Question 1 of 3 · No calculator needed
Evaluate 8C3.
8C3 = 8!/(3!5!) = (8×7×6)/(3×2×1) = 56.
Cambridge International · A LevelMathematics,
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Specification 9709. Explore a topic, practise the skill, and see where to focus next.
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Question 2 of 3 · No calculator needed
For which values of k does x² + kx + 9 = 0 have equal roots?
Equal roots require discriminant zero: k² - 4(1)(9) = 0. Thus k² = 36 and k = ±6.
Question 3 of 3 · No calculator needed
Simplify sin x/(1 + cos x) by rationalising the trigonometric denominator.
Multiply by (1-cos x)/(1-cos x). This gives sin x(1-cos x)/sin²x = (1-cos x)/sin x = tan(x/2).
These tiles show your answers to three questions. They are a starting point, not a mastery score or grade prediction.
Permutations and combinations
Correct answer: 56
8C3 = 8!/(3!5!) = (8×7×6)/(3×2×1) = 56.
Quadratics
Correct answer: k = ±6
Equal roots require discriminant zero: k² - 4(1)(9) = 0. Thus k² = 36 and k = ±6.
Trigonometry
Correct answer: tan(x/2)
Multiply by (1-cos x)/(1-cos x). This gives sin x(1-cos x)/sin²x = (1-cos x)/sin x = tan(x/2).
For parents: look at the explanation together. A correct answer is encouraging; a missed answer gives you something specific to work on. Broader practice over time is needed to understand progress.
YOUR TOPIC MAP
38 areasFind your starting point.
01Quadratics
Specification reference: 9709 1.1 Quadratics
Quadratics requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Quadratics requires. Keep exact values until the final step and check signs, units and reasonableness.
02Functions
Specification reference: 9709 1.2 Functions
Functions requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Functions requires. Keep exact values until the final step and check signs, units and reasonableness.
03Coordinate geometry
Specification reference: 9709 1.3 Coordinate geometry
Coordinate geometry requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Coordinate geometry requires. Keep exact values until the final step and check signs, units and reasonableness.
04Circular measure
Specification reference: 9709 1.4 Circular measure
Circular measure requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Circular measure requires. Keep exact values until the final step and check signs, units and reasonableness.
05Trigonometry
Specification reference: 9709 1.5 Trigonometry
Trigonometry requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Trigonometry requires. Keep exact values until the final step and check signs, units and reasonableness.
06Series
Specification reference: 9709 1.6 Series
Series requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Series requires. Keep exact values until the final step and check signs, units and reasonableness.
07Differentiation
Specification reference: 9709 1.7 Differentiation
Differentiation requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Differentiation requires. Keep exact values until the final step and check signs, units and reasonableness.
08Integration
Specification reference: 9709 1.8 Integration
Integration requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Integration requires. Keep exact values until the final step and check signs, units and reasonableness.
09Algebra
Specification reference: 9709 2.1 Algebra
Algebra requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Algebra requires. Keep exact values until the final step and check signs, units and reasonableness.
10Logarithmic and exponential functions
Specification reference: 9709 2.2 Logarithmic and exponential functions
Logarithmic and exponential functions requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Logarithmic and exponential functions requires. Keep exact values until the final step and check signs, units and reasonableness.
11Trigonometry
Specification reference: 9709 2.3 Trigonometry
Trigonometry requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Trigonometry requires. Keep exact values until the final step and check signs, units and reasonableness.
12Differentiation
Specification reference: 9709 2.4 Differentiation
Differentiation requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Differentiation requires. Keep exact values until the final step and check signs, units and reasonableness.
13Integration
Specification reference: 9709 2.5 Integration
Integration requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Integration requires. Keep exact values until the final step and check signs, units and reasonableness.
14Numerical solution of equations
Specification reference: 9709 2.6 Numerical solution of equations
Numerical solution of equations requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Numerical solution of equations requires. Keep exact values until the final step and check signs, units and reasonableness.
15Algebra
Specification reference: 9709 3.1 Algebra
Algebra requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Algebra requires. Keep exact values until the final step and check signs, units and reasonableness.
16Logarithmic and exponential functions
Specification reference: 9709 3.2 Logarithmic and exponential functions
Logarithmic and exponential functions requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Logarithmic and exponential functions requires. Keep exact values until the final step and check signs, units and reasonableness.
17Trigonometry
Specification reference: 9709 3.3 Trigonometry
Trigonometry requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Trigonometry requires. Keep exact values until the final step and check signs, units and reasonableness.
18Differentiation
Specification reference: 9709 3.4 Differentiation
Differentiation requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Differentiation requires. Keep exact values until the final step and check signs, units and reasonableness.
