A-Level Maths (Legacy) Help & Practice

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  1. 1Surds
    1. 1.1Square and square roots
    2. 1.2Cubic and cube roots
    3. 1.3Evaluating and simplifying surds
    4. 1.4Converting radicals to mixed radicals
    5. 1.5Converting radicals to entire radicals
    6. 1.6Adding and subtracting surds
    7. 1.7Multiplying and dividing surds
    8. 1.8Rationalize the denominator
  2. 2Factorisation
    1. 2.1Factorise by taking out the greatest common factor
    2. 2.2Factorise by grouping
    3. 2.3Factorising difference of squares: x^2 - y^2
    4. 2.4Factorising trinomials
    5. 2.5Factoring difference of cubes
    6. 2.6Factoring sum of cubes
  3. 3Quadratic Functions
    1. 3.1Introduction to quadratic functions
    2. 3.2Transformations of quadratic functions
    3. 3.3Quadratic function in general form: y = ax^2 + bx + c
    4. 3.4Quadratic function in vertex form: y = a(x-p)^2 + q
    5. 3.5Completing the square
    6. 3.6Converting from general to vertex form by completing the square
    7. 3.7Shortcut: Vertex formula
    8. 3.8Graphing quadratic functions: General form VS. Vertex form
    9. 3.9Finding the quadratic functions for given parabolas
    10. 3.10Applications of quadratic functions
  4. 4Quadratic Equations
    1. 4.1Solving quadratic equations by factorising
    2. 4.2Solving quadratic equations by completing the square
    3. 4.3Using quadratic formula to solve quadratic equations
    4. 4.4The discriminant: Nature of roots of quadratic equations
    5. 4.5Applications of quadratic equations
  5. 5Inequalities
    1. 5.1Express linear inequalities graphically and algebraically
    2. 5.2Solving one-step linear inequalities
    3. 5.3Solving multi-step linear inequalities
    4. 5.4Compound inequalities
    5. 5.5Solving quadratic inequalities
    6. 5.6Solving absolute value inequalities
  6. 6Simultaneous Equations
    1. 6.1Simultaneous linear equations
    2. 6.2Simultaneous linear-quadratic equations
    3. 6.3Simultaneous quadratic-quadratic equations
    4. 6.4Solving 3 variable simultaneous equations by substitution
    5. 6.5Solving 3 variable simultaneous equations by elimination
    6. 6.6Solving 3 variable simultaneous equations with no or infinite solutions
    7. 6.7Word problems relating 3 variable simultaneous equations
  7. 7Operations with Algebraic Fractions
    1. 7.1Simplifying algebraic fractions and restrictions
    2. 7.2Adding and subtracting algebraic fractions
    3. 7.3Multiplying algebraic fractions
    4. 7.4Dividing algebraic fractions
    5. 7.5Solving equations with algebraic fractions
    6. 7.6Applications of equations with algebraic fractions
    7. 7.7Simplifying complex fractions
    8. 7.8Partial fraction decomposition
  8. 8Algebraic Division
    1. 8.1Polynomial long division
    2. 8.2Polynomial synthetic division
    3. 8.3Remainder theorem
    4. 8.4Factor theorem
  9. 9Linear Functions
    1. 9.1Distance formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
    2. 9.2Midpoint formula: M=(x1+x22,y1+y22)M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)
    3. 9.3Gradient equation: m=y2−y1x2−x1m = \frac{y_2-y_1}{x_2- x_1}
    4. 9.4Gradient intercept form: y = mx + b
    5. 9.5General form: Ax + By + C = 0
    6. 9.6Gradient-point form: y−y1=m(x−x1)y - y_1 = m (x - x_1)
    7. 9.7Rate of change
    8. 9.8Graphing linear functions using table of values
    9. 9.9Graphing linear functions using x- and y-intercepts
    10. 9.10Graphing from slope-intercept form y=mx+b
    11. 9.11Graphing linear functions using a single point and gradient
    12. 9.12Word problems of graphing linear functions
    13. 9.13Parallel and perpendicular lines in linear functions
    14. 9.14Applications of linear relations
  10. 10Functions
    1. 10.1Function notation
    2. 10.2Operations with functions
    3. 10.3Adding functions
    4. 10.4Subtracting functions
    5. 10.5Multiplying functions
    6. 10.6Dividing functions
    7. 10.7Composite functions
    8. 10.8Inverse functions
    9. 10.9Domain and range of a function
    10. 10.10One to one functions
    11. 10.11Difference quotient: applications of functions
  11. 11Transformations of Functions
    1. 11.1Transformations of functions: Horizontal translations
    2. 11.2Transformations of functions: Vertical translations
    3. 11.3Reflection across the y-axis: y = f(-x)
    4. 11.4Reflection across the x-axis: y = -f(x)
    5. 11.5Transformations of functions: Horizontal stretches
    6. 11.6Transformations of functions: Vertical stretches
    7. 11.7Combining transformations of functions
  12. 12Reciprocal Functions
    1. 12.1Graphing reciprocals of linear functions
    2. 12.2Graphing reciprocals of quadratic functions
  13. 13Rational Functions
    1. 13.1What is a rational function?
