Cambridge AS and A Level Mathematics (9709) Syllabus

Mathematics (9709) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics (9709) in Cambridge AS and A Level, not a summary of it.

1. 1

   Quadratics

   3 topics

   • Completing the square and the discriminant
   • Solving quadratic equations and inequalities
   • Quadratic graphs and transformations
2. 2

   Functions

   3 topics

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

   Coordinate Geometry

   3 topics

   • Equation of a straight line
   • The circle
   • Points of intersection
4. 4

   Trigonometry

   3 topics

   • Trigonometric ratios and graphs
   • Trigonometric identities and equations
   • Radian measure, arc length and sector area
5. 5

   Differentiation

   3 topics

   • Differentiation from first principles and rules
   • Stationary points and second derivatives
   • Connected rates of change
6. 6

   Integration

   3 topics

   • Indefinite and definite integration
   • Area under a curve and volumes of revolution
   • Integration techniques
7. 7

   Series

   2 topics

   • Arithmetic and geometric progressions
   • Binomial expansion
8. 8

   Exponentials and Logarithms

   3 topics

   • Laws of logarithms
   • Exponential and logarithmic functions
   • Reduction to linear form
9. 9

   Further Pure Topics

   4 topics

   • Numerical solution of equations
   • Vectors in three dimensions
   • Complex numbers
   • Differential equations
10. 10

    Mechanics

    4 topics

    • Forces and equilibrium
    • Kinematics of motion in a straight line
    • Newton's laws of motion
    • Momentum and energy
11. 11

    Probability and Statistics

    6 topics

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

Mathematics (9709) flashcards for Cambridge AS and A Level

19 of 50 cards from the Mathematics (9709) deck — real questions with worked 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.

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