AS Level Mathematics (9709) topical questions with answers
Every AS Level Mathematics topic, with free past-paper-style questions and full mark schemes. Pick a chapter to practise its questions, or generate a complete 9709-format paper in seconds, from easy warm-ups to A*-level "brutal" papers.
Paper 1 · Pure Mathematics 1 (75 marks · 1h 50m); Paper 2 · Pure Mathematics 2 (50 marks · 1h 15m); Paper 4 · Mechanics (50 marks · 1h 15m); Paper 5 · Probability & Statistics 1 (50 marks · 1h 15m).
Sample AS Level Mathematics questions
A mixed set from across the syllabus. Questions marked A* challenge come from the hardest bank.
Question 1Structured · 6 marksA* challenge
(a)
Prove the identity 1 + sin θcos θ + cos θ1 + sin θ ≡ 2cos θ.
[3]
(b)
Hence solve the equation 1 + sin θcos θ + cos θ1 + sin θ = 4 for 0° < θ < 360°.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
combine over common denominator; (1 + sin θ)2 + cos2 θ = 2 + 2 sin θ; cancel (1 + sin θ)
common denominator cos θ(1 + sin θ) M1 use sin2 θ + cos2 θ = 1 to get 2(1 + sin θ) M1 cancel to 2 ÷ cos θ, fully shown A1
3
(b)
60°, 300°
cos θ = 12M1 one correct angle A1 60°, 300° and no others A1
3
Question 2Structured · 8 marksA* challenge
The first, second and third terms of a geometric progression are respectively the first, 4th and 10th terms of an arithmetic progression with first term a and common difference d, where d ≠ 0.
(a)
Show that a = 3d.
[3]
(b)
Find the common ratio of the geometric progression.
[1]
(c)
Given that d = 4, find the sum of the first 20 terms of the arithmetic progression and the sum of the first 8 terms of the geometric progression.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
a = 3d
(a + 3d)2 = a(a + 9d) M1 9d2 = 3ad A1 a = 3d since d ≠ 0 A1
3
(b)
2
r = (a + 3d)/a = 2 B1
1
(c)
1000; 3060
a = 12 B1 AP sum formula used M1 1000 A1 GP sum = 3060 A1
4
Question 3Structured · 4 marks
A car of mass 1300 kg travels on a straight horizontal road against a resistance of 1200 N. The engine works at 38 kW. Find the acceleration when the speed is 17 m s−1.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
0.796 m s−2
Driving force = P ÷ v M1 2240 N A1 F − R = ma M1 0.796 A1
4
Question 4Structured · 4 marks
Find the gradient of the curve y = x2e2x at the point where x = 2. Give your answer correct to 3 significant figures.
The masses, X g, of apples are normally distributed with mean 70 and standard deviation 12.
(a)
Find P(X < 67.6).
[3]
(b)
Find P(67.6 < X < 90.4).
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.4207
Standardise M1 z = −0.200 A1 0.4207 A1
3
(b)
0.5347
Φ(z2) − Φ(z1) M1 0.5347 A1
2
Question 6Structured · 6 marks
The masses, in grams, of 9 eggs are 19 27 31 39 41 47 48 54 58
(a)
Find the median and the interquartile range.
[3]
(b)
Calculate the mean and standard deviation.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
median 41; IQR 22
median 41 B1 quartiles M1 IQR 22 A1
3
(b)
40.4; 12.1
mean B1 Σx2/n − mean2M1 sd A1
3
Frequently asked questions
Are these real 9709 past paper questions?
No. They are original questions written to match the AS Level Mathematics 9709 syllabus and the style of real Cambridge papers, so you can practise beyond the past papers you have already done.
Do the Mathematics questions have answers?
Yes. Every question on these pages has its answer, and every generated paper comes with a full Cambridge-style mark scheme you can download as a PDF.
Can I make a full AS Level Mathematics practice paper?
Yes. Choose the paper (Paper 1 Pure Mathematics 1, Paper 2 Pure Mathematics 2, Paper 4 Mechanics, Paper 5 Probability & Statistics 1), the topics, the marks and the difficulty, and the generator builds a fresh paper with the same number and style of questions as the real exam.
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Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in