8. A-level 수학 (Maths)

'영국 교육제도와 수학 공부' 전체보기

• Pure mathematics Year 1
• Pure mathematics Year 2
• Statistics and Mechanics Year 1
• Statistics and Mechanics Year 2

Pure mathematics Year 1

1. ALGEBRAIC EXPRESSIONS

1.1 Index laws

1.2 Expanding brackets

1.3 Factorising

1.4 Negative and fractional indices

1.5 Surds

1.6 Rationalising denominators

1.7 Mixed exercise 1

1.8 Review exercise for chapter 1

 

2. QUADRATICS

2.1 Solving quadratic equations

2.2 Completing the square

2.3 Functions

2.4 Quadratic graphs

2.5 The discriminant

2.6 Modelling with quadratics

2.7 Mixed exercise 2

2.8 Review exercise for chapter 2

 

3. EQUATIONS AND INEQUALITIES

3.1 Linear simultaneous equations

3.2 Quadratic simultaneous equations

3.3 Simultaneous equations on graphs

3.4 Linear inequalities

3.5 Quadratic inequalities

3.6 Inequalities on graphs

3.7 Regions

3.8 Mixed exercise 3

3.9 Review exercise for chapter 3

 

4. GRAPHS AND TRANSFORMATIONS

4.1 Cubic graphs

4.2 Quartic graphs

4.3 Reciprocal graphs

4.4 Points of intersection

4.5 Translating graphs

4.6 Stretching graphs

4.7 Transforming functions

4.8 Mixed exercise 4

4.9 Review exercise for chapter 4

 

5. STRAIGHT LINE GRAPHS

5.1 $y=mx+c$

5.2 Equations of straight lines

5.3 Parallel and perpendicular lines

5.4 Length and area

5.5 Modelling with straight lines

5.6 Mixed exercise 5

5.7 Review exercise for chapter 5

 

6. CIRCLES

6.1 Mid-points and perpendicular bisectors

6.2 Equations of a circle

6.3 Intersections of straight lines and circles

6.4 Use tangent and chord properties

6.5 Circles and triangles

6.6 Mixed exercise 6

6.7 Review exercise for chapter 6

 

7. ALGEBRAIC METHODS

7.1 Algebraic fractions

7.2 Dividing polynomials

7.3 The factor theorem

7.4 Mathematical proof

7.5 Methods of proof

7.6 Mixed exercise 7

7.7 Review exercise for chapter 7

 

8. THE BINOMIAL EXPANSION

8.1 Pascal’s triangle

8.2 Factorial notation

8.3 The binomial expansion

8.4 Solving binomial problems

8.5 Binomial estimation

8.6 Mixed exercise 8

8.7 Review exercise for chapter 8

 

9. TRIGONOMETRIC RATIOS

9.1 The cosine rule

9.2 The sine rule

9.3 Areas of triangles

9.4 Solving triangle problems

9.5 Graphs of sine, cosine and tangent

9.6 Transforming trigonometric graphs

9.7 Mixed exercise 9

9.8 Review exercise for chapter 9

 

10. TRIGONOMETRIC IDENTITIES AND EQUATIONS

10.1 Angles in all four quadrants

10.2 Exact values of trigonometric ratios

10.3 Trigonometric identities

10.4 Simple trigonometric equations

10.5 Harder trigonometric equations

10.6 Equations and identities

10.7 Mixed exercise 10

10.8 Review exercise for chapter 10

 

11. VECTORS

11.1 Vectors

11.2 Representing vectors

11.3 Magnitude and direction

11.4 Position vectors

11.5 Solving geometric problems

11.6 Modelling with vectors

11.7 Mixed exercise 11

11.8 Review exercise for chapter 11

 

12. DIFFERENTIATION

12.1 Gradients of curves

12.2 Finding the derivative

12.3 Differentiating $x^n$

12.4 Differentiating quadratics

12.5 Differentiating functions with two or more terms

12.6 Gradients, tangents and normal

12.7 Increasing and decreasing functions

12.8 Second order derivatives

12.9 Stationary points

12.10 Sketching gradient functions

12.11 Modelling with differentiation

12.12 Mixed exercise 12

12.13 Review exercise for chapter 12

 

13. INTEGRATION

13.1 Integrating $x^n$

13.2 Indefinite integrals

13.3 Finding functions

13.4 Definite integrals

13.5 Areas under curves

13.6 Areas under the x-axis

13.7 Areas between curves and lines

13.8 Mixed exercise 13

13.9 Review exercise for chapter 13

 

