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Cambridge AS and A Level Mathematics (9709) Syllabus

The Mathematics (9709) syllabus for Cambridge AS and A Level in full: 11 chapters and 37 topics, every one listed below, plus 50 Mathematics (9709) flashcards with answers.

11Chapters
37Topics
50Flashcards

Mathematics (9709) syllabus: all 11 chapters

Expand any chapter to see its topics and sub-topics.

  1. Quadratics

     3 topics
    • Completing the square and the discriminant
    • Solving quadratic equations and inequalities
    • Quadratic graphs and transformations

  2. Functions

     3 topics
    • Domain, range and composite functions
    • Inverse functions
    • Graph transformations

  3. Coordinate Geometry

     3 topics
    • Equation of a straight line
    • The circle
    • Points of intersection

  4. Trigonometry

     3 topics
    • Trigonometric ratios and graphs
    • Trigonometric identities and equations
    • Radian measure, arc length and sector area

  5. Differentiation

     3 topics
    • Differentiation from first principles and rules
    • Stationary points and second derivatives
    • Connected rates of change

  6. Integration

     3 topics
    • Indefinite and definite integration
    • Area under a curve and volumes of revolution
    • Integration techniques

  7. Series

     2 topics
    • Arithmetic and geometric progressions
    • Binomial expansion

  8. Exponentials and Logarithms

     3 topics
    • Laws of logarithms
    • Exponential and logarithmic functions
    • Reduction to linear form

  9. Further Pure Topics

     4 topics
    • Numerical solution of equations
    • Vectors in three dimensions
    • Complex numbers
    • Differential equations

  10. Mechanics

     4 topics
    • Forces and equilibrium
    • Kinematics of motion in a straight line
    • Newton's laws of motion
    • Momentum and energy

  11. Probability and Statistics

     6 topics
    • Representation and measures of data
    • Permutations and combinations
    • Probability
    • Discrete random variables
    • The normal distribution
    • Hypothesis testing

Cambridge AS and A Level Mathematics (9709) flashcards — 50 questions with answers

  1. What is the result of completing the square for x² + bx + c?

    (x + b/2)² + c − (b/2)², i.e. (x + b/2)² − b²/4 + c.

  2. In the completed-square form a(x − h)² + k, what do h and k represent?

    The coordinates of the vertex (turning point) of the parabola are (h, k).

  3. What is the discriminant of ax² + bx + c = 0, and its symbol?

    The discriminant is Δ = b² − 4ac.

  4. What does the discriminant tell you about the roots of a quadratic when b² − 4ac > 0?

    Two distinct real roots.

  5. What does b² − 4ac = 0 indicate about a quadratic's roots?

    One repeated (equal) real root; the graph touches the x-axis at one point.

  6. What does b² − 4ac < 0 indicate about a quadratic's roots?

    No real roots (two complex roots); the graph does not cross the x-axis.

  7. State the quadratic formula for solving ax² + bx + c = 0.

    x = (−b ± √(b² − 4ac)) / (2a).

  8. For a quadratic to be a perfect square (tangent to x-axis), what condition must the discriminant satisfy?

    b² − 4ac = 0.

  9. When solving the inequality ax² + bx + c > 0 with a > 0 and two real roots p < q, what is the solution set?

    x < p or x > q (the regions outside the roots).

  10. When solving ax² + bx + c < 0 with a > 0 and roots p < q, what is the solution set?

    p < x < q (the region between the roots).

  11. What is the first step to solve a quadratic inequality?

    Rearrange so one side is zero, find the critical values (roots), then test the sign of each interval (or use a sketch).

  12. A quadratic graph y = ax² + bx + c opens upward when ___ and downward when ___.

    Opens upward when a > 0; opens downward when a < 0.

  13. What is the equation of the axis of symmetry of y = ax² + bx + c?

    x = −b/(2a).

  14. How is the graph of y = f(x) + a transformed from y = f(x)?

    Translation by a units in the positive y-direction (upward), vector (0, a).

  15. How is the graph of y = f(x + a) transformed from y = f(x)?

    Translation by a units in the negative x-direction (left), vector (−a, 0).

  16. How does y = f(ax) transform y = f(x)?

    Horizontal stretch with scale factor 1/a (parallel to the x-axis).

  17. How does y = a·f(x) transform y = f(x)?

    Vertical stretch with scale factor a (parallel to the y-axis).

  18. What transformation does y = −f(x) represent?

    Reflection in the x-axis.

  19. What transformation does y = f(−x) represent?

    Reflection in the y-axis.

  20. Define the domain of a function.

    The set of all input values (x-values) for which the function is defined.