19Integration
Specification reference: 9709 3.5 Integration
Integration requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Integration requires. Keep exact values until the final step and check signs, units and reasonableness.
20Numerical solution of equations
Specification reference: 9709 3.6 Numerical solution of equations
Numerical solution of equations requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Numerical solution of equations requires. Keep exact values until the final step and check signs, units and reasonableness.
21Vectors
Specification reference: 9709 3.7 Vectors
Vectors requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Vectors requires. Keep exact values until the final step and check signs, units and reasonableness.
22Differential equations
Specification reference: 9709 3.8 Differential equations
Differential equations requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Differential equations requires. Keep exact values until the final step and check signs, units and reasonableness.
23Complex numbers
Specification reference: 9709 3.9 Complex numbers
Complex numbers requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Complex numbers requires. Keep exact values until the final step and check signs, units and reasonableness.
24Forces and equilibrium
Specification reference: 9709 4.1 Forces and equilibrium
Forces and equilibrium requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Forces and equilibrium requires. Keep exact values until the final step and check signs, units and reasonableness.
25Kinematics of motion in a straight line
Specification reference: 9709 4.2 Kinematics of motion in a straight line
Kinematics of motion in a straight line requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Kinematics of motion in a straight line requires. Keep exact values until the final step and check signs, units and reasonableness.
26Momentum
Specification reference: 9709 4.3 Momentum
Momentum requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Momentum requires. Keep exact values until the final step and check signs, units and reasonableness.
27Newton's laws of motion
Specification reference: 9709 4.4 Newton's laws of motion
Newton's laws of motion requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Newton's laws of motion requires. Keep exact values until the final step and check signs, units and reasonableness.
28Energy, work and power
Specification reference: 9709 4.5 Energy, work and power
Energy, work and power requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Energy, work and power requires. Keep exact values until the final step and check signs, units and reasonableness.
29Representation of data
Specification reference: 9709 5.1 Representation of data
Representation of data requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Representation of data requires. Keep exact values until the final step and check signs, units and reasonableness.
30Permutations and combinations
Specification reference: 9709 5.2 Permutations and combinations
Permutations and combinations requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Permutations and combinations requires. Keep exact values until the final step and check signs, units and reasonableness.
31Probability
Specification reference: 9709 5.3 Probability
Probability requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Probability requires. Keep exact values until the final step and check signs, units and reasonableness.
32Discrete random variables
Specification reference: 9709 5.4 Discrete random variables
Discrete random variables requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Discrete random variables requires. Keep exact values until the final step and check signs, units and reasonableness.
33The normal distribution
Specification reference: 9709 5.5 The normal distribution
The normal distribution requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what The normal distribution requires. Keep exact values until the final step and check signs, units and reasonableness.
34The Poisson distribution
Specification reference: 9709 6.1 The Poisson distribution
The Poisson distribution requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what The Poisson distribution requires. Keep exact values until the final step and check signs, units and reasonableness.
35Linear combinations of random variables
Specification reference: 9709 6.2 Linear combinations of random variables
Linear combinations of random variables requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Linear combinations of random variables requires. Keep exact values until the final step and check signs, units and reasonableness.
36Continuous random variables
Specification reference: 9709 6.3 Continuous random variables
Continuous random variables requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Continuous random variables requires. Keep exact values until the final step and check signs, units and reasonableness.
37Sampling and estimation
Specification reference: 9709 6.4 Sampling and estimation
Sampling and estimation requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Sampling and estimation requires. Keep exact values until the final step and check signs, units and reasonableness.
38Hypothesis tests
Specification reference: 9709 6.5 Hypothesis tests
Hypothesis tests requires the correct mathematical rule, exact working and a final check that the result satisfies the conditions in the question.
Watch for: Do not select a familiar formula before identifying what Hypothesis tests requires. Keep exact values until the final step and check signs, units and reasonableness.
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Cambridge International · A Level · Mathematics · 9709
Version 4, December 2025, for examinations in 2026-2027; current checked 2026-08-27
Source checked: 2026-08-27. The current official specification controls assessment requirements and option choices.
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It shows the outcome of these three answers and gives a specific skill to discuss or practise next. It is not a full assessment, a mastery score or a grade prediction. Broader practice over time is needed to understand progress.
What does the Cambridge International A Level Mathematics route cover?
LearningP currently maps 38 assessed areas for specification 9709. The visible topic map below is derived from the verified route; the current official specification remains controlling.
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The verified source bank contains 401 original LearningP question records for this route. Every mapped area meets the current publication minimum and passed the latest blocker and review audit.
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