14. EXPONENTIALS AND LOGARITHMS

14.1 Exponential functions

14.2 $y=\textrm{e}^x$

14.3 Exponential modelling

14.4 Logarithms

14.5 Laws of logarithms

14.6 Solving equations using logarithms

14.7 Working with natural logarithms

14.8 Logarithms and non-linear data

14.9 Mixed exercise 14

14.10 Review exercise for chapter 14

Pure mathematics Year 2

1. ALGEBRAIC METHODS

1.1 Proof by contradiction

1.2 Algebraic fractions

1.3 Partial fractions

1.4 Repeated factors

1.5 Algebraic division

1.6 Mixed exercise 1

1.7 Review exercise for chapter 1

 

2. FUNCTIONS AND GRAPHS

2.1 The modulus function

2.2 Functions and mappings

2.3 Composite functions

2.4 Inverse functions

2.5 $y=\vert f(x)\vert$ and $y=f(\vert x\vert)$

2.6 Combining transformations

2.7 Solving modulus problems

2.8 Mixed exercise 2

2.9 Review exercise for chapter 2

 

3. SEQUENCES AND SERIES

3.1 Arithmetic sequences

3.2 Arithmetic series

3.3 Geometric sequences

3.4 Geometric series

3.5 Sum to infinity

3.6 Sigma notation

3.7 Recurrence relations

3.8 Modelling with series

3.9 Mixed exercise 3

3.10 Review exercise for chapter 3

 

4. BINOMIAL EXPANSION

4.1 Expanding $(1+x)^n$

4.2 Expanding $(a+bx)^n$

4.3 Using partial fractions

4.4 Mixed exercise 4

4.5 Review exercise for chapter 4

 

5. RADIANS

5.1 Radian measure

5.2 Arc length

5.3 Areas of sectors and segments

5.4 Solving trigonometric equations

5.5 Small angle approximations

5.6 Mixed exercise 5

5.7 Review exercise for chapter 5

 

6. TRIGONOMETRIC FUNCTIONS

6.1 Secant, cosecant and cotangent

6.2 Graphs of $\sec x$, $\csc x$ and $\cot x$

6.3 Using $\sec x$, $\csc x$ and $\cot x$

6.4 Trigonometric identities

6.5 Inverse trigonometric functions

6.6 Mixed exercise 6

6.7 Review exercise for chapter 6

 

7. TRIGONOMETRY AND MODELLING

7.1 Addition formulae

7.2 Using the angle addition formulae

7.3 Double-angle formulae

7.4 Solving trigonometric equations

7.5 Simplifying $a \cos x \pm b \sin x$ (Harmonic forms/identities)

7.6 Proving trigonometric identities

7.7 Modelling with trigonometric functions

7.8 Mixed exercise 7

7.9 Review exercise for chapter 7

 

8. PARAMETRIC EQUATIONS

8.1 Parametric equations

8.2 Using trigonometric identities

8.3 Curve sketching

8.4 Points of interection

8.5 Modelling with parametric equations

8.6 Mixed exercise 8

8.7 Review exercise for chapter 8

 

9. DIFFERENTIATION

9.1 Differentiating $\sin x$ and $\cos x$

9.2 Differentiating exponentials and logarithms

9.3 The chain rule

9.4 The product rule

9.5 The quotient rule

9.6 Differentiating trigonometric functions

9.7 Parametric differentiation

9.8 Implicit differentiation

9.9 Using second derivatives

9.10 Rates of change

9.11 Mixed exercise 9

9.12 Review exercise for chapter 9

 

10. NUMERICAL METHODS

10.1 Locating roots

10.2 Iteration

10.3 The Newton-Raphson method

10.4 Applications to modelling

10.5 Mixed exercise 10

10.6 Review exercise for chapter 10

 

11. INTEGATION

11.1 Integrating standard functions

11.2 Reverse chain rule 1: Integrating ${\rm f}(ax+b)$

11.3 Using trigonometric identities

11.4 Reverse chain rule 2: Integrating ${\rm f}'(x)/{\rm f}(x)$ and ${\rm f}'(x)[{\rm f}(x)]^n$

11.5 Integration by substitution (Reverse chain rule)

11.6 Integration by parts (Reverse product rule)

11.7 Partial fractions

11.8 Finding areas

11.9 The trapezium rule

11.10 Solving differential equations

11.11 Modelling with differential equations

11.12 Mixed exercise 11

11.13 Review exercise for chapter 11

 

12. VECTORS

12.1 3D coordinates

12.2 Vectors in 3D

12.3 Solving geometric problems

12.4 Application to mechanics

12.5 Mixed exercise 12

12.6 Review exercise for chapter 12

Mechanics Year 1

1. MODELLING IN MECHANICS

1.1 Constructing a model

1.2 Modelling assumptions

1.3 Quantities and units

1.4 Working with vectors

1.5 Mixed exercise 1

1.6 Review exercise for chapter 1

 