    2. 13.2Point of discontinuity
    3. 13.3Vertical asymptote
    4. 13.4Horizontal asymptote
    5. 13.5Slant asymptote
    6. 13.6Graphs of rational functions
    7. 13.7Solving rational inequalities
  14. 14Absolute Value Functions
    1. 14.1Absolute value functions
    2. 14.2Solving absolute value equations
  15. 15Laws of Indices
    1. 15.1Indices: Product rule (a^x)(a^y) = a^(x+y)
    2. 15.2Indices: Division rule (a^x / a^y) = a^(x-y)
    3. 15.3Indices: Power rule (a^x)^y = a^(x * y)
    4. 15.4Negative indices
    5. 15.5Zero index: a^0 = 1
    6. 15.6Rational indices
  16. 16Exponentials Functions
    1. 16.1Solving exponential equations using laws of indices
    2. 16.2Graphing exponential functions
    3. 16.3Graphing transformations of exponential functions
    4. 16.4Finding an exponential function given its graph
  17. 17Logarithms
    1. 17.1What is a logarithm?
    2. 17.2Converting from logarithmic form to exponential form
    3. 17.3Evaluating logarithms without a calculator
    4. 17.4Common logarithms
    5. 17.5Natural log: ln
    6. 17.6Evaluating logarithms using change-of-base formula
    7. 17.7Converting from exponential form to logarithmic form
    8. 17.8Solving exponential equations with logarithms
    9. 17.9Product rule of logarithms
    10. 17.10Quotient rule of logarithms
    11. 17.11Combining product rule and quotient rule in logarithms
    12. 17.12Evaluating logarithms using logarithm rules
    13. 17.13Solving logarithmic equations
    14. 17.14Graphing logarithmic functions
    15. 17.15Finding a logarithmic function given its graph
  18. 18Trigonometry
    1. 18.1Use sine ratio to calculate angles and sides (Sin = oh \frac{o}{h} )
    2. 18.2Use cosine ratio to calculate angles and sides (Cos = ah \frac{a}{h} )
    3. 18.3Use tangent ratio to calculate angles and sides (Tan = oa \frac{o}{a} )
    4. 18.4Combination of SohCahToa questions
    5. 18.5Solving expressions using 45-45-90 special right triangles
    6. 18.6Solving expressions using 30-60-90 special right triangles
    7. 18.7Word problems relating ladder in trigonometry
    8. 18.8Word problems relating guy wire in trigonometry
    9. 18.9Other word problems relating angles in trigonometry
  19. 19Trigonometric Ratios and Angle Measure
    1. 19.1Unit circle
    2. 19.2Converting between degrees and radians
    3. 19.3Trigonometric ratios of angles in radians
    4. 19.4Radian measure and arc length
  20. 20Sine Rule and Cosine Rule
    1. 20.1Sine rule
    2. 20.2Cosine rule
    3. 20.3Sine rule and cosine rule word problems
  21. 21Bearings
    1. 21.1Introduction to bearings
    2. 21.2Bearings and direction word problems
    3. 21.3Angle of elevation and depression
  22. 22Graphing Trigonometric Functions
    1. 22.1Sine graph: y = sin x
    2. 22.2Cosine graph: y = cos x
    3. 22.3Tangent graph: y = tan x
    4. 22.4Cotangent graph: y = cot x
    5. 22.5Secant graph: y = sec x
    6. 22.6Cosecant graph: y = csc x
    7. 22.7Graphing transformations of trigonometric functions
    8. 22.8Determining trigonometric functions given their graphs
  23. 23Trigonometric Identities
    1. 23.1Quotient identities and reciprocal identities
    2. 23.2Pythagorean identities
    3. 23.3Sum and difference identities
    4. 23.4Cofunction identities
    5. 23.5Double-angle identities
  24. 24Solving Trigonometric Equations
    1. 24.1Solving first degree trigonometric equations
    2. 24.2Determining non-permissible values for trig expressions
    3. 24.3Solving second degree trigonometric equations
    4. 24.4Solving trigonometric equations involving multiple angles
    5. 24.5Solving trigonometric equations using pythagorean identities
    6. 24.6Solving trigonometric equations using sum and difference identities
    7. 24.7Solving trigonometric equations using double-angle identities
  25. 25Inverse Trigonometric Functions
    1. 25.1Finding inverse trigonometric function from its graph
    2. 25.2Evaluating inverse trigonometric functions
    3. 25.3Finding inverse reciprocal trigonometric function from its graph