Show the other 30 Mathematics (9709) flashcards

  1. Define the range of a function.

    The set of all possible output values (y-values) the function can produce.

  2. What does the composite function fg(x) (or f∘g) mean?

    Apply g first, then apply f to the result: fg(x) = f(g(x)).

  3. For the composite fg(x) to be defined, what must be true?

    The range of g must lie within (be a subset of) the domain of f.

  4. Is function composition commutative? i.e. does fg(x) = gf(x) in general?

    No; in general fg(x) ≠ gf(x).

  5. What condition must a function satisfy to have an inverse?

    It must be one-to-one (one-one) over its domain (each output comes from exactly one input).

  6. How are the domain and range of f⁻¹ related to those of f?

    The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f.

  7. What is the geometric relationship between the graphs of y = f(x) and y = f⁻¹(x)?

    They are reflections of each other in the line y = x.

  8. What is the process to find the inverse function f⁻¹(x)?

    Write y = f(x), swap x and y (or solve for x), then make y the subject; rename as f⁻¹(x).

  9. What are ff⁻¹(x) and f⁻¹f(x) equal to?

    Both equal x (over the appropriate domains).

  10. State the equation of a straight line in gradient-intercept form.

    y = mx + c, where m is the gradient and c is the y-intercept.

  11. How do you find the gradient of a line through points (x₁, y₁) and (x₂, y₂)?

    m = (y₂ − y₁) / (x₂ − x₁).

  12. State the point-gradient form of the equation of a straight line.

    y − y₁ = m(x − x₁).

  13. What is the relationship between the gradients of two parallel lines?

    Their gradients are equal (m₁ = m₂).

  14. What is the relationship between the gradients of two perpendicular lines?

    The product of their gradients is −1 (m₁ · m₂ = −1), i.e. m₂ = −1/m₁.

  15. State the distance formula between two points (x₁, y₁) and (x₂, y₂).

    d = √((x₂ − x₁)² + (y₂ − y₁)²).

  16. State the midpoint formula for the segment joining (x₁, y₁) and (x₂, y₂).

    Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

  17. State the equation of a circle with centre (a, b) and radius r.

    (x − a)² + (y − b)² = r².

  18. State the general expanded form of a circle's equation.

    x² + y² + 2gx + 2fy + c = 0, with centre (−g, −f) and radius √(g² + f² − c).

  19. What is the geometric property linking a tangent to a circle and the radius at the point of contact?

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

  20. How do you find the points of intersection of a line and a curve algebraically?

    Solve their equations simultaneously (usually by substitution), giving a single equation whose solutions are the intersection coordinates.

  21. When a line meets a circle, what does the discriminant of the resulting quadratic tell you?

    Δ > 0: line is a secant (two intersection points); Δ = 0: line is a tangent (one point); Δ < 0: line misses the circle.

  22. Give the exact values of sin, cos and tan of 30°.

    sin30° = 1/2, cos30° = √3/2, tan30° = 1/√3.

  23. Give the exact values of sin, cos and tan of 45°.

    sin45° = 1/√2, cos45° = 1/√2, tan45° = 1.

  24. Give the exact values of sin, cos and tan of 60°.

    sin60° = √3/2, cos60° = 1/2, tan60° = √3.

  25. State the periods of the graphs y = sin x, y = cos x and y = tan x (in degrees).

    sin and cos have period 360°; tan has period 180°.

  26. State the Pythagorean trigonometric identity.

    sin²θ + cos²θ = 1.

  27. State the identity relating tan θ to sin θ and cos θ.

    tan θ = sin θ / cos θ (for cos θ ≠ 0).

  28. How do you convert degrees to radians?

    Multiply by π/180; equivalently, 180° = π radians.

  29. State the formula for the arc length of a sector with radius r and angle θ in radians.

    Arc length s = rθ.

  30. State the formula for the area of a sector with radius r and angle θ in radians.

    Area = ½ r²θ.

Cambridge AS and A Level Mathematics (9709) FAQ

What is in the Cambridge AS and A Level Mathematics (9709) syllabus?

Cambridge AS and A Level Mathematics (9709) is split into 11 chapters: Quadratics, Functions, Coordinate Geometry, Trigonometry, Differentiation, Integration, Series, Exponentials and Logarithms, Further Pure Topics, Mechanics and Probability and Statistics. Together they contain 37 topics.

How many chapters are in Cambridge AS and A Level Mathematics (9709)?

11 chapters, containing 37 topics. The largest chapter is Probability and Statistics, with 6 topics.

Are there flashcards for Cambridge AS and A Level Mathematics (9709)?

Yes — a 50-card Mathematics (9709) deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.