2. CONSTANT ACCELERATION

2.1 Displacement-time graphs

2.2 Velocity-time graphs

2.3 Constant acceleration formulae 1

2.4 Constant acceleration formulae 2

2.5 Vertical motion under gravity

2.6 Mixed exercise 2

2.7 Review exercise for chapter 2

 

3. FORCES AND MOTION

3.1 Force diagrams

3.2 Forces as vectors

3.3 Forces and acceleration

3.4 Motion in 2 dimensions

3.5 Connected particles

3.6 Pulleys

3.7 Mixed exercise 3

3.8 Review exercise for chapter 3

 

4. VARIABLE ACCELERATION

4.1 Functions of time

4.2 Using differentiation

4.3 Maxima and minima problems

4.4 Using integration

4.5 Constant acceleration formulae

4.6 Mixed exercise 4

4.7 Review exercise for chapter 4

 

Mechanics Year 2

 

1. MOMENTS

1.1 Moments

1.2 Resultant moments

1.3 Equilibrium

1.4 Centres of mass

1.5 Tilting

1.6 Mixed exercise 1

1.7 Review exercise for chapter 1

 

2. FORCES AND FRICTION

2.1 Resolving forces

2.2 Inclined planes

2.3 Friction

2.4 Mixed exercise 2

2.5 Review exercise for chapter 2

 

3. PROJECTILES

3.1 Horizontal projection

3.2 Horizontal and vertical components

3.3 Projection at any angle

3.4 Projectile motion formulae

3.5 Mixed exercise 3

3.6 Review exercise for chapter 3

 

4. APPLICATIONS OF FORCES

4.1 Static particles

4.2 Modelling with statics

4.3 Friction and static particles 4.4 Static rigid particles

4.5 Dynamics and inclined planes

4.6 Connected particles

4.7 Mixed exercise 4

4.8 Review exercise for chapter 4

 

5. FURTHER KINEMATICS

5.1 Vectors in kinematics

5.2 Vector methods with projectiles

5.3 Variable acceleration in one dimension

5.4 Differentiating vectors

5.5 Integrating vectors

5.6 Mixed exercise 5

5.7 Review exercise for chapter 5

Statistics Year 1

1. DATA COLLECTION

1.1 Populations and samples

1.2 Sampling

1.3 Non-random sampling

1.4 Types of data

1.5 The large data set

1.6 Mixed exercise 1

1.7 Review exercise for chapter 1

 

2. MEASURES OF LOCATION AND SPREAD

2.1 Measures of central tendency

2.2 Other measures of location

2.3 Measures of spread

2.4 Variance and standard deviation

2.5 Coding

2.6 Mixed exercise 2

2.7 Review exercise for chapter 2

 

3. REPRESENTATIONS OF DATA

3.1 Outliers

3.2 Box plots

3.3 Cumulative frequency

3.4 Histograms

3.5 Comparing data

3.6 Mixed exercise 3

3.7 Review exercise for chapter 3

 

4. CORRELATION

4.1 Correlation

4.2 Linear regression

4.3 Mixed exercise 4

4.4 Review exercise for chapter 4

 

5. PROBABILITY

5.1 Calculating probabilities

5.2 Venn diagrams

5.3 Mutually exclusive and independent events

5.4 Tree diagrams

5.5 Mixed exercise 5

5.6 Review exercise for chapter 5

 

6. STATISTICAL DISTRIBUTIONS

6.1 Probability distributions

6.2 The binomial distribution

6.3 Cumulative probabilities

6.4 Mixed exercise 6

6.5 Review exercise for chapter 6

 

7. HYPOTHESIS TESTING

7.1 Hypothesis testing

7.2 Finding critical values

7.3 One-tailed tests

7.4 Two-tailed tests

7.5 Mixed exercise 6

7.6 Review exercise for chapter 6

Statistics 2

1. REGRESSION, CORRELATION AND HYPOTHESIS TESTING

1.1 Exponential models

1.2 Measuring correlation

1.3 Hypothesis testing for zero correlation

1.4 Mixed exercise 1

1.5 Review exercise for chapter 1

 

2. CONDITIONAL PROBABILITY

2.1 Set notation

2.2 Conditional probability

2.3 Conditional probabilities in Venn diagrams

2.4 Probability formulae

2.5 Tree diagrams

2.6 Mixed exercise 2

2.7 Review exercise for chapter 2

 

3. THE NORMAL DISTRIBUTION

3.1 The normal distribution

3.2 Finding probabilities for normal distributions

3.3 The inverse normal distribution function

3.4 The standard normal distribution

3.5 Finding $\mu$ and $\sigma$

3.6 Approximating a binomial distribution

3.7 Hypothesis testing with the normal distribution

3.8 Mixed exercise 3

3.9 Review exercise for chapter 3