    4. 25.4Finding exact value of inverse reciprocal trig functions
  26. 26Circle Theorems
    1. 26.1Angles in a circle
    2. 26.2Chord properties
    3. 26.3Tangent properties
    4. 26.4Circles and circumference
    5. 26.5Arcs of a circle
    6. 26.6Areas and sectors of circles
    7. 26.7Inscribed quadrilaterals in circles
    8. 26.8Central and inscribed angles in circles
    9. 26.9Circles in coordinate plane
  27. 27Sequences and Series
    1. 27.1Arithmetic sequences
    2. 27.2Arithmetic series
    3. 27.3Geometric sequences
    4. 27.4Geometric series
    5. 27.5Infinite geometric series
    6. 27.6Sigma notation
    7. 27.7Arithmetic mean vs. Geometric mean
  28. 28Binomial Expansion
    1. 28.1Factorial notation
    2. 28.2Pascal's triangle
    3. 28.3Binomial theorem
  29. 29Differentiation
    1. 29.1Power rule
    2. 29.2Gradient and equation of tangent line
    3. 29.3Chain rule
    4. 29.4Derivative of trigonometric functions
    5. 29.5Derivative of exponential functions
    6. 29.6Product rule
    7. 29.7Quotient rule
    8. 29.8Derivative of inverse trigonometric functions
    9. 29.9Derivative of logarithmic functions
    10. 29.10Higher order derivatives
    11. 29.11Critical number & maximum and minimum values
    12. 29.12Curve sketching
  30. 30Integration
    1. 30.1Antiderivatives
    2. 30.2Definite integral
    3. 30.3U-Substitution
    4. 30.4Integration by parts
    5. 30.5Trigonometric substitution
    6. 30.6Integration of rational functions by partial fractions
    7. 30.7Volumes of solids of revolution - Disc method
    8. 30.8Volumes of solids of revolution - Shell method
  31. 31Parametric Equations
    1. 31.1Defining curves with parametric equations
    2. 31.2Tangent and concavity of parametric equations
  32. 32Differential Equations
    1. 32.1Introduction to differential equations
    2. 32.2Separable equations
    3. 32.3Applications to differential equations
  33. 33Vectors
    1. 33.1Introduction to vectors
    2. 33.2Magnitude of a vector
    3. 33.3Direction angle of a vector
    4. 33.4Scalar multiplication
    5. 33.5Equivalent vectors
    6. 33.6Adding and subtracting vectors in component form
    7. 33.7Operations on vectors in magnitude and direction form
    8. 33.8Unit Vector
    9. 33.9Word problems on vectors
  34. 34Statistics
    1. 34.1Median and mode
    2. 34.2Mean
    3. 34.3Range and outliers
    4. 34.4Application of averages
    5. 34.5Spread of a data set - standard deviation & variance
  35. 35Data and Graphs
    1. 35.1Reading and drawing bar graphs
    2. 35.2Reading and drawing histograms
    3. 35.3Reading and drawing line graphs
    4. 35.4Box-and-whisker plots and scatter plots
    5. 35.5Reading and drawing Venn diagrams
    6. 35.6Stem-and-leaf plots
  36. 36Probability
    1. 36.1Addition rule for "OR"
    2. 36.2Multiplication rule for "AND"
    3. 36.3Conditional probability
  37. 37Discrete Probabilities
    1. 37.1Probability distribution - histogram, mean, variance & standard deviation
    2. 37.2Binomial distribution
    3. 37.3Mean and standard deviation of binomial distribution
    4. 37.4Poisson distribution
  38. 38Normal Distribution and Confidence Intervals
    1. 38.1Introduction to normal distribution
    2. 38.2Normal distribution and continuous random variable
    3. 38.3Z-scores and random continuous variables
    4. 38.4Sampling distributions
    5. 38.5Central limit theorem
    6. 38.6Confidence levels and critical values
    7. 38.7Confidence intervals to estimate population mean
    8. 38.8Student's t-distribution
  39. 39Correlation and Regression
    1. 39.1Bivariate, scatter plots and correlation
    2. 39.2Regression analysis
    3. 39.3Equation of the best fit line
  40. 40Hypothesis Testing
    1. 40.1Null hypothesis and alternative hypothesis
    2. 40.2Proving claims
    3. 40.3Confidence levels, significance levels and critical values
    4. 40.4Test statistics
    5. 40.5Traditional hypothesis testing
    6. 40.6P-value hypothesis testing
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