Differentiating implicitly and parametrically, integration by substitution and by parts, integrating a partial fraction, and separating the variables of a differential equation.

3 sections·~11 min·แคลคูลัส (Calculus)

Vectors in space, the scalar product and the angle it measures, the vector equation of a line, and deciding whether two lines meet, are parallel, or are skew.

3 sections·~10 min·การวัดและเรขาคณิต (Measurement and Geometry)·CC N-VM

The number $i$ and what it fixes, arithmetic with complex numbers, modulus–argument form, and reading a locus off an Argand diagram instead of solving for it.

3 sections·~11 min·จำนวนและพีชคณิต (Number and Algebra)·CC N-CN

Displaying data and measuring its centre and spread, counting arrangements and selections, and the probability rules — including the conditional one that most questions are secretly about.

3 sections·~13 min·สถิติและความน่าจะเป็น (Statistics and Probability)·CC S-ID, S-CP

Discrete random variables and their expectation, the binomial and geometric distributions, the normal distribution and its standardisation, and when a binomial may be approximated by a normal.

3 sections·~13 min·สถิติและความน่าจะเป็น (Statistics and Probability)·CC S-MD, S